Peterson's SAT Prep Guide

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PETERSON’S®
®

SAT Prep Guide:
The Ultimate Guide to
Mastering the SAT®

About Peterson’s®
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TABLE OF CONTENTS
PART IV: WRITING STRATEGIES FOR THE
SAT® EXAM

Introduction to Peterson’s SAT   ® Prep Guide	������������������vii
The Peterson’s Suite of SAT  ® Products	�������������������������xi

PART I: BASICS FACTS ABOUT THE SAT®

1

4

ALL ABOUT THE SAT®

How the SAT ® Is Used for College Admissions	�����������4
When to Take the SAT ® (and SAT Subject Tests™)	������4
How Your Scores Are Reported	����������������������������������������5
How Often to Take the SAT ®	���������������������������������������������5
Registering for the SAT ®	�����������������������������������������������������5
Get to Know the Current SAT ® Exam Format	��������������7
Get to Know the SAT ® Question Types	��������������������������8
Evidence-Based Reading Test Section	��������������������������8
Writing and Language Test Section	�������������������������������14
Math Test Sections	���������������������������������������������������������������17
Grid-Ins	�������������������������������������������������������������������������������������20
SAT ® Essay (Optional)	����������������������������������������������������������24
The SAT ® Answer Sheet	������������������������������������������������������24
How the SAT ® Is Scored	������������������������������������������������������25
Strategies for SAT ® Success	�����������������������������������������������26
Make an SAT ® Study Plan	���������������������������������������������������27
Measuring Your Progress	���������������������������������������������������28
Simulate Test-Taking Conditions	�������������������������������������29
The Night Before and Day of the Exam	�������������������������31
Top 10 Strategies to Raise Your Score	����������������������������32
Summing It Up	����������������������������������������������������������������������33

5

PART II: THE DIAGNOSTIC TEST

2

THE DIAGNOSTIC TEST

Introduction to the Diagnostic Test	�������������������������������37
Answer Sheets	�����������������������������������������������������������������������39
Section 1: Reading Test	�������������������������������������������������������45
Section 2: Writing and Language Test	���������������������������57
Section 3: Math Test—No Calculator	�����������������������������69
Section 4: Math Test—Calculator	������������������������������������77
Essay	������������������������������������������������������������������������������������������93
Answer Keys and Explanations	����������������������������������������97
Computing Your Scores	������������������������������������������������������120

6

3

A Closer Look at the Evidence-Based
Reading Test	���������������������������������������������������������������������������129
Basic Steps for Answering Evidence-Based
Reading Questions	���������������������������������������������������������������130
Tips for Taking the Reading Test	��������������������������������������136
Strategies for Answering Specific Question Types	���138
Exercise: Evidence-Based Reading Test	������������������������139
Summing It Up	����������������������������������������������������������������������150

A Closer Look at the Writing and Language Test	������155
The Three Most Common Multiple-Choice
Editing Questions	�����������������������������������������������������������������156
Expression of Ideas Questions: Words In Context	�����157
Expression of Ideas Questions: Adding or
Deleting Text	��������������������������������������������������������������������������158
Expression of Ideas Questions:
Reordering Sentences	���������������������������������������������������������159
Expression of Ideas Questions: Combining
Sentencesand Using Transitional Words and
Phrases Correctly	������������������������������������������������������������������160
Graphic Organizer Questions	�������������������������������������������160
Exercise: Writing and Language Test	�����������������������������162
Summing It Up	����������������������������������������������������������������������171

ENGLISH LANGUAGE CONVENTIONS

Sentence Formation	������������������������������������������������������������174
Verb Tense, Mood, and Voice	��������������������������������������������181
Conventions of Usage	���������������������������������������������������������184
Agreement	������������������������������������������������������������������������������188
Frequently Confused Words	���������������������������������������������192
Conventions of Punctuation	���������������������������������������������225
Summing It Up	����������������������������������������������������������������������240

PART V: ESSAY WRITING STRATEGIES
FOR THE SAT®
THE SAT® ESSAY

A Closer Look at the Essay Question	������������������������������244
Pacing Your Writing	��������������������������������������������������������������244
Prewriting	��������������������������������������������������������������������������������244
Writing the Introduction	����������������������������������������������������246
Developing Your Ideas	��������������������������������������������������������248
Writing the Conclusion	�������������������������������������������������������250
The Scoring Rubric for the SAT® Essay	���������������������������252
Exercise: Practicing Your Essay Skills	������������������������������253
Additional Essay Writing Practice	�����������������������������������259
Summing It Up	����������������������������������������������������������������������262

PART III: READING STRATEGIES
FOR THE SAT®
EVIDENCE-BASED READING
TEST STRATEGIES

WRITING AND LANGUAGE
TEST STRATEGIES

7

PART VI: MATH STRATEGIES FOR
THE SAT®
MULTIPLE-CHOICE MATH

Why Multiple-Choice Math Is Easier	������������������������������267
Question Format	�������������������������������������������������������������������268
Solving Multiple-Choice Math Questions	��������������������269
Know When to Use Your Calculator	��������������������������������270
Learn the Most Important Multiple-Choice
Math Tips	���������������������������������������������������������������������������������271
Exercises: Multiple-Choice Math	�������������������������������������284
Summing It Up	����������������������������������������������������������������������296

8
9

GRID-IN STRATEGIES

Why Grid-Ins Are Easier Than You Think	�����������������������297
How to Record Your Answers	�������������������������������������������298
Guessing on Grid-Ins Can’t Hurt You	�����������������������������302
Exercises: Grid-Ins	�����������������������������������������������������������������303
Summing It Up	����������������������������������������������������������������������317

12

Functions	���������������������������������������������������������������������������������473
Exercises: Functions	�������������������������������������������������������������479
Integer and Rational Exponents	��������������������������������������482
Exercises: Integer and Rational Exponents	������������������485
Solving Complex Equations	����������������������������������������������488
Exercises: Solving Complex Equations	��������������������������492
Linear, Quadratic, and Exponential Functions	�����������496
Exercises: Linear, Quadratic, and
Exponential Functions	��������������������������������������������������������505
Summing It Up	����������������������������������������������������������������������511

NUMBERS AND OPERATIONS

Operations with Fractions	�������������������������������������������������320
Tests for Divisibility	��������������������������������������������������������������320
Exercises: Operations with Fractions	�����������������������������323
Word Problems Involving Fractions	�������������������������������325
Exercises: Word Problems Involving Fractions	�����������327
Complex Numbers	���������������������������������������������������������������333
Exercises: Complex Numbers	�������������������������������������������336
Direct and Inverse Variation	����������������������������������������������341
Exercises: Direct and Inverse Variation	��������������������������343
Finding Percents	�������������������������������������������������������������������347
Exercises: Finding Percents	�����������������������������������������������353
Percent Word Problems	������������������������������������������������������356
Exercises: Percent Word Problems	����������������������������������360
Summing It Up	����������������������������������������������������������������������365

13

10 BASIC ALGEBRA

Signed Numbers	�������������������������������������������������������������������367
Exercises: Signed Numbers	�����������������������������������������������368
Linear Equations	�������������������������������������������������������������������372
Simultaneous Equations	����������������������������������������������������380
Exercises: Simultaneous Equations	��������������������������������387
Exponents	�������������������������������������������������������������������������������392
Exercises: Exponents	�����������������������������������������������������������394
Quadratic Equations	������������������������������������������������������������397
Exercises: Quadratic Equations	����������������������������������������400
Literal Equations	�������������������������������������������������������������������403
Exercises: Literal Equations	�����������������������������������������������404
Roots and Radicals	���������������������������������������������������������������407
Exercises: Roots and Radicals	�������������������������������������������410
Monomials and Polynomials	��������������������������������������������413
Exercises: Monomials and Polynomials	������������������������416
Problem Solving in Algebra	����������������������������������������������419
Exercises: Problem Solving in Algebra	��������������������������424
Inequalities	�����������������������������������������������������������������������������427
Exercises: Inequalities	���������������������������������������������������������431
Summing It Up	����������������������������������������������������������������������434

11

GEOMETRY

Geometric Notation	������������������������������������������������������������435
Angle Measurement	����������������������������������������������������������437
Intersecting Lines	�����������������������������������������������������������������439
Area	��������������������������������������������������������������������������������������������441
Circles	����������������������������������������������������������������������������������������442
Volume	�������������������������������������������������������������������������������������445
Triangles	�����������������������������������������������������������������������������������447
Parallel Lines	���������������������������������������������������������������������������453
Coordinate Geometry	���������������������������������������������������������454
Exercises: Geometry	������������������������������������������������������������462
Summing It Up	����������������������������������������������������������������������472

FUNCTIONS AND INTERMEDIATE
ALGEBRA

DATA ANALYSIS, STATISTICS,
AND PROBABILITY

Averages	����������������������������������������������������������������������������������513
Weighted Average	����������������������������������������������������������������516
Exercises: Averages	��������������������������������������������������������������517
Probability	�������������������������������������������������������������������������������523
Exercises: Probability	�����������������������������������������������������������526
Data Interpretation	��������������������������������������������������������������532
Exercises: Date Interpretation	������������������������������������������538
Statistics	�����������������������������������������������������������������������������������545
Exercises: Statistics	���������������������������������������������������������������552
Summing It Up	����������������������������������������������������������������������558

PART VII: PRACTICE TESTS FOR
THE SAT® EXAM

14

SAT® PRACTICE TESTS

Introduction to the Practice Tests	�����������������������������������561
Practice Test 1	�������������������������������������������������������������������������563
Practice Test 1—Answer Keys and Explanations	�������619
Computing Your Scores	������������������������������������������������������643
Practice Test 2	�������������������������������������������������������������������������651
Practice Test 2—Answer Keys and Explanations	�������709
Computing Your Scores	������������������������������������������������������734
Practice Test 3	�������������������������������������������������������������������������741
Practice Test 3—Answer Keys and Explanations	�������801
Computing Your Scores	������������������������������������������������������827
Practice Test 4	�������������������������������������������������������������������������835
Practice Test 4—Answer Keys and Explanations	�������897
Computing Your Scores	������������������������������������������������������922
Practice Test 5	�������������������������������������������������������������������������929
Practice Test 5—Answer Keys and Explanations	�������987
Computing Your Scores	������������������������������������������������������1013

PART VIII: APPENDIX
Parents’ Guide to College Admission Testing	�������������1023

INTRODUCTION TO PETERSON’S SAT   ® PREP GUIDE
Whether you have three long months or just four short weeks to prepare for the exam, Peterson’s® SAT® Prep Guide can help you
develop a study plan that caters to your individual needs and personal timetable. These step-by-step plans are easy to follow and
remarkably effective. No matter which plan you select, begin by taking a diagnostic practice test.

The Diagnostic Practice Test and Process
The diagnostic practice test does more than give you testing experience. Easy-to-use diagnostic tables help you track your performance, identify your strengths, and pinpoint areas for improvement. At the end of the diagnostic testing process, you will know
which question formats are giving you the most difficulty. You will also know which topics to review in depth and which ones you
can spend less time on, whether they are algebra or geometry, literary analysis, or reading charts and graphs. By understanding your
testing profile, you can immediately address your weak areas by working through the relevant review chapters, learning the important
test-taking tips, and completing the additional practice exercises.

Five Full Practice Tests in This Book
When you have completed your formal review, take the practice tests to sharpen your skills further. Even if you understand the
redesigned SAT® perfectly, you still need to practice applying the methods you have learned in Peterson’s® SAT® Prep Guide. Take the
practice tests under simulated test conditions. Find a quiet place where you won’t be interrupted, set a timer for the required time
for each section, and work through each test as though it were test day. This will help you to get used to the time limits and to learn
to pace yourself. If you don’t have time to take full-length practice tests, Peterson’s® SAT® Prep Guide explains how to use timing drills
to take shorter sections of the exams to combat your weaknesses, work on your pacing, and increase your level of confidence.

Comprehensive Answer Explanations
At the end of each practice session, read all the answers and explanations, even for the questions that you answered correctly. There
are comprehensive explanations for every one of the book’s 1,600+ questions! By reading the answer explanations, you can learn
from your mistakes.
You’ll also find that Peterson’s® SAT® Prep Guide discusses all of the “big picture issues” other books ignore. For example, it addresses
questions such as:

• How is the SAT® really used for college admission?
• When should you take the test?

Peterson's SAT® Prep Guide

  vii

• How many times should you plan to take the SAT® exam?
• Do all SAT® exam sscores “count” in college admissions?
By addressing these questions, Peterson’s® SAT® Prep Guide debunks prevailing myths and helps you put the SAT® into its proper
perspective. It also serves as your “college guidance counselor,” giving you the expert advice you need to apply to college. And when
you think about it, that’s our number-one goal here. Our objective is to help you dramatically raise your scores so that you can
maximize the likelihood of getting into the college of your choice.

How This Book Is Organized
Peterson’s SAT® Prep Guide is divided into eight parts to facilitate your study.

• Part I explains everything you need to know about the SAT® exam and provides an overview with examples of the different
question types you’ll find on the actual test.

• Part II offers a diagnostic test to help you identify your areas of strength and those areas where you need to spend more time in
your review sessions.

• Part III explores the Evidence-Based Reading Test section and offers expert strategies for answering each type of question.
• Part IV goes into detail about the different types of questions you’ll see on the Writing and Language Test section of the SAT®
exam. You’ll also find a helpful review of Standard English Conventions.

• Part V describes the optional Essay part of the SAT® exam and provides strategies for developing a well-supported and coherent
response to the essay prompt in this section.

• Part VI offers a thorough review of all math topics you’ll see on the Math Test–No Calculator and Math Test–Calculator sections.
You’ll find helpful information on multiple-choice and grid-in math strategies, plus helpful reviews of numbers and operations,
basic algebra, geometry, functions and intermediate algebra, and data analysis, statistics, and probability.

• Part VII has five more tests that provide you with practice for the SAT® exam so you can simulate taking the test under timed
conditions. Each of the practice tests has detailed answer explanations plus instructions on how to determine your scores for
the Evidence-Based Reading and Writing portion and the two Math sections. You’ll also be able to calculate your subscores in
the categories of Expression of Ideas, Standard English Conventions, Words in Context, Command of Evidence, Heart of Algebra,
Problem Solving and Data Analysis, and Passport to Advanced Math as well as the cross-test scores for the Analysis in History/
Social Studies and Analysis in Science questions.

• Part VIII is the Appendix: Parents’ Guide to College Admission Testing, offering great information for parents in creating a plan to
help their teen prepare for college-admissions tests. It discusses the various roles parents play, how to approach teens on this
subject matter, and how to work with the guidance counselor. It also provides great tips on how to help teens improve their
time management—essential when preparing for standardized tests like the SAT® exam.

  viii

www.petersons.com

Special Study Features
You will find four kinds of special study features scattered throughout the book. Each study feature highlights specific types of
information:
Tips point out valuable information you need to know when taking the SAT® exam. Tips provide quick and
simple hints for selecting the correct answers for the most common SAT® question types.
Alerts identify potential pitfalls in the testing format or question types that can cause common mistakes in
selecting answers.
NOTES:

Notes address information about the test structure itself.

Cautions provide warnings, such as common grammatical errors or possible errors in computation or formulas
that can result in choosing incorrect answers.

How to Use This Book
It’s understandable that all of the information you’ll be reading about the SAT® exam might seem a little overwhelming. But even if
you are feeling confused by everything the exam requires, take some comfort in the knowledge that you are holding a great resource
to help you do well on test day. This book’s job is not to make you a genius; its job is to make sure you are prepared to take the SAT®
exam. If you become a genius in the process, consider that a bonus.
The following four steps will help you get the most out of using this guide:

Step 1: First Things First—Get to Know the Exam
You will get the most out of Peterson’s SAT® Prep Guide by using this book as it is organized. You may want to skip Chapter 1: All About
the SAT® because you’re anxious to get right to the lessons, since that’s where the real preparation begins, but Chapter 1 is very useful
for giving you a picture of the exam’s content as a whole.
ALERT: If you skip Chapter 1, you won’t learn about when and how many times you should take the SAT®; how to register
for the SAT®; how the SAT® is scored and how scores are reported; strategies for SAT® success; the SAT® test format and
question types, including a first look at kinds of questions on the Evidence-Based Reading Test, the Writing and Language
Test, and the Math Tests (Calculator and No Calculator); and much more!

Step 2: The Diagnostic Test Is Your Friend—Don’t Skip It!
Once you’ve learned the essential information about the SAT® in general, you will need to take the first step toward getting your
scores where you want them to be by taking the diagnostic test. The diagnostic test is a full-length practice test that you take before
you start studying or reviewing any subject material.
Understandably, taking a long diagnostic test may not seem the ideal way to get started on your test-preparation path. However,
the point of a diagnostic test is to give you an idea of what your strengths and weaknesses are before you dive into your SAT® preparation. By taking the diagnostic test and analyzing your answers, you may discover that you retained more information from your
English classes than you realized. You might also learn that you aren’t quite the math expert you thought you were. Or maybe you

Peterson's SAT® Prep Guide

  ix

will find that most of your math skills are really strong, but you need some help when it comes to quadratic equations.
Getting a clear idea of your strengths and weaknesses will help you know which chapters of this book really demand
your focus. The diagnostic test will help you get that clear idea.

Step 3: Build Your Skills—Practice, Practice, Practice!
After evaluating your diagnostic test results, you should
know the skills on which you need to focus. By diligently
studying the information presented, you will become
familiar with not only the question types that will appear
on the SAT®, but also the language of the exam to help
you be prepared for how the test will be worded.

Familiarizing yourself with the way certain questions
are worded on the SAT® may help you figure out the
kind of question you are answering, which may help
you select the best answer.

Throughout each review chapter (Chapters 3–13), you will find numerous practice questions, which will help familiarize
you with the language and presentation of the SAT® exam. The exercises and quizzes in these review chapters are a
great way to practice, and the thorough answer explanations will help you understand why an answer is right—or more
importantly why an answer is wrong. This can hopefully keep you from making a similar mistake again when you take
the real test.

Step 4: See What You Have Learned—Test Yourself Again
Near the end of the book, in Part VII, you will have the opportunity to take five complete SAT® practice tests. With each
practice test, you should see an improvement in your score since taking the diagnostic test.
You should take these tests under the same circumstances you will encounter on test day. That means completing each
test section in the exact same time that will be allotted for the actual test:

• Reading Test: 65 minutes
• Writing and Language Test: 35 minutes
• Math Test—No Calculator: 25 minutes
• Math Test—Calculator: 55 minutes
• Essay: 50 minutes

If you begin using this book to prepare for the SAT® well in
advance of test day, you might want to hold off taking the
last practice test until a few days before taking the actual SAT®
to re-familiarize yourself with the test’s format and content.

  x

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THE PETERSON’S SUITE OF SAT® PRODUCTS
In addition to Peterson’s SAT® Prep Guide, Peterson’s has an array of cutting-edge SAT® preparation resources designed to give you the
best test preparation possible. Our online course and interactive practice tests can be used alone or combined with other Peterson’s
SAT®-focused products to help you succeed and get the test scores you want. Take a few minutes to discover what’s available in
Peterson’s suite of SAT® products or visit our website at www.petersons.com/sat.

Peterson’s SAT® Prep Guide
Peterson’s SAT® Prep Guide features content and strategies that will help you master the SAT®. It contains a full-length diagnostic test
and access to eight full-length practice tests—five within the book and three online. The expert subject review and skill-specific
exercises in Peterson’s SAT® Prep Guide can help familiarize you with the unique content, structure, and format of the test. Test-taking
tips and advice guide you smoothly from your first day of test preparation to test day.
In addition, taking online practice tests is ideal because you get immediate feedback and automated scoring. Peterson’s SAT® Prep
Guide gives you access to three full-length online SAT® practice tests, with detailed feedback to help you understand the concepts
presented. The content in these three practice tests was created by the test-prep experts at Peterson’s to help you boost your test-prep
confidence so you can score high on test day. You can access these three practice tests at www.petersons.com/sat.

Peterson’s Practice Tests for the SAT®
Looking for even more online test practice? You can find it with Peterson’s Practice Tests for the SAT®. This test-prep tool provides three
full-length tests with immediate feedback and explanations for each question. Your purchase allows you 90-day access to these tests,
which feature the Essay Self-Score option—you can compare your essay to samples provided to give you an idea of how your essay
will be evaluated based on official scoring guidelines. Equipped with on-the-spot feedback and sample essays for comparison, you
can be confident that you’re getting the guidance you need to improve your score in all sections of the SAT®.

Peterson’s SAT® Online Course
Peterson’s SAT® Online Course is a comprehensive test prep course that is customized for you. In addition to practice tests, the online
course allows access to supplemental content, including additional subject-specific strategies and lessons, tips, and college search
options tailored to your projected test scores and interests.
Here’s how the course works:
1.
2.
3.
4.

An initial diagnostic pretest determines your strengths and weaknesses.
Based on your diagnostic test results, interactive lessons teach you the subject areas you need to learn.
Quizzes after each lesson gauge how well you have learned the materials just taught.
Full-length practice tests allow you to apply all the skills you’ve learned and monitor your progress.

Peterson’s SAT® Online Course gives you the opportunity to solidify your understanding and build your confidence about any concept
you may encounter on the SAT®—no matter how close it is to test day!

Interested in going the extra mile and using additional online practice tests or the online course? Take advantage of
customer-friendly discounts available only to customers who purchase Peterson’s SAT® Prep Guide. For more information and to
obtain your 20% discount on Peterson’s online courses, go to www.petersons.com/sat and enter the code: SAT2017.

Peterson's SAT® Prep Guide

  xi

Give Us Your Feedback
• Peterson’s publishes a full line of books—test prep, career preparation, education exploration, and financial aid. Peterson’s
publications can be found at high school guidance offices, college libraries and career centers, and your local bookstore and
library. Peterson’s books are also available online at www.petersonsbooks.com.

• We welcome any comments or suggestions you may have about this publication. Your feedback will help us make educational
dreams possible for you—and others like you.

Good
Luck!

  xii

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Now that you know why Peterson’s SAT® Prep Guide is an essential resource to prepare you to take a very important
test, it’s time to make the most of this powerful preparation tool. Turn the page and find out everything you need
to know about the SAT®!

part i : basic facts
about the sat ®
Chapter 1: All About the SAT

®

Chapter 1:

All About The SAT®
OVERVIEW
How the SAT® Is Used for College Admissions
When to Take the SAT® (and SAT Subject Tests™)
How Your Scores Are Reported
How Often to Take the SAT®
Registering for the SAT®
Get to Know the Current SAT® Exam Format
Get to Know the SAT® Question Types
Evidence-Based Reading Test Section
Writing and Language Test Section
Math Test Sections
SAT® Essay (Optional)

3

The SAT® Answer Sheet

Chapter 1

How the SAT® Is Scored

All About
the SAT®

Strategies for SAT® Success
Make an SAT® Study Plan
Measuring Your Progress
Simulate Test-Taking Conditions
The Night Before and Day of the Exam
Summing It Up

Peterson’s SAT® Prep Guide 2017

HOW THE SAT ® IS USED FOR COLLEGE ADMISSIONS
The explicitly stated purpose of the SAT® exam is to predict how students will perform academically as first-year college students.
But the more practical purpose of the test is to help college admissions officers make acceptance decisions. When you think
about it, admissions officers have a difficult job, particularly when they are asked to compare the academic records of students
from different high schools in different parts of the country taking different classes. It’s not easy to figure out how one student’s
grade point average (GPA) in New Mexico correlates with that of another student in Florida. Even though admissions officers
can do a good deal of detective work to fairly evaluate candidates, they benefit a great deal from the SAT®. The SAT® provides a
single, standardized means of comparison. After all, virtually every student takes the SAT®, and the SAT® is the same for everyone.
It doesn’t matter whether you hail from Maine, Maryland, or Montana.
So the SAT® is an important test. But it is not the be-all, end-all. Keep it in perspective! It is only one of several important pieces
of the college admissions puzzle. Other factors that weigh heavily into the admissions process include GPA, difficulty of course
load, level of extracurricular involvement, and the strength of the college application itself.

WHEN TO TAKE THE SAT ® (AND SAT SUBJECT TESTS™)
When you decide which schools you’re going to apply to, find out if they require the SAT®. Most do! Your next step is to determine
when they need your SAT® scores. Write that date down. That’s the one you really don’t want to miss.
You do have some leeway in choosing your test date. The SAT® is typically offered on one Saturday morning in October, November,
December, January, March (or April, alternating), May, and June. Check the exact dates to see which ones meet your deadlines.
Tests are offered on a Sunday, usually the day after each Saturday test date, for students who cannot take the test on Saturday
due to religious observance.

4
Chapter 1
All About
the SAT®

What if you don’t know which schools you want to apply to? Don’t panic! Even if you take the exam in December or January of
your senior year, you’ll probably have plenty of time to send your scores to most schools.
When you plan to take the SAT®, there is something even more important than the application deadlines of particular schools.
You need to select a test date that works best with your schedule. Ideally, you should allow yourself at least two to three months
to use this book to prepare. Many students like to take the test in March of their junior year. That way, they take the SAT® several
months before final exams, the prom, and end-of-the-year distractions. Taking the test in March also gives students early feedback
as to how they are scoring. If they are dissatisfied with their scores, there is ample opportunity to take the test again in the spring
or following fall. But your schedule might not easily accommodate a March testing. Maybe you’re involved in a winter sport or
school play that will take too much time away from studying. Maybe you have a family reunion planned over spring break in
March. Or maybe you simply prefer to prepare during a different time of year. If that’s the case, just pick another date.
If the schools you’ve decided on also require SAT Subject Tests™, here’s one good piece of advice: Try to take SAT Subject
Tests™ immediately after you finish the subject(s) in school. For most of you, this means taking the SAT Subject Tests™ in June.
By taking the exam then, you’ll save an awful lot of review work. Remember this, too: You have to register for the SAT Subject
Tests™ separately, and you can’t take the Subject Tests on the same day as the SAT®. So check the dates, think ahead, and plan
it out. It’s worth it in the end.
Are you starting to prepare a little later than you had planned? Don’t get upset;
it happens. Using the accelerated plan, you should be able to cover most of the
material within a month. You probably won’t have much time to practice, but you’ll
get the most important facts about the test and be able to take a few sample exams.

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HOW YOUR SCORES ARE REPORTED
After you have taken the SAT®, College Board scores your test and creates a score report. We will discuss in detail how the SAT®
is scored later in this chapter. You and your high school receive score reports from each SAT® and SAT Subject Test™ that you
decide to take.
At the time of registration, you can pick four colleges or universities to receive your score report. College Board will send your
scores to these four schools for free. Within nine days of taking the test, you can change your school selection. If you want to
send more than four reports or change your mind more than nine days after your test date, you will have to pay to do so.
If you decide to take the SAT®, or any SAT Subject Test™, more than once, you have the option to decide which scores to send
to the schools you’ve picked—scores from one, several, or all test dates.
You may only designate the test date or dates for your score reports; you cannot designate individual test sections. In other
words, if you take the SAT® exam in October, December, and March, you cannot pick the Evidence-Based Reading and Writing
section score from October, Math score from December, and Essay score from March and ask to have those results sent to the
schools of your choice. You can only choose whether to send your complete results from one, two, or all three test dates.
If you choose not to take advantage of this option, all of your scores will be sent to the schools you’ve selected. However, no
score reports will ever be sent without your specific consent. You and your counselor will receive e-mail reminders, asking which
scores you want to send. You can find more information about this and how colleges and universities use your score reports on
the website https://sat.collegeboard.org/register/sat-score-choice.

HOW OFTEN TO TAKE THE SAT ®
Different colleges evaluate the exam in different ways. Some take your highest math, evidence-based reading and writing, and
essay scores, even if they were earned on different test days. So if you nailed the math portion in March, the evidence-based
reading and writing section in October, and the essay in December, the colleges will combine those scores to maximize your
overall score. However, many other colleges won’t do that. Some pay most attention to your highest combined score from a
single day. Many others will average all of your scores or lend equal weight to all of them.

5
Chapter 1
All About
the SAT®

So what does this mean? It means that you should only take the SAT® when you are truly prepared. There is nothing wrong with
taking the SAT® two or three times, as long as you are confident that your scores will improve substantially each time. Let’s say
that you scored an 1100 on your first SAT®. If you would have been thrilled to have hit 1120, it’s probably not worth taking the
test again. Most colleges look at SAT® scores in ranges and will not hold 20 points against you. They understand that scoring an
1100 means that you were only one or two questions away from 1120. But if you scored an 1100 and expected to score closer
to 1200 or 1300 based on practice testing, then you should probably retake the exam. In other words, it is of little value to take
the SAT® multiple times if you expect to earn roughly the same score. But it is worthwhile if you expect to score significantly
higher on a second or third try. For more advice about this, see your high school guidance counselor.

REGISTERING FOR THE SAT ®
You should register for the SAT® at least six weeks before your testing date. That way you will avoid late registration fees
and increase your chances of taking the exam at your first-choice testing center. You can register through the mail by completing the SAT® registration form found inside The SAT® and SAT Subject TestsTM Student Registration Booklet, which can be
found in your guidance counselor’s office or online (printable PDF) at https://collegereadiness.collegeboard.org/pdf/
sat-registration-booklet-students.pdf. Registering online is probably the quickest and easiest method, and you will receive

Peterson’s SAT® Prep Guide 2017

immediate registration confirmation. You will need to pay by credit card, and you will need to upload a photo with your
registration. The photo you provide will become part of your Admission Ticket on test day. For more information, visit
https://sat.collegeboard.org/register.

Photo ID
The photo you provide (either uploaded with your online registration or mailed in with the printed registration) becomes
part of your Admission Ticket on test day.
Photos must be properly focused with a full-face view. The photo must be clearly identifiable as you, and it must match your
appearance on test day. IMPORTANT: If you are not easily recognizable in your photo, you will not be admitted to the test
center.
Choose a photo that:

•
•
•
•

Shows only you—no other people in the shot
Shows a head-and-shoulders view, with the entire face, both eyes, and hair clearly visible
Is properly focused and has no dark spots or shadows
Shows a head covering only if it is worn for religious purposes

Visit https://sat.collegeboard.org/register/photo-requirements for more information about the required photo ID.
In addition, you are responsible for bringing an acceptable form of identification.
Some acceptable examples include:

6
Chapter 1
All About
the SAT®

•
•
•
•
•
•
•

State-issued driver’s license
State-issued nondriver ID
School identification card
Passport (required in India, Ghana, Nepal, Nigeria, and Pakistan)
Government-issued ID
School ID Form* prepared by your school
Talent Identification Program ID/Authorization to Test Form (grades 7 and 8 only); photo not required

* Your school can prepare an ID form for you. This form must include a recognizable photo, and the school seal must overlap the photo. Sign
the ID form in the presence of your counselor or principal. You will be asked to sign the ID form again at the test center. This form must be
dated and is good only for one year.

ALERT: You must provide a photo when you sign up for the SAT®. The photo will be part of your Admission Ticket, and it will
be checked against your photo ID on test day.

Registration Fees
At the time of this book’s printing, the fee for the SAT® (no essay) is $45. If you are planning to take the SAT® with the Essay section,
you will need to pay $57. To determine if you are eligible for a fee waiver, visit https://collegereadiness.collegeboard.org/sat/
register/fees/fee-waivers.

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GET TO KNOW THE CURRENT SAT ® EXAM FORMAT
The SAT® consists of sections on math, evidence-based reading and writing, and an optional essay. The sections are timed to
range from 25 to 65 minutes. The whole test takes 3 hours, plus 50 minutes for the optional essay. Don’t worry. There are breaks.
The following chart gives you an idea of what to expect. The test sections appear in the following order: Reading Test, Writing
and Language Test, Math Test—No Calculator, Math Test—Calculator, and the Essay.

Format of the SAT® Test
Evidence-Based Reading and Writing
Evidence-Based Reading questions based on:

• Passages in U.S. and world literature, history/social
•
•
•

studies, and science
Paired passages
Lower and higher text complexities
Words in context, command of evidence, and analysis

Score 200–800
Time—Total: 100 Minutes

• Reading Test (65 minutes)
• Writing and Language Test (35 minutes)
Question Types

• Multiple-choice with 4 answer choices

Writing and Language questions based on:

• Passages in careers, history/social studies, humanities,
•
•

and science
Argument, informative/explanatory, and nonfiction
narrative passages
Words in context, grammar, expression of ideas,
and analysis

Math

7

Score 200–800

Real-world problems solved using:

•
•
•
•
•
•

Algebra
Problem solving and data analysis
Advanced math
Area and volume calculations
Lines, triangles, and circles using theorems
Trigonometric functions

Time—Total: 80 Minutes

• Math Test—No Calculator (25 minutes)
• Math Test—Calculator (55 minutes)
Question Types

Chapter 1
All About
the SAT®

• Multiple-choice with 4 answer choices
• Student-produced responses (grid-ins)

Optional Essay
What’s involved:

• Read an argument-type passage written for a
•

general audience
Analyze the passage in terms of how the writer uses
evidence, reasoning, and stylistic elements to build
an argument to persuade his or her audience

Time—Total: 50 minutes
Question Type

• One prompt that emphasizes analyzing the
argument presented in the passage
Score: 3–12 (Reading: 1–4 scale, Analysis: 1–4 scale,
Writing: 1–4 scale)

Peterson’s SAT® Prep Guide 2017

GET TO KNOW THE SAT ® QUESTION TYPES
The question types in the SAT® don’t cover a wide variety of topics. They are very limited—no science, no world languages, no
social studies. You’ll find only questions testing reading comprehension, writing skills, and math skills—skills that you’ve been
working on since kindergarten.
Most of the questions are multiple choice. That’s good, because it means the correct answer is right there on the page for you.
You just have to find it—easier said than done sometimes, but true. Only the math grid-ins and the essay are student-produced
answers. For the grid-ins, you’ll need to do the calculations and then fill in circles on the answer sheet to show your answers.
(More about the answer circles later in this chapter.) The following pages provide you with a closer look at the question types
and question formats that you will find in each section of the SAT®.

On the SAT®, all questions count the same. You won’t get more points for answering a
really difficult question than you will get for answering a very simple one. Remember
that when you’re moving through the test. The more time you spend wrestling with the
answer to one “stumper,” the less time you have to whip through several easier questions.

EVIDENCE-BASED READING TEST SECTION

8
Chapter 1
All About
the SAT®

The reading section tests your knowledge of words in context, command of evidence in the passages, and your analysis of the
passage, including graphics. All the questions are multiple choice. The questions in the Evidence-Based Reading section are all
passage-based. All passages are from previously published sources and cover topics in U.S. and world literature, history/social
studies, and science.

Words in Context
Just as the name implies, words-in-context questions assess your ability to determine the meaning of words or phrases in the
context of an extended passage. If you do not recognize the meaning of the word, its meaning may be determined by context.
Your job is to read the passage and the question, and then analyze the answer choices to figure out which one makes the most
sense based on the words around it. That means you must look for clues in the passage.
Here is an excerpt from a passage on the opah fish. There are three sample words-in-context questions. Read the passage excerpt
and try to answer each question on your own before you read the answer explanations on the next page.

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The silvery fish, roughly the size of a large automobile

Nicholas Wegner of NOAA Fisheries’ Southwest Fisheries Science
Center in La Jolla, California, is lead author of a new paper on the
opah, or moonfish. He and his coauthor, biologist Owyn Snodgrass,
discovered that the opah has the unusual ability to keep its body
warm, even in the cold depths of the ocean. An excerpt on their
findings follows.

tire, is known from oceans around the world and dwells
hundreds of feet beneath the surface in chilly, dimly lit
waters. . . .
10

Courtesy: NOAA Fisheries

Fish that typically inhabit such cold depths tend to be
slow and sluggish, conserving energy by ambushing prey
instead of chasing it. But the opah’s constant flapping of its

Warm Blood Makes Opah an Agile Predator

fins heats its body, speeding its metabolism, movement and

New research by NOAA Fisheries has revealed the opah, or
moonfish, as the first fully warm-blooded fish that circulates
heated blood throughout its body much like mammals and

reaction times, scientists report today in the journal
15 Science. . . .

Line birds, giving it a competitive advantage in the cold ocean

“Before this discovery, I was under the impression

5 depths.

this was a slow-moving fish, like most other fish in cold
environments,” Wegner said. “But because it can warm its
body, it turns out to be a very active predator that chases
20 down agile prey like squid and can migrate long distances.”

Questions:
1. As used in the first paragraph, “competitive advantage” refers to
A.

a way to seek out a mate.

B.

an ability to outperform rivals.

C.

an aptitude for keeping itself moving.

D.

a capacity to conceal itself from predators.

9
Chapter 1
All About
the SAT®

2. As it is used in paragraph 3, “ambushing” most nearly means
A.

pursuing for long distances.

B.

moving slowly at first.

C.

hiding and then attacking.

D.

weakening and then killing.

3. As it is used in paragraph 4, “agile” most nearly means
A.

nimble.

B.

inactive.

C.

strong.

D.

clever.

Peterson’s SAT® Prep Guide 2017

Answer Explanations:
1. Clues to the meaning of the phrase don’t appear until the fourth paragraph: “I was under the impression . . . like
most other fish in cold environments” and “But . . . it turns out to be a very active predator.” Here, you’re told that the
opah is unlike other fish in that it can swim faster and farther and catch more prey. Choices A, C, and D are specific
traits that might help the fish in its environment. But choice B is the only one that makes sense in the context of the
passage. The correct answer is choice B.
2. The biggest clue to the meaning of ambushing is “instead of chasing it.” Because you know that the fish don’t chase
their prey, you can exclude choice A. Choices B and D don’t make sense in the context of the sentence because neither
is a method for capturing prey, as chasing is. Choice C, however, makes sense when you consider the context clue.
The correct answer is choice C.
3. The clue “very active predator” is your clue that agile must mean that the squid provides a challenge for the opah.
This eliminates choice B. Choice D can also be eliminated because the context emphasizes physical, not mental,
abilities. Likewise, you can eliminate choice C because the level of activity, not strength, is the focus. Choice A fits
the context, as it suggests that the squid is able to move quickly and easily. The correct answer is choice A.

In SAT® evidence-based reading questions, the answers will always be
directly stated or implied in the passage.

10
Chapter 1
All About
the SAT®

Command of Evidence
The evidence-based reading and writing sections of the SAT® require you to interpret information or ideas in a passage and then
use evidence to support your conclusion. This element of the Reading Test, which makes up 20 percent of the questions, works
like this: You answer a multiple-choice question in which you analyze a portion of the passage or pair of passages. You then answer
a second question requiring you to cite the best evidence in the text for the answer.
The passages include literary texts from U.S. and world literature, as well as nonfiction texts in science and history/social studies.
In some cases, related passages are paired and require you to make connections between the texts.
The following is an example of how these “command of evidence” questions work. The passage is a continuation of the NOAA
article cited previously, “Warm Blood Makes Opah an Agile Predator.”

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Questions 1–2 refer to the following passage.

“There has never been anything like this seen in a fish’s
gills before,” Wegner said. “This is a cool innovation by these

Gills Show Unusual Design

animals that gives them a competitive edge. The concept
of counter-current heat exchange was invented in fish long

Courtesy: NOAA Fisheries

20 before we thought of it.”

Wegner realized the opah was unusual when a coauthor
The researchers collected temperature data from opah

of the study, biologist Owyn Snodgrass, collected a sample

caught during surveys off the West Coast, finding that

of its gill tissue. Wegner recognized an unusual design:

their body temperatures were regularly warmer than the

Line Blood vessels that carry warm blood into the fish’s gills wind

surrounding water. They also attached temperature monitors

5 around those carrying cold blood back to the body core

after absorbing oxygen from water.

25 to opah as they tracked the fish on dives to several hundred

feet and found that their body temperatures remained
The design is known in engineering as “counter-current

steady even as the water temperature dropped sharply. The

heat exchange.” In opah it means that warm blood leaving

20 fish had an average muscle temperature about 5 degrees

the body core helps heat up cold blood returning from the
10 respiratory surface of the gills, where it absorbs oxygen.

C above the surrounding water while swimming about 150
30 to 1,000 feet below the surface, the researchers found. . . .

Resembling a car radiator, it’s a natural adaptation that
conserves heat. The unique location of the heat exchange

A few other fish . . . warm certain parts of their bodies . . .

within the gills allows nearly the fish’s entire body to

boosting their swimming performance. But internal organs,

maintain an elevated temperature, known as endothermy,

including their hearts, cool off quickly and begin to slow
down when they dive into cold depths, forcing them to

15 even in the chilly depths.

11

35 return to shallower depths to warm up.

Chapter 1
All About
the SAT®

Questions:
1. The author discusses the adaptations of some fish in the last paragraph mainly to show that
A.

opah swim faster because they are able to keep themselves warm.

B.

some fish maintain a body temperature warmer than the sea water.

C.

biologists have found evidence that some fish are warm-blooded.

D.

opah have a distinctive design that keeps them warm at greater depths.

2. Which choice provides the best evidence for the answer to the previous question?
A.

Lines 12–15 (“The unique . . . chilly depths.”)

B.

Lines 18–20 (“The concept . . . of it.”)

C.

Lines 21–24 (“The researchers . . . surrounding water.”)

D.

Lines 27–30 (“The 20 fish . . . researchers found.”)

Peterson’s SAT® Prep Guide 2017

Answer Explanations:
1. In the first question, the author’s intention is to contrast the warming ability of other fish with the warming ability of
the opah. Though the passage does note that some fish maintain a body temperature warmer than seawater for a
short period, this is not the reason the author includes details about other fish. Thus, choice B is not correct. Choices
A and C are incorrect because neither idea is noted in the text. The correct answer is choice D.
2. The second question asks you to determine which of four segments of the passage provides the best evidence to
support your answer to the first question. Choices B, C, and D do not provide textual support for the contrast the
author makes in the last paragraph. The correct answer is choice A.

Analysis and Graphics
Two passages in the SAT® Reading Test include a graphic. Your job is to analyze the passage and interpret the information in the
graphic as it relates to the passage. Questions based on the graphic are multiple choice. Here is a sample reading passage with
an accompanying graphic, in this case a map, and a question that requires your analysis.
From “About John Snow,” by Professor Paul Fine, London School of Hygiene & Tropical Medicine and The John Snow Society.
John Snow (1813–1858)
John Snow is an iconic figure in epidemiology and public

12

Line

health, best known for his work on cholera, for a famous map,

interested in the physiology of respiration in recognition of

and for organizing the removal of a pump handle in Soho.

the major problem of asphyxia of the newborn.

Less well-known are his important contributions to

5 anesthesia and to epidemiological methods, and his

Chapter 1
All About
the SAT®

set up general practice in Soho. Early in his career he became

These interests led him to be invited to witness one of the
25 first applications of ether anesthesia in the UK in December

engagement in public debates of the time. The breadth and

1846. He immediately recognized the importance of ambient

depth of Snow’s activities provide a model for population

temperature and within one month published tables of

researchers concerned not only with sound method but also

the vapor pressure of ether. This initiated an important line

with bringing their results to public benefit.

of research on instruments for administering anesthetics
30 and led to his becoming the most prominent authority on

10

Indeed, though epidemiology is often described as the
study of health-related aspects of populations, its methods

anesthesia in the UK. He administered chloroform to Queen
Victoria at the birth of Prince Leopold in 1853.

are applicable to studies of virtually anything in populations,
and disciplines which now acknowledge the methods and
terminology of epidemiology range from education to crime
15 science and economics.

The second great cholera epidemic arrived in London
in 1848, and many attributed its cause to an atmospheric
35 “effluence” or “miasma.” Snow’s firsthand experience of the

disease in 1832, combined with studies of respiration, led
Snow was born in York on 15 March 1813, one of eight
children in a family of modest means. He apprenticed with
a surgeon-apothecary in Newcastle from 1827 to 1833, and
there witnessed the first epidemic of cholera in the UK. He
20 then moved to London, qualified as physician in 1843 and

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him to question miasma theories and to publish the first
edition of On the Mode of Communication of Cholera in 1849,
in which he proposed that cholera was attributable to a
40 self-replicating agent which was excreted in the cholera

evacuations and inadvertently ingested, often, but not

Though the epidemic was already in decline by that date,

necessarily, through the medium of water.

the rapidity of his action, the logic of the analysis, and the
60 pragmatism of the response has made this a classic event

When cholera returned in 1853, Snow recognized an

in the history of public health, well known to students and

ideal opportunity to test his hypothesis by comparing

practitioners the world over. The combination of these

45 cholera mortality rates in populations of south London

studies provided overwhelming evidence for an infectious

supplied by water drawn from sewage-contaminated versus

agent, known now as Vibrio cholerae.

uncontaminated regions of the Thames. He personally carried
out a cohort study to make this comparison, recognizing the
need to confirm the water source of each case and to assure
50 comparability of the populations concerned. On 30 August

65

Snow described this work in the second edition of On the
Mode of Communication of Cholera. He then expanded his
public health interests by becoming involved in debates over

1854 while involved in these studies, a dramatic cholera

legislation concerning nuisance industries in London, while

epidemic began near his home in Soho, leading to more than

maintaining his research and practice in anesthesia until his

550 deaths within two weeks. Analysis of the addresses of

70 death in 1858.

the cholera deaths and interviews of residents of the area
55 led him to suspect that water from a pump on Broad Street

was responsible—and he prevailed upon the local council to
remove the handle of the pump on 8 September 1854.

The 200th anniversary of Snow’s birth provides an
occasion to celebrate his achievements, to consider their
original context, to discuss their place in contemporary
epidemiology, and consider their likely future, not only as
75 the armamentarium of public health, but as a framework of

method for science and society.

13
Chapter 1
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Peterson’s SAT® Prep Guide 2017

Question:
On John Snow’s map, deaths from the 1854 cholera epidemic are represented by stacks of black lines. Based on the
image and the passage, what can we assume about Blenheim Street (shown magnified in the upper-left side of the
map)?
A.

No one lived there.

B.

Broad Street did not supply their water.

C.

Its residents were as affected as the rest of the neighborhood.

D.

Relatives came to stay there to avoid the cholera outbreak.

Answer Explanation:
The passage explains that Snow determined that the pump on Broad Street supplied contaminated water to nearby
residents. You can infer that people living on streets containing stacks of bars used the Broad Street pump. Based on
the map, then, people on Blenheim Street likely did not use the Broad Street pump. The correct answer is choice B.

WRITING AND LANGUAGE TEST SECTION
The SAT® Writing and Language Test consists of multiple-choice questions based on passages. The multiple-choice questions test
how well you understand and use Standard Written English, as well as recognize words in context and command of evidence in
the passages. Analysis of the passages and graphics is also included.

14
Chapter 1
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the SAT®

Standard Written English and Words in Context
The Standard Written English questions require you to act as an editor and revise text so that it conforms to the standard rules for
punctuation, sentence structure, and usage. In most instances, you will be given a multiparagraph passage that includes several
errors. The most common question format asks you to choose the best alternative to a potential error, identified as an underlined
portion of the passage. Here is a sample question that concerns sentence structure:
Question:

Scientists conducted a series of experiments with

1

chimpanzees in the 1 Democratic Republic of the

Which choice most effectively combines the sentences
at the underlined portion?

Congo. The results were astounding. The conclusion, that

A.

Democratic Republic of the Congo, and the
results were astounding.

B.

Democratic Republic of the Congo, the results
were astounding.

C.

Democratic Republic of the Congo: the results
were astounding.

D.

Democratic Republic of the Congo, but the results
were astounding.

chimpanzees would eventually learn to cook if provided an
oven, could help explain how and when early humans began
to cook their food.

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Answer Explanation:
Choice B creates a comma splice, which is a form of a run-on sentence, so that’s not correct. Likewise, the colon in
choice C is not correct, as the clause it introduces does not really explain the first part of the sentence. Introducing
but in choice D changes the meaning of the sentences by setting up a contrasting scenario. Only choice A maintains
the two sentences’ meanings and combines them without confusion. The correct answer is choice A.
The words-in-context questions on the test measure your ability to choose appropriate words based on the context of the passage.
These questions are multiple choice and include the option to keep the word that is used.
Question:

There is a debate about whether early humans had

2

the mental capacity to cook. Though it may not seem
sophisticated, cooking requires planning, an ability to
2 interrupt gratification, and the complicated use of tools.

A.

NO CHANGE

B.

apprehend

C.

delay

D.

restrain

Answer Explanation:
Here, you must choose the word that makes the most sense in the context. The words interrupt, apprehend, and restrain
don’t convey what is meant here—to hold off. Only delay, choice C, conveys that sense. The correct answer is choice C.

Command of Evidence

15
Chapter 1

To answer the Command of Evidence questions in the Writing and Language Test section, you need to carefully read the passage
in question. Here is an example of this type of question. The excerpt comes from the passage “About John Snow”.

All About
the SAT®

Question:

The second great cholera epidemic arrived in London in

3

1848, and many attributed its cause to an atmospheric

Which choice best summarizes the main idea of the
paragraph?

“effluence” or “miasma.” Snow’s firsthand experience of the

A.

In 1848, many people were incorrect to blame
atmospheric miasma for the spread of cholera.

B.

John Snow’s study, On the Mode of Communication
of Cholera, was read by scholars worldwide.

C.

John Snow was curious, and he never took
anything at face value.

D.

Snow’s questioning of the miasma theory and
theories on water contamination changed the
conversation of disease circulation.

disease in 1832, combined with studies of respiration, led
him to question miasma theories and to publish the first
edition of On the Mode of Communication of Cholera in 1849,
in which he proposed that cholera was attributable to a
self-replicating agent which was excreted in the cholera
evacuations and inadvertently ingested, often, but not
necessarily, through the medium of water. 3

Peterson’s SAT® Prep Guide 2017

Answer Explanation:
The question asks you to determine which sentence best summarizes the main idea of the paragraph. Choices A, B,
and C all contain ideas that are important in the paragraph. But choice D contains the crux of the paragraph: that Snow
questioned the prevailing wisdom and then proposed his own theory about how cholera was transmitted. The correct
answer is choice D.

Analysis and Graphics
One or two of the passages in the Writing and Language Test section include graphics. You will be asked to determine how the
passage needs to be revised based on the information in the graphic. Here is an excerpt from another paragraph in “About John
Snow” and a sample question to help illustrate this concept.
Question:

Snow’s mapping of the outbreak showed few surprising
results. Little Pulteney Street is a case in point. The street
is blocks from Broad Street and closer to two other water
pumps. Among residents of the street, there 4 were no
cases of cholera reported during the outbreak.

16
Chapter 1
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the SAT®

www.petersons.com

4

Which choice completes the sentence using accurate data
based on the map?
A.

NO CHANGE

B.

was one new case of cholera

C.

were fewer than ten new cases of cholera

D.

were more than twenty new cases of cholera

Answer Explanation:
Here, you are being asked to interpret the information in the passage based on the map. If you look closely at the area
in question, Little Pulteney Street, you’ll see that there are about six bars, and we can infer that each bar represents a
case of cholera. The correct answer is choice C.

MATH TEST SECTIONS
The questions in the math sections (Math Test—No Calculator and Math Test—Calculator) address concepts, skills, and practices
that are most useful for students after they graduate from high school. There are two question formats for math questions: multiple-choice and grid-ins (student-produced responses).

A four-function, battery-powered, scientific or graphing calculator is allowed for
the Math Test—Calculator section of the SAT®. You may not use the following:
handheld mini-computers, laptop computers, pocket organizers, calculators
that print or “talk,” or calculators with letters on the keyboard.

Multiple-Choice Math
SAT® multiple-choice math questions look like all the other standard multiple-choice math questions you’ve ever seen. A problem
is given in algebra, problem solving, data analysis, advanced math, or additional topics, and four choices are presented from
which you must choose the correct answer. The major concepts that you might need in order to solve math problems are given
in the test section. You don’t need to worry about memorizing these facts, but you do need to know when to use each one.
The directions are similar to the following:

DIRECTIONS: For Questions 1–15, solve each problem, select the best answer from the choices provided, and fill in the
corresponding circle on your answer sheet. For Questions 16–20, solve the problem and enter your answer in the grid on
the answer sheet. The directions before Question 16 will provide information on how to enter your answers in the grid.

Circle:

Rectangle:

r
C = 2�r
A = �r2

l
V = lwh

h

l
A = lw

r

c

a
b

x 2

x

b

1
A = bh
2

Cylinder:

w

Chapter 1
All About
the SAT®

Triangle:

w

Rectangular
Solid:

17

x

x

Special Right Triangles

a2 + b2 = c2
Sphere:

2x

x 3

Cone:

Rectangular-Based
Pyramid:

h
r

V = �r2h

h

h
r

4
V = �r3
3

The number of degrees of arc in a circle is 360.
The number of radians in the arc of a circle is 2 .
The sum of the measures in degrees of the angles of a triangle is 180.

1
V = �r2h
3

l

w
1
3

V = lwh

Peterson’s SAT® Prep Guide 2017

ADDITIONAL INFORMATION:
1.  	 The use of a calculator in this section is not permitted (is permitted for the Math Test—Calculator section).
2.  	 All variables and expressions used represent real numbers unless otherwise indicated.
3.  	 Figures provided in this test are drawn to scale unless otherwise indicated.
4.  	 All figures lie in a plane unless otherwise indicated.
5.  	 Unless otherwise specified, the domain of a given function f is the set of all real numbers x for which f(x) is a real
number.

Here are some sample multiple-choice math questions. Try them yourself before looking at the solutions that are given.
Example:
Michele is at the airport renting a car that costs $39.95 per day plus tax. A tax of 7% is applied to the rental rate, and
an additional one-time untaxed fee of $5.00 is charged by the airport where she picks up the car. Which of the
following represents Michele’s total charge c(x), in dollars, for renting a car for x days?
A.

c(x) = (39.95 + 0.07x) + 5

B.

c(x) = 1.07(39.95x) + 5

C.

c(x) = 1.07(39.95x + 5)

D.

c(x) = 1.07(39.95 + 5)x

Solution:

18
Chapter 1

The total cost, c(x), can be found by multiplying any daily charges by the number of days, x, and then adding any
one-time charges. The daily charges include the $39.95 daily rate and the 7% tax. This can be computed by:
$39.95 + 0.07($39.95) = 1($39.95) + 0.07($39.95) = 1.07($39.95)

All About
the SAT®

Multiply the daily charge by x and add the one-time charge of $5 to obtain the function rule:
c(x) = 1.07(39.95)x + 5
The correct answer is choice B.

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Example:
The graph of y = (3x + 9)(x – 5) is a parabola in the xy-plane. In which of the following equivalent equations do the
x- and y-coordinates of the vertex of the parabola appear as constants or coefficients?
A.

y = 3x2 – 6x – 45

B.

y = 3x(x – 2) – 45

C.

y = 3(x – 1)2 + (– 48)

D.

y = (x + 3)(3x – 15)

Solution:
The equation y = (3x + 9)(x – 5) can be written in vertex form y = a(x – h)2 + k, where the vertex of the parabola is
(h, k). To put the equation in vertex form, first multiply the factors, then complete the square. The correct answer is
choice C.
Example:
The same final exam is given to two separate groups of students taking the same class. The students who took the
exam on the first floor had a mean score of 84. The students who took the exam on the second floor had a mean
score of 78. Which of the following represents the mean score x of both groups of students?
A.

x = 81

B.

x < 81

C.

x > 81

D.

78 < x < 84

19
Chapter 1
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the SAT®

Solution:
Many students will select choice A as the answer because 81 is the mean of 78 and 84, but there is no information
about the size of the two groups that are being averaged. If the groups were equal in size, choice A would be correct. If
there were more students on the second floor, then choice B would be the correct answer. Similarly, if there were more
students on the first floor, then choice C would be correct. Since we don’t know which floor has more students taking
the exam or if the number of students is equal, we can only say that choice D is true. The correct answer is choice D.

Peterson’s SAT® Prep Guide 2017

GRID-INS
Unlike multiple-choice math, the grid-in section of the SAT® does not give you the answers. You have to compute the answer and
then fill in your answer in the circles on your answer sheet. You may use the Reference Information table that appeared earlier
in this chapter for these problems also.
On the SAT®, each set of grid-in questions starts with directions that look approximately like this:

DIRECTIONS: For these questions, solve the problem and enter your answer in the grid, as described below, on the
answer sheet.

1. Although not required, it is suggested that you write your answer in the boxes at the top of the columns to help you fill in
the circles accurately. You will receive credit only if the circles are filled in correctly.
2. Mark no more than one circle in any column.
3. No question has a negative answer.
4. Some problems may have more than one correct answer. In such cases, grid only one answer.
7
1
5. Mixed numbers such as 3 must be gridded as 3.5 or .
2
2
		
If 3

20

1
is entered into the grid as
2

, it will be interpreted as

1
31
, not 3 .
2
2

6. Decimal answers: If you obtain a decimal answer with more digits than the grid can accommodate, it may be either rounded
or truncated, but it must fill the entire grid.

Chapter 1
All About
the SAT®

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7
12

Answer:

Answer: 2.5

Write answer
in boxes.

.

Fraction
line
0

Grid in
result.

0

0

0

0
1

1

2

2

3

3

3

4

4

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1

2

Decimal
point

0

Answer: 201
Either position is correct.

0
1

1

0
2

2

0

0

1

1

1

1

2

2

2

2

2

1

3

3

3

3

3

3

3

3

4

4

4

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4

4

4

4

Acceptable ways to grid

2
are:
3

.

1

0

21

.

0

0

0

0

0

0

0

0

0

1

1

1

1

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6

Chapter 1
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the SAT®

6

Once you understand the following six rules, you can concentrate on just solving the math problems in this section.
1. Write your answer in the boxes at the top of the grid.
2. Mark the corresponding circles, one per column.
3. Start in any column.
4. Work with decimals or fractions.
5. Express mixed numbers as decimals or improper fractions.
6. If more than one answer is possible, grid any one.

Peterson’s SAT® Prep Guide 2017

NOTE: Don’t use a comma in a number larger than 999. Just fill in the four digits and the corresponding circles.
You only have circles for numbers, decimal points, and fraction slashes; there aren’t any for commas.

Now let’s look at these rules in more detail:
1.

Write your answer in the boxes at the top of the grid. Technically, this isn’t required by the SAT®. Realistically, it gives you
something to follow as you fill in the circles. Do it—it will help you.

2.

Make sure to mark the circles that correspond to the answer you entered in the boxes, one per column. The machine that
scores the test can only read the circles, so if you don’t fill them in, you won’t get credit. Just entering your answer in the
boxes is not enough!

3.

You can start entering your answer in any column, if space permits. Unused columns should be left blank; don’t put in zeroes.
Here are some examples of these kinds of problems:

Examples:
Use the grids provided to try the following grid-in quesions.
1.

The circumference of a circle is 20π. If the area of a sector of the circle with a central angle of
3p
is aπ, what is the value of a?
2

22

2.

There are 70 students in a school who participate in the music program. If 35% of the students participate in the music
program, how many students are in the school?

3.

What is one possible solution to the equation

Chapter 1

1.

2.

All About
the SAT®

www.petersons.com

1
3
+
= −4 ?
x x −1

/
.

/
.

.

0

0

0

0

1

1

1

2

2

3

3

4
5

4
5

3.
/
.

/
.

.

/
.

/
.

0

0

0

0

.

0

0

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0

1

1

1

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1

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9

Solutions:
1. C = 2pr = 20 p
r = 10
3
3
A = pr 2 = p(10 ) 2 = 75p = ap
4
4
a = 75
2.

35 70
=
100 x
35 x = 7, 000
x = 200

3.

1
3
+
= −4
x x −1
x − 1+ 3 x = −4 x ( x − 1)
4 x − 1 = −4 x 2 +4 x
0 = −4 x 2 +1
0 = ( −2 x + 1)(2 x + 1)
x = ±0.5
Only 0.5 or

1
(1/2) can be entered in the grid because, as the directions stated, no answer requires a minus sign.
2

1.

2.
/
.

/
.

3.

.

/
.

/
.

OR
/
.

.

/
.

.

.

/
.

/
.

.
0

23

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Chapter 1

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All About
the SAT®

4
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You will learn more about grid-ins in Chapter 8: “Grid-In Strategies.”

Peterson’s SAT® Prep Guide 2017

SAT ® ESSAY (OPTIONAL)
For the essay, you will be given a previously published passage that examines ideas in the sciences and arts, as well as in civic,
cultural, and political life. The passages are written for a broad-based audience, and prior knowledge of the topic is not expected.
Your task in writing the essay is to read and comprehend the text sufficiently to write a thoughtful analysis of the passage.
Though the passage contents may vary from test to test, the prompt will not change. You will be asked to explain how the author
of the passage builds an argument to persuade an audience. The prompt will likely look something like this:

As you read the following passage, consider how the author uses the following:

• Evidence, such as facts, statistics, or examples, to support claims.
• Reasoning to develop ideas and to connect claims and evidence.
• Stylistic or persuasive elements, such as word choice or appeals to emotion, to add power to
the ideas expressed.
Your response will be evaluated based on your comprehension of the text, as well as on the quality of your analysis and writing.
This means that you must show thoughtful understanding of the source text and appropriate use of textual evidence to support
your arguments. You will also be expected to organize your ideas in a coherent way and to express them clearly, using the conventions of Standard Written English. The essay does not elicit your opinion or ask you to use your imagination to write creatively.
Instead, your response should depend entirely on the source text to support your analysis. You can learn more about the optional
Essay in Chapter 6: “The SAT® Essay.”

24
Chapter 1
All About
the SAT®

THE SAT ® ANSWER SHEET
On the day of the test when you are given your test booklet, you’ll also be given a separate answer sheet. For each multiple-choice
question, you’ll see a corresponding set of answer circles. The circles are labeled A, B, C, and D. Remember the following about
the answer sheet:

• Answer sheets are read by machines—and machines can’t think. That means it’s up to you to make sure you’re in the right place
on the answer sheet every time you record an answer. The machine won’t know that you really meant to answer Question 25
when you marked the space for Question 26.

• If you skip a question, list the number on your scratch paper. Don’t mark the answer sheet in any way as a reminder. Any stray
marks may affect how the machine scores your answer sheet.

• Always check to see that the answer space you have filled in corresponds to the question you are answering.
• Be sure to fill in the answer circles completely so that there can be no mistake about which answers you chose.
These seem like simple things, but you’d be surprised how many students fail to do them, especially keeping track of answer
lines if they skip a question.

Make sure you’re in the right place! Always check to see that the answer space you
fill in corresponds to the question you are answering.

As you just read in the “Grid-Ins” section of this chapter, grid-inresponses are only for questions you will see in the math sections.
You’ll still be filling in circles, but they will look a little different from the multiple-choice circles. Again, here’s a sample of the
special grid you will use.

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/
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/
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boxes to write your numerical answer
fraction lines—use one at most per answer
decimal points—use one at most per answer

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At the top of the grid, you’ll write in the numerical answer. The slashes that appear in the second row are used for answers with
fractions. If you need one of these fraction lines in your answer, darken one of the circles. The circles with the dots are for answers
with decimal points—use these circles just as you do the fraction line circles. In the lower part of the grid, fill in the numbered
circles that correspond to the numbers in your answer.
Here are some examples. Note that for grid-in responses, answers can begin on the left or the right.

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25
Chapter 1
All About
the SAT®

HOW THE SAT ® IS SCORED
OK, you’ve filled in all your answer circles and perhaps written an essay. The 3 hours (and 50 minutes if you are taking the optional
essay portion) are up (not a minute too soon), and you’ve turned in your answer sheet and your essay sheet. What next? Off your
answers go to the machines at College Board and to the high school and college teachers who have been trained to read and
score the essays. The machines can scan the bubble sheets in seconds and calculate a score for most of your test. If you are taking
the optional essay portion, two readers will score it based on three criteria: reading, analysis, and writing. Their scores will be
combined and reported separately from the main portion of the SAT®.

Peterson’s SAT® Prep Guide 2017

In scoring the multiple-choice and grid-in sections of the SAT®, the machines give one point for each correct answer. Incorrect
answers have no effect on your score. Each reader of your essay uses a rubric against which he or she reads your essay. Each reader
then gives your essay a score from 1 to 4. The two scores will be combined to give you an essay subscore.
The result of these calculations for each part of the SAT®—Math and Evidence-Based Reading and Writing—is your raw score.
This is then converted to a scaled score between 400 and 1600. Your essay will be given a score ranging between 6 and 24, and
it will be reported separately. These scores will be reported to you and to the colleges you have chosen.
Remember, if you take the SAT® more than once, you can choose whether the schools you are applying to receive the scores
from each test date or just some of them.

NOTE: Because the SAT® can vary in format, scaled scores allow the test-maker to account for differences from
one version of the SAT® to another. Using scaled scores ensures that a score of 500 on one SAT® is equivalent to
500 on another.

STRATEGIES FOR SAT ® SUCCESS
What makes some people better test-takers than others? The secret isn’t just knowing the subject; it’s knowing specific test-taking
strategies that can add up to extra points. This means psyching out the test, knowing how the test-makers think and what they’re
looking for, and using this knowledge to your advantage. Smart test-takers know how to use pacing and guessing to add points
to their score.

Pace Yourself

26
Chapter 1

Suppose there are 20 questions in one of the math sections and they need to be answered in 25 minutes. That means that you
have 1 minute and 15 seconds to answer each question. But smart test-takers know that’s not the best way to use their time. If
you use less than a minute to answer the easier questions, you’ll have extra time to answer the more difficult ones. That’s why
learning to pace yourself is so important.

All About
the SAT®

ALERT: Don’t spin your wheels by spending too much time on any one question. Give it some thought, take your best shot,
and move along.

Question Sets in Math Usually Go from Easiest to Most Difficult—You Should, Too
A question set is one set of similar questions within the larger math and evidence-based reading and writing sections. In the
math sections, SAT® questions follow the pattern of easiest to hardest. Work your way through the easier questions as quickly as
you can. That way you’ll have more time for the more difficult ones.
But two words of caution: First, what may be easy to the test-writer may not be to you. Don’t panic if Question 3 seems hard. Try
to work through it, but don’t spend too much time on it if it’s a topic such as factoring that has just never been easy for you to
understand. Second, work quickly but carefully. Don’t work so fast that you make a silly mistake and lose a point that you should
have gained.

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You Can Set Your Own Speed Limit
All right, how will you know what your speed limit is? Use the practice tests to check your timing and see how it affects your
answers. If you’ve answered most of the questions in the time limit but have a lot of incorrect answers, you’d better slow down.
On the other hand, if you are very accurate in your answers but aren’t answering every question in a section, you can probably
pick up the pace a bit.

It’s Smart to Keep Moving
It’s hard to let go, but sometimes you have to. Don’t spend too much time on any one question before you’ve tried all the questions in a section. There may be questions later on in the test that you can answer easily, and you don’t want to lose points just
because you didn’t get to them.

The Easy Choice Isn’t Always Best
Are you at the end of a math section? Remember, that’s where you’ll usually find the hardest questions, which means that the
answers are more complex. Look carefully at the choices and really think about what the question is asking.

You Don’t Have to Read the Directions
What? Yes, you read it correctly the first time—you don’t have to read the directions. By the time you actually sit down to take the
SAT®, you’ve read this book, you’ve taken all the practice tests you could find, and you’ve read enough SAT® directions to fill a library.
So when the exam clock starts ticking, don’t waste time rereading directions you already know. Instead, go directly to Question 1.

27

You’re Going to Need a Watch
If you’re going to pace yourself, you need to keep track of the time—and what if there is no clock in your room or if the only clock
is out of your line of vision? That’s why it’s a good idea to bring a watch to the test. A word of warning: Don’t use a watch alarm
or your watch will end up on the proctor’s desk.

Chapter 1
All About
the SAT®

MAKE AN SAT ® STUDY PLAN
As with almost any form of learning, preparing for the SAT® is an investment of time. The more you have, the better your chances
of boosting your score significantly. Next, we’ll walk you through two different study plans, each tailored to a specific amount of
preparation time. Choose the plan that fits your circumstances and adapt it to your needs.
Regardless of how much time you have before the actual exam, your first step should be to take the Diagnostic Test in Part II of
this book. After you score it, compute your category percentages to assess your relative strengths and weaknesses. Hang on to
the scoring sheet so you know where to get started.

The Complete Plan
If you have three or more months to prepare, you should congratulate yourself! This will give you sufficient time to familiarize
yourself with the test, learn critical strategies, review grammar and math fundamentals, practice writing, and take full-length tests.

Peterson’s SAT® Prep Guide 2017

You’ll get the most out of your SAT® preparation if you:

•
•
•
•
•
•

Reread this chapter to ensure that you understand the format, structure, and scoring of the SAT®.
Take the Diagnostic Test and identify your areas that need improvement.
Read each and every strategy and review chapter.
Work through all the examples, exercises, and practice exams.
Read all the answer explanations.
Focus on the chapters where your scores show you need to improve.

The Accelerated Plan
If you have one month or less to prepare for the SAT® or if you cannot devote a lot of time to studying for any other reason, follow
the accelerated plan. You’ll get the most out of this plan if you:

• Reread this chapter to ensure that you understand the format, structure, and scoring of the SAT®.
• Take the Diagnostic Test and identify your areas that need improvement.
• Focus on the chapters that cover material that is most problematic for you and work through all the examples and
exercises in these chapters.

• Work through as many practice exams as you can.
• Read all the answer explanations.

28

NOTE: You may be wondering how you can possibly wade through all this information in time for the test. Don’t be
discouraged! We wrote this book knowing that some of you would be on very condensed schedules. The information
in this section will help you construct a study plan that works for you—one that will help you boost your score no
matter how limited your time may be. Remember, though, that practice and targeted study are essential elements
of that score boosting, so invest as much time as possible in your SAT® preparation.

Chapter 1
All About
the SAT®

MEASURING YOUR PROGRESS
It does seem as if you’re on a treadmill sometimes, doesn’t it? Question after question after question—are you really getting
anywhere? Is all of this studying really working?
The way to find out is to monitor your progress throughout the preparation period, whether it’s three months or four weeks. By
taking a diagnostic examination at the beginning, you’ll establish your skill baseline, and you’ll be able to craft the study plan
that’s right for you. Then, you can either start to read the entire book (if you are using the complete plan) or go directly to the
chapters that address your weaknesses (if you are using the accelerated plan). At the end of each chapter, complete the exercises
and compare your percentages to your original diagnostic percentages. How have you improved? Where do you still need work?
Even if you haven’t reached your ultimate performance goal, are you at least applying new test-taking methods?

Here’s an important point: You don’t have to go through the book in order. You might want to start with
the topic that you find most difficult, such as functions or grammar, or the question type that you’re
most unsure about, such as grid-ins. Then move to the next most difficult and so on down the line,
saving the easiest topics or question types until the end. If you use the accelerated plan, you should
definitely take this approach.

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SIMULATE TEST-TAKING CONDITIONS
The five full-length practice exams in Chapter 14 can help you prepare for the experience of taking a timed, standardized test.
Taking these tests will improve your familiarity with the SAT®, reduce your number of careless errors, and increase your overall
level of confidence. To make sure that you get the most out of this practice, you should do everything in your power to simulate
actual test-taking conditions.

Find a Block of Time
Because the SAT® is administered in one long block of time, the best way to simulate test-taking conditions is to take an entire
practice exam in one sitting. This means that you should set aside 3½–4 hours of consecutive time.
If you find it difficult to find approximately 4 quiet hours at home, maybe take the test in the library. If you decide to take a test
at home, take precautions. Let your friends know you are taking a practice test, put your phone in another room, and convince
siblings to stay out of your room. Easier said than done, right? Although infrequent interruptions won’t completely invalidate
your testing experience, you should try to avoid them.

Work at a Desk and Wear a Watch
Don’t take a practice test while you are lounging on your bed. After all, the SAT® proctors won’t let you take the test lying down!
Clear off sufficient space on a desk or table to work comfortably. Wear a watch to properly administer the sections under timed
conditions. Or use a timer. The time for each section is marked on the section, so check the beginning of each section and set
your timer or your watch for that amount of time.
You are not allowed to explore other sections on the test while you are supposed to be working on a particular one. So when you
take your practice tests, don’t look ahead or back. Take the full time to complete each section.

29
Chapter 1

If you’re worried that you won’t be able to resist the temptation to check the answer keys
during the practice tests, cover them up before you take a test. Don’t allow yourself to become
dependent upon a sneak look now and then. You won’t have answer keys available on test
day, and the main purpose of the practice tests is to prepare you for the real experience.

All About
the SAT®

Practice on a Weekend Morning
Since the SAT® is typically administered at 8:30 a.m. on Saturday (or Sunday for religious observers), why not take the practice test
at the exact same time on a weekend morning? You should be most energetic in the morning anyway. If you are not a morning
person, now is a good time to become one since that’s when you’ll have to take the actual SAT®! When you take the practice test,
allow yourself two breaks. Give yourself a 5-minute break after Section 2: Writing and Language Test to run to the bathroom, eat
a snack, and re-sharpen your pencils. You can take another 5-minute break after Section 4: Math Test—Calculator.
Remember that your goal is to take these practice tests in as true an environment as possible so that you’re prepared to take the
real SAT®. You will be accustomed to sitting for a long period of time, but you will get two breaks. This knowledge will make you
considerably less anxious on test day.

Peterson’s SAT® Prep Guide 2017

The following table puts all of the SAT® section times in a one-stop format, so you can refer to it often when planning your SAT®
study time.

Section

Number of Questions

Time Allowed

Reading

52

65 minutes

Writing and Language

44

35 minutes

Math Test—No Calculator

20

25 minutes

Math Test—Calculator

38

55 minutes

Essay

--

50 minutes

One Third of the Way Through Your Study
When you are approximately one third of the way through your plan of study—this can be after ten days or a month—it’s time
to take one of the practice tests. When you have finished scoring and reading the answer explanations, compare your scores with
your original diagnostic scores. Hopefully, you’re doing better. But if you’re not, don’t panic. At this point in test preparation, it’s
not unusual to score about the same as you did at the beginning.
What’s more important than what you scored is how you took the practice test. Did you really use the test-taking strategies to
which you’ve been introduced? If you didn’t, go back to the strategy chapters and either reread them, if you are doing the complete plan, or at least reread the summaries, if you are on the accelerated plan. Then continue your review. Read more review
chapters and complete the exercises.

30
Chapter 1
All About
the SAT®

Two Thirds of the Way Through Your Study
After you have worked through most of the review chapters (under the complete plan) or all of the material relating to your
areas of weakness (under the accelerated plan), it’s time to take another practice test. By now, you should be seeing some real
improvement in your scores. If you are still having trouble with certain topics, review the problematic material again.

The Home Stretch
For the most part, the last phase of study should involve less learning and more practice. Take more practice tests! By now, you
probably understand how to take the exam. What you need is more practice taking the test under simulated test-day conditions
to work on your pacing and test-taking strategies.
When you take additional practice exams, be sure to do so in a near-test environment. Keep analyzing your scores to ensure that
all of this practice is working. Determine which areas need additional work. If you skipped over any of the review chapters in this
book, go back and use the exercises to improve your skills.

The Final Week
One last word of advice: No matter which study plan you select, you should probably take one full, timed practice SAT® the week
before you take the actual SAT®. This will get you ready for the big day. But don’t take the practice test the day before the real
exam. That’s a time when you should be relaxing, not cramming.

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THE NIGHT BEFORE AND DAY OF THE EXAM
If you follow the guidelines in this book, you will be extremely well prepared. You will know the format inside and out; you will
know how to approach every type of question; you will have worked hard to strengthen your weak areas; and you will have taken
multiple practice tests under simulated testing conditions. The last 24 hours before the SAT® exam is not the time to cram—it’s
actually the time to relax. Remember that the SAT® is primarily a test of how you think, not what you know. So last-minute
cramming can be more confusing than illuminating.
In the morning, take a shower to wake up and then eat a sensible breakfast. If you are a person who usually eats breakfast, you
should probably eat your customary meal. If you don’t usually eat breakfast, don’t gorge yourself on test day, because it will be
a shock to your system. Eat something light (like a granola bar and a piece of fruit) and pack that snack.

Test Day Checklist
On the night before the big day, find a diversion to keep yourself from obsessing about the SAT®. Maybe stay home
and watch some of your favorite television shows. Or go out to an early movie. Do whatever is best for you. Just
make sure you get plenty of sleep.
You should also lay out the following items before you go to bed:
Test ticket
A cceptable photo ID
S harp pencils with erasers

NOTE: Make sure you allow enough time to arrive at the
test site at least 15 minutes before the 8 a.m. arrival time.
You don’t want to raise your level of anxiety by having to
rush to get there.

P ermissible calculator
S nack and bottle of water
A  sweater or sweatshirt in case of cooler test center conditions (optional)

31
Chapter 1
All About
the SAT®

Peterson’s SAT® Prep Guide 2017

TOP 10 STRATEGIES TO RAISE YOUR SCORE
When it comes to taking the SAT®, some test-taking skills will do you more good than others. There are concepts you can learn
and techniques you can follow that will help you do your best. Here’s our pick for the top 10 strategies to raise your score:

32
Chapter 1
All About
the SAT®

1.

Create a study plan and follow it. The right SAT® study plan will help you get the most out of this book in whatever time
you have.

2.

Don’t get stuck on any one question. Since you have a specific amount of time to answer questions, you can’t afford to
spend too much time on any one problem.

3.

Learn the directions in advance. If you already know the directions, you won’t have to waste your time reading them. You’ll
be able to jump right in and start answering questions as soon as the testing clock begins.

4.

If you choose to take the essay portion of the test, it’s important to develop your ideas and express them clearly, using
examples to back them up. Your essay doesn’t have to be grammatically perfect, but it does have to be focused and organized, and it should explain how the author develops his or her argument.

5.

For the Writing and Language Test multiple-choice questions, think about the simplest, clearest way to express an idea.
If an answer choice sounds awkward or overly complicated, chances are good that it’s wrong.

6.

For relevant words in context, be sure to read the sentences around the word carefully. The SAT® is no longer testing obscure
words but instead is focusing on defining words in the context of a passage.

7.

For Evidence-Based Reading Test questions, first skim the passage to see what it’s about. Look for the main ideas, and then
tackle the questions that direct you straight to the answer by referring you to a specific line in the passage. Then work on
the detailed questions that require a closer reading of the passage.

8.

For the math multiple-choice questions, it can help if you know how to approach the problems. If you’re stuck, try substituting numbers for variables. You can also try plugging in numbers from the answer choices. Start with one of the middle
numbers. That way, if it doesn’t work, you can strategically choose one that’s higher or lower.

9.

For the math grid-ins, you determine the answer and fill it into a grid. Be sure to make your best guess, even if you’re
not sure.

10. Finally, relax the night before the test. Don’t cram. Studying at the last minute will only stress you out. Go to a movie or
hang out with a friend—anything to get your mind off the test!

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SUMMING IT UP
• Learning the question types is the best way to prepare for the SAT® exam. Knowing the test format and question types will
relieve test anxiety, because you’ll know exactly what to expect on test day.
1.	Evidence-Based Reading and Writing: The answer to every question will be either directly stated or implied in the
passage.
2.	Standard Written English: These question sets test your ability to spot and correct grammatical errors, usage problems,
and wordiness. You will also be expected to answer questions to improve the development and organization of a particular paragraph or the passage as a whole.
3. Multiple-Choice Math: A set of reference formulas is given at the beginning of each math section, so you don’t have
to worry about forgetting an important formula.
4. Grid-ins: You have to calculate the answer and then fill in circles on the grids provided on the answer sheet. Only the
circles count, so fill in each one correctly.
5. Essay (Optional): You will have 50 minutes to write your essay, which tests reading, analysis, and writing skills. You will
be asked to produce a written analysis of a provided source text. You will need to explain how the author has effectively
built an argument to persuade his or her audience. The readers are trained to evaluate the essays as first drafts, not
polished final products.

• When you take a full practice examination, try to simulate test-taking conditions:
ºº You’ll need to set aside approximately 4 quiet hours.
ºº Work at a desk and wear a watch.
ºº Cover the answers before you start the test.
ºº Whenever possible, take the full practice tests on weekend mornings.

• You may use a calculator on the SAT®, but only in one math section. For some questions, you will need to decide if the
calculator will help you or slow you down.

33

• Every SAT® exam question is worth 1 point, whether it is an easy question or a difficult one. So nail the easier questions—and
Chapter 1

quickly accumulate points.

• Fill in the answer circles cleanly and completely, or you won’t get credit for your answers.
• Random guessing will have little effect on your score, but educated guessing can boost your score.

All About
the SAT®

• Pace yourself and move through the test relatively quickly.
• Relax the evening before the SAT®, but also be sure you’re prepared.
ºº Assemble the supplies you will need for the test.
ºº Pick out what you’ll wear and remember to layer your clothes.
ºº Be sure your calculator has fresh batteries.

• On the morning of the exam, eat breakfast, pack your snack, and leave for the test site in plenty of time to get there at least
15 minutes before the start time.

Peterson’s SAT® Prep Guide 2017

ONLINE
PREP

Want to Know More?

Access more practice questions, valuable lessons, helpful tips, and expert strategies from the following introductory lessons about
the SAT® in Peterson’s SAT® Online Course:

• About the SAT®
• Demystifying the SAT®
• Scoring
To purchase and access the course, go to www.petersons.com/sat.

34
Chapter 1
All About
the SAT®

www.petersons.com

part ii: the diagnostic test

Chapter 2: The Diagnostic Test

Chapter 2:
The Diagnostic Test
OVERVIEW
Introduction to the Diagnostic Test
Answer Sheets
Section 1: Reading Test
Section 2: Writing and Language Test
Section 3: Math Test—No Calculator
Section 4: Math Test—Calculator
Section 5: Essay
Answer Keys and Explanations
Computing Your Scores

INTRODUCTION TO THE DIAGNOSTIC TEST
Before you begin preparing for the SAT® exam, it’s important to know your strengths and the areas where you need improvement.
If you find the questions easy for the Reading Test, for example, it would be a mistake to dedicate hours practicing them. Taking the
Diagnostic Test in this chapter and then working out your scores will help you determine how you should apportion your study time.

Chapter 2
The
Diagnostic
Test

Preparing To Take the Diagnostic Test
If possible, take the test in one sitting. Give yourself at least 4 hours to complete the Diagnostic Test. The actual test is 3 hours and
45 minutes, and you’ll be allowed to take three short breaks—you may even want to have some healthy snacks nearby for a quick
break you’ll want to take. Simulating the test this way will give you an idea of how long the sections are and how it feels to take the
entire test. You will also get a sense of how long you can spend on each question in each section, so you can begin to work out a
pacing schedule for yourself.
First, assemble all the things you will need to take the test. These include:

•
•
•
•

37

No. 2 pencils, at least three
A calculator with fresh batteries
A timer
The answer sheets and the lined paper for the essay—provided on the following pages

Set a timer for the time specified for each section, which is noted at the top of the first page of each test section. Stick to that time,
so you are simulating the real test. At this point, it’s as important to know how many questions you can answer in the time allotted
as it is to answer questions correctly. Good luck!

Peterson’s SAT® Prep Guide 2017

Diagnostic Test—Answer Sheet
Section 1: Reading Test
1.

12.

23.

33.

43.

2.

13.

24.

34.

44.

3.

14.

25.

35.

45.

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15.

26.

36.

46.

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37.

47.

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48.

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49.

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30.

40.

50.

9.

20.

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41.

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10.

21.

32.

42.

52.

11.

22.

Section 2: Writing and Language Test
1.

10.

19.

28.

37.

2.

11.

20.

29.

38.

3.

12.

21.

30.

39.

4.

13.

22.

31.

40.

5.

14.

23.

32.

41.

6.

15.

24.

33.

42.

7.

16.

25.

34.

43.

8.

17.

26.

35.

44.

9.

18.

27.

36.

Section 3: Math Test—No Calculator
1.

4.

7.

10.

13.

2.

5.

8.

11.

14.

3.

6.

9.

12.

15.

Peterson's SAT® Prep Guide 2017

  39

Diagnostic Test—Answer Sheet
Section 3: Math Test—No Calculator
16.

17.

18.

19.

20.

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Diagnostic Test—Answer Sheet
Section 5: Essay
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Diagnostic Test—Answer Sheet

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Diagnostic Test—Answer Sheet

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Diagnostic Test—Answer Sheet

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Diagnostic Test
SECTION 1: READING TEST
65 Minutes—52 Questions
TURN TO SECTION 1 OF YOUR ANSWER SHEET TO ANSWER THE QUESTIONS IN THIS SECTION.
DIRECTIONS: Each passage (or pair of passages) in this section is followed by a number of multiple-choice questions. After
reading each passage, select the best answer to each question based on what is stated or implied in the passage or passages
and in any supplementary material, such as a table, graph, chart, or photograph.

John James Audubon (1785–1851) is known primarily for his bird
studies, but as this passage from Ornithological Biography shows,
he wrote about the behavior of other animals as well.

35

Black Bear

Line
5

10

15

20

25

30

		 The Black Bear (Ursus americanus), however clumsy
in appearance, is active, vigilant, and persevering;
possesses great strength, courage, and address; and
undergoes with little injury the greatest fatigues and
hardships in avoiding the pursuit of the hunter. Like the
Deer, it changes its haunts with the seasons, and for the
same reason, namely, the desire of obtaining suitable
food, or of retiring to the more inaccessible parts, where
it can pass the time in security, unobserved by man,
the most dangerous of its enemies. During the spring
months, it searches for food in the low rich alluvial lands
that border the rivers, or by the margins of such inland
lakes as, on account of their small size, are called by us
ponds. There it procures abundance of succulent roots,
and of the tender juicy stems of plants, on which it chiefly
feeds at that season. During the summer heat, it enters
the gloomy swamps, passes much of its time wallowing
in the mud, like a hog, and contents itself with crayfish,
roots, and nettles, now and then, when hard pressed
by hunger, seizing on a young pig, or perhaps a sow, or
even a calf. As soon as the different kinds of berries which
grow on the mountain begin to ripen, the Bears betake
themselves to the high grounds, followed by their cubs.
In such retired parts of the country where there are no
hilly grounds, it pays visits to the maize fields, which
it ravages for a while. After this, the various species of
nuts, acorns, grapes, and other forest fruits, that form
what in the western country is called mast, attract its
attention. The Bear is then seen rambling singly through
the woods to gather this harvest, not forgetting to rob

40

45

50

55

60

every Bee tree it meets with, Bears being, as you well
know, expert at this operation. You also know that they
are good climbers, and may have been told, or at least
may now be told, that the Black Bear now and then
houses itself in the hollow trunks of the larger trees for
weeks together, when it is said to suck its paws. You
are probably not aware of a habit in which it indulges,
and which, being curious, must be interesting to you.
At one season, the Black Bear may be seen examining
the lower part of the trunk of a tree for several minutes
with much attention, at the same time looking around,
and snuffing the air, to assure itself that no enemy is
near. It then raises itself on its hind legs, approaches
the trunk, embraces it with its forelegs, and scratches
the bark with its teeth and claws for several minutes
in continuance. Its jaws clash against each other, until
a mass of foam runs down both sides of the mouth.
After this it continues its rambles. In various portions
of our country, many of our woodsmen and hunters
who have seen the Bear performing the singular operation just described, imagine that it does so for the
purpose of leaving behind an indication of its size and
power. They measure the height at which the scratches
are made, and in this manner, can, in fact, form an
estimate of the magnitude of the individual. My own
opinion, however, is different. It seems to me that the
Bear scratches on the trees, not for the purpose of
showing its size or its strength, but merely for that of
sharpening its teeth and claws, to enable it better to
encounter a rival of its own species during the amatory
season. The Wild Boar of Europe clashes its tusks and
scrapes the earth with its feet, and the Deer rubs its
antlers against the lower part of the stems of young
trees or bushes, for the same purpose.

Diagnostic Test — reading

Questions 1–10 are based on the following passage.

CONTINUE
Peterson's SAT® Prep Guide 2017

  45

Diagnostic Test
1

2

3

4

5

  46

As used in line 3, “address” refers to
A.

habitat.

B.

anxiety.

C.

skill.

D.

direction.

What is the most likely reason that Audubon wrote about
the black bear?
A.

He wanted to provide more information about
another animal to his readers.

B.

He was fascinated by mammals.

C.

He wanted to prove he had interests other than
birds.

D.

He wanted to show the commonalities in
behavioral patterns of bears and birds.

Which of the following quotes from the text provides
evidence to support the idea that certain dissimilar forest
mammals share some of the same traits?

6

7

8

Which choice provides the best evidence for the answer
to the previous question?
A.

Lines 16–17 (“During . . . swamps”)

B.

Lines 17–18 (“passes . . . mud”)

C.

Lines 19–20 (“now and . . . hunger”)

D.

Lines 20–21 (“seizing on . . . calf.”)

According to Audubon, how are the claws of the black bear
like the tusks of the wild boar?
A.

Both are parts of the body that warn other animals
that they are predators.

B.

Both animals use these parts of their bodies to
forage for crayfish and roots.

C.

Both animals use these body parts to defend
themselves from human predators.

D.

Both are parts of the body that the animal sharpens
to better compete for a mate.

The fact that Audubon calls man the bear’s “most dangerous” enemy (line 10) indicates that he
A.

is a hunter himself.

A.

Lines 5–6 (“Like . . . seasons”)

B.

has some sympathy for hunted bears.

B.

Line 14 (“it procures . . . roots”)

C.

does not believe that bears are dangerous.

C.

Lines 18–20 (“contents itself . . . pig”)

D.

thinks bears are more dangerous than people.

D.

Lines 30–32 (“not forgetting . . . operation”)

Huntsmen and woodsmen claim that the bear scratches
tree bark with its teeth and claws to
A.

sharpen its teeth.

B.

mark the tree for winter hibernation.

C.

ward off potential predators by showing its size.

D.

mark the tree so that other animals can’t harvest its
nuts and acorns.

9

Which choice provides the best evidence for the answer
to the previous question?
A.

Lines 5–6 (“Like the Deer . . . seasons”)

B.

Lines 6–8 (“for the same . . . food”)

C.

Line 8 (“retiring . . . parts”)

D.

Lines 9–10 (“man, . . . enemies.”)

10 The author indicates that he believes that the reader
A.

knows absolutely nothing about bears.

What is the main rhetorical effect of the author’s description of how black bears behave in swamps, lines 16–21?

B.

already has some knowledge of bears.

C.

needs help overcoming a fear of bears.

A.

To show that the bear is an exceptional predator

D.

believes bears can climb trees as well as apes.

B.

To explain why humans might want to hunt bears

C.

To impress the reader with how varied a bear’s
diet is

D.

To create an image of a bear placidly foraging for
food

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Diagnostic Test
Questions 11–21 are based on the following passage.
Thomas Jefferson wrote in 1787 to his nephew, Peter Carr, a student
at the College of William and Mary.
Paris, August 10, 1787

Line
5

10

15

		 Dear Peter, I have received your two letters of
December 30 and April 18 and am very happy to find by
them, as well as by letters from Mr. Wythe,* that you have
been so fortunate as to attract his notice and good will: I
am sure you will find this to have been one of the more
fortunate events of your life, as I have ever been sensible
it was of mine. I enclose you a sketch of the sciences to
which I would wish you to apply in such order as Mr. Wythe
shall advise: I mention also the books in them worth your
reading, which submit to his correction. Many of these
are among your father’s books, which you should have
brought to you. As I do not recollect those of them not in
his library, you must write to me for them, making out a
catalogue of such as you think you shall have occasion for
in 18 months from the date of your letter, and consulting
Mr. Wythe on the subject. To this sketch I will add a few
particular observations.
		

1. Italian. I fear the learning of this language will

ploughman and a professor. The former will decide it
as well, and often better than the latter, because he has
not been led astray by artificial rules.
*George Wythe, a well-respected scholar, the first American
law professor, and one of the signatories of the Declaration of
Independence, became an important teacher and mentor to
Thomas Jefferson.

11 What is the best description of Mr. Wythe and his relationship to the Jefferson family?
A.

Teacher

B.

Cousin

C.

Family friend

D.

Public servant

12 What is the purpose of Jefferson’s letter to his nephew?
A.

To advise him about his education

B.

To advise him about leading a moral life

C.

To make sure he will learn a second language

D.

To keep in touch with his family while abroad

20 confound your French and Spanish. Being all of them

degenerated dialects of the Latin, they are apt to mix in
conversation. I have never seen a person speaking the
three languages who did not mix them. It is a delightful
language, but late events having rendered the Spanish
25 more useful, lay it aside to prosecute that.
		 2. Spanish. Bestow great attention on this, and
endeavor to acquire an accurate knowledge of it. Our
future connections with Spain and Spanish America will
render that language a valuable acquisition. The ancient
30 history of a great part of America too is written in that
language. I send you a dictionary.
		 3. Moral philosophy. I think it lost time to attend
lectures in this branch. He who made us would have been
a pitiful bungler if he had made the rules of our moral
35 conduct a matter of science. For one man of science, there
are thousands who are not. What would have become of
them? Man was destined for society. His morality therefore
was to be formed to this object. He was endowed with a
sense of right and wrong merely relative to this. This sense
40 is as much a part of his nature as the sense of hearing,
seeing, feeling; it is the true foundation of morality. . . .The
moral sense, or conscience, is as much a part of man as
his leg or arm. It is given to all human beings in a stronger
or weaker degree, as force of members is given them
45 in a greater or less degree. . . . State a moral case to a

13 What does Jefferson suggest his reader do about studying
the Italian language?
A.

Ignore it to pursue Spanish instead

B.

Ignore it to study French instead

C.

Use it as a building block to studying Spanish

D.

Use it in conversation with Spanish people

14 Which choice provides the best evidence for the answer
to the previous question?
A.

Lines 19–20 (“I fear . . . Spanish.”)

B.

Lines 20–22 (“Being all . . . conversation.”)

C.

Lines 22–23 (“I have . . . mix them.”)

D.

Lines 23–25 (“It is a . . . prosecute that.”)

CONTINUE
Peterson's SAT® Prep Guide 2017

  47

Diagnostic Test
15 In lines 42–43, Jefferson compares conscience to a physical
limb of the body to show

18 Jefferson tells his nephew to lay Italian aside because it is
A.

a degenerated dialect.

A.

that it is natural and present in all human beings.

B.

not necessary since he already knows French.

B.

how easily we take it for granted.

C.

not useful to be multilingual.

C.

that without it, humans are powerless.

D.

too easy to get it mixed up with Spanish.

D.

how mental and physical states are integrated.

16 Based on the passage, what country does Jefferson think
will most closely align with the newly independent colonies in the future?
A.

England

B.

France

C.

Italy

D.

Spain

19 Which of the following best summarizes Jefferson’s overall
view of morality?
A.

Morality is a science that can be taught by professors
and scholars.

B.

Moral philosophy is self-taught.

C.

A sense of morality is part of human nature.

D.

Humans are moral beings who need rules to guide
their behavior.

17 By “lost time” (line 32), Jefferson means
A.

wasted time.

B.

the past.

C.

missing time.

D.

youth.
Three Generations of the Jefferson Family
Peter Jefferson (b: 1708; d: 1757) - [spouse] Jane Randolph (b: 1721; d: 1776)
[children: 10]

Thomas (1743 - 1826) - [spouse] Martha Wales (1748 - 1782) Martha (1746 - 1811) - [spouse] Dabney Carr (1743 - 1773)
[children: 6]

[children: 6]

(others died before reaching adulthood)

Martha (b: 1772; d: 1836)

Mary (b: 1778; d: 1782)

20 Based on information in the family tree, Peter may have
relied on advice from his uncle Thomas because Peter

21 Which of the following best describes the tone of Jefferson’s letter to his nephew?

considered Thomas to be a better parental figure
than his father was.

A.

Invested and warm

B.

Anxious and worried

B.

did not receive reliable advice from his own mother.

C.

Objective and matter-of-fact

C.

did not have a father, and his mother was busy with
her 5 other children.

D.

Distant and preoccupied

D.

wanted the best education so he could help take
care of his 5 siblings.

A.

  48

Peter (b: 1770; d: 1815)

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Diagnostic Test
Questions 22–32 are based on the following passage
and supplementary material.
This excerpt is from the article “New Link in the Food Chain? Marine
Plastic Pollution and Seafood Safety,” by Nate Seltenrich. It has been
reproduced from the journal Environmental Health Perspectives.
		 World plastics production has experienced almost
constant growth for more than half a century, rising from
approximately 1.9 tons in 1950 to approximately 330
Line million tons in 2013. The World Bank estimates that 1.4
5 billion tons of trash are generated globally each year,
10% of it plastic. The International Maritime Organization
has banned the dumping of plastic waste (and most
other garbage) at sea. However, an unknown portion of
the plastic produced each year escapes into the envi10 ronment—instead of being landfilled, incinerated, or
recycled—and at least some of it eventually makes its way
to sea.
		 Plastics that reach the ocean will gradually break
down into ever-smaller pieces due to sunlight exposure,
15 oxidation, and the physical action of waves, currents, and
grazing by fish and birds. So-called microplastics—variably
defined in the scientific literature and popular press as
smaller than 1 or 5 mm in diameter—are understood to
be the most abundant type of plastic in the ocean. The 5
20 Gyres’ authors* found microplastics almost everywhere
they sampled, from near-shore environments to the open
ocean, in varying concentrations, and they estimated
that particles 4.75 mm or smaller—about the size of a
lentil—made up roughly 90% of the total plastic pieces
25 they collected.
		 But the degradation of larger pieces of plastic is not
the only way microplastics end up in the ocean. Nurdles—
the plastic pellets used as a feedstock for producing
plastic goods—can spill from ships or land-based sources,
30 and “microbeads” used as scrubbing agents in personal
care products such as skin cleansers, toothpastes, and
shampoos, can escape water-treatment facilities and pass
into water-sheds with treated water. (In June 2014, Illinois
became the first US state to ban the manufacture and
35 sale of products containing microbeads, which have been
documented in the Great Lakes and Chicago’s North Shore
Channel.)
		 Marine organisms throughout the food chain
commonly consume plastics of various sizes. The tiniest
40 microplastics are small enough to be mistaken for food by
zooplankton, allowing them to enter the food chain at very
low trophic levels. Some larger predators are thought to
confuse nurdles (which typically measure less than 5 mm
in diameter) with fish eggs or other food sources.

45 		

Once plastics have been consumed, laboratory tests
show that chemical additives and adsorbed pollutants and
metals on their surface can desorb (leach out) and transfer
into the guts and tissues of marine organisms. . . .
		

Research has shown that harmful and persistent

50 substances can both bioaccumulate (or increase in

concentration as exposures persist) and biomagnify (or
increase in concentration at higher trophic levels) within
organisms as they assume some of the chemical burden
of their prey or environment. Yet again, no research has yet
55 demonstrated the bioaccumulation of sorbed pollutants
in the environment.
		 Three key questions remain to be determined. To
what extent do plastics transfer pollutants and additives to
organisms upon ingestion? What contribution are plastics
60 making to the contaminant burden in organisms above
and beyond their exposures through water, sediments, and
food? And, finally, what proportion of humans’ exposure
to plastic ingredients and environmental pollutants occurs
through seafood? Researchers are moving carefully in the
65 direction of answers to these questions. . . .
		 New laws . . . could require handling plastics more
responsibly at the end of their useful life through recycling,
proper disposal, and extended producer responsibility.
		

Rolf Halden, director of the Center for Environmental

70 Security at the Biodesign Institute at Arizona State Uni-

versity, advocates for another solution: manufacturing
more sustainable plastics from the start. “We need to
design the next generation of plastics to make them more
biodegradable so that they don’t have a long half-life,
75 they don’t accumulate in the oceans, and they don’t have
the opportunity to collect chemicals long-term,” he says.
“There’s just no way we can shield people from all exposures that could occur. Let’s design safer chemicals and
make the whole problem moot.”
*The 5 Gyres Institute addresses plastic pollution in the ocean.

22 According to the passage, plastic is
A.

wasted more than any other material.

B.

responsible for a massive amount of waste.

C.

not being produced as much as it once was.

D.

the single most dangerous material to the planet.

CONTINUE
Peterson's SAT® Prep Guide 2017

  49

Diagnostic Test
23 Which choice provides the best evidence for the answer
to the previous question?
A.

Lines 1–2 (“World . . . century”)

A.

Passing laws to mandate more rigorous recycling

B.

Lines 4–6 (“The World . . . it plastic.”)

B.

Developing plastics that are biodegradable

C.

Lines 6–8 (“The International . . . at sea.”)

C.

Making plastics that are safe to ingest

D.

Lines 8–10 (“However . . . environment”)

D.

Banning the production of new plastic products

24 Which best describes the overall tone of the article?

  50

25 What solution does Rolf Halden support to decrease the
effects of pollution from plastics on humans?

A.

Neutral and scientific

B.

Emotional and persuasive

C.

Personal and human

D.

Subjective and opinionated

www.petersons.com

26 This article was most likely written to
A.

offer a theory about why the environment is in
danger.

B.

inform the public of the problems of plastic in the
ocean.

C.

start a movement to halt all plastic production.

D.

inspire readers to clean up the oceans personally.

Diagnostic Test
27 Which choice provides the best evidence for the answer
to the previous question?

30 Which of the following words would be most helpful in
figuring out the meaning of the word “adsorbed” (line 46)?

A.

Lines 45–48 (“Once plastics . . . organisms . . . ”)

A.

Absolve

B.

Lines 49–51 (“Research . . . biomagnify”)

B.

Adhere

C.

Lines 54–56 (“Yet again . . . environment.”)

C.

Absorb

D.

Line 57 (“Three key . . . determined.”)

D.

Sorbet

28 Which of the following statements could be learned from
the diagram on the previous page about how plastics
disrupt the food chain?
A.

Plastics enter the food chain when small fish eat
small pieces of plastic.

B.

Sunlight breaks down larger pieces of plastic that
have been tossed in the ocean.

C.

Emptying wastewater directly into the ocean is
illegal in many places.

D.

People should only eat a limited amount of seafood
because it may contain unsafe contaminants.

29 Why did Illinois ban the sale of certain personal care
products?
A.

Residues from the products were ending up in the
ocean.

B.

The containers couldn’t be recycled.

C.

The products were determined to be carcinogenic.

D.

The products contained microbeads that were
getting into the water system.

31 One of the questions the author raises in lines 57–65 deals
with
A.

how much plastic waste from the oceans people
might be ingesting.

B.

how sea creatures happen to ingest plastic waste in
the oceans.

C.

what kinds of plastic waste can be found in the
oceans.

D.

why people are so careless about dumping plastic
waste into the oceans.

32 Which choice provides the best evidence for the answer
to the previous question?
A.

Lines 57–59 (“To what . . . ingestion?”)

B.

Lines 59–62 (“What contribution . . . food?”)

C.

Lines 62–64 (“And, finally . . . seafood.”)

D.

Line 64–65 (“Researchers . . . questions . . . ”)

CONTINUE
Peterson's SAT® Prep Guide 2017

  51

Diagnostic Test
Questions 33–42 are based on the following passage.
Angel Decora was born Hinookmahiwikilinaka on the Winnebago
Reservation in Nebraska in 1871. She worked as a book illustrator,
particularly on books by and about Native Americans, and lectured
and wrote about Indian art. The story from which this excerpt is
taken, “The Sick Child,” may be autobiographical.
		 It was about sunset when I, a little child, was sent
with a handful of powdered tobacco leaves and red
feathers to make an offering to the spirit who had caused
Line the sickness of my little sister. It had been a long, hard
5 winter, and the snow lay deep on the prairie as far as
the eye could reach. The medicine-woman’s directions
had been that the offering must be laid upon the naked
earth, and that to find it I must face toward the setting
sun.
10 		

I was taught the prayer: “Spirit grandfather, I offer
this to thee. I pray thee restore my little sister to health.”
Full of reverence and a strong faith that I could appease
the anger of the spirit, I started out to plead for the life
of our little one.

15 		

But now where was a spot of earth to be found in
all that white monotony? They had talked of death at
the house. I hoped that my little sister would live, but I
was afraid of nature.

		 I reached a little spring. I looked down to its pebbly
20 bottom, wondering whether I should leave my offering
there, or keep on in search of a spot of earth. If I put my
offering in the water, would it reach the bottom and
touch the earth, or would it float away, as it had always
done when I made my offering to the water spirit?
25 		

Once more I started on in my search of the bare
ground.

		 The surface was crusted in some places, and
walking was easy; in other places I would wade through
a foot or more of snow. Often I paused, thinking to clear
30 the snow away in some place and there lay my offering.
But no, my faith must be in nature, and I must trust to it
to lay bare the earth.
		

It was a hard struggle for so small a child.

		 I went on and on; the reeds were waving their tas35 selled ends in the wind. I stopped and looked at them.
A reed, whirling in the wind, had formed a space round
its stem, making a loose socket. I stood looking into the
opening. The reed must be rooted in the ground, and
the hole must follow the stem to the earth. If I poured
40 my offerings into the hole, surely they must reach the

  52

www.petersons.com

ground; so I said the prayer I had been taught, and
dropped my tobacco and red feathers into the opening
that nature itself had created.
		

No sooner was the sacrifice accomplished than a

45 feeling of doubt and fear thrilled me. What if my offering

should never reach the earth? Would my little sister die?
		 Not till I turned homeward did I realize how cold I
was. When at last I reached the house they took me in
and warmed me, but did not question me, and I said
50 nothing. Everyone was sad, for the little one had grown
worse.
		 The next day the medicine woman said my little
sister was beyond hope; she could not live. Then bitter
remorse was mine, for I thought I had been unfaithful,
55 and therefore my little sister was to be called to the
spirit-land. I was a silent child, and did not utter my
feelings; my remorse was intense.
		 My parents would not listen to what the medicine-woman had said, but clung to hope. As soon as
60 she had gone, they sent for a medicine-man who lived
many miles away.
		 He arrived about dark. He was a large man, with
a sad, gentle face. His presence had always filled me
with awe, and that night it was especially so, for he
65 was coming as a holy man. He entered the room where
the baby lay, and took a seat, hardly noticing any one.
There was silence saving only for the tinkling of the
little tin ornaments on his medicine-bag. He began to
speak: “A soul has departed from this house, gone to the
70 spirit-land. As I came I saw luminous vapor above the
house. It ascended, it grew less, it was gone on its way
to the spirit-land. It was the spirit of the little child who
is sick; she still breathes, but her spirit is beyond our
reach . . .
33 The narrator wants to place her offering correctly because
she
A.

will have to explain her choice to everyone else.

B.

wants to be trusted with similar tasks in the future.

C.

thinks doing so will save her little sister’s life.

D.

is afraid of being punished if she does it incorrectly.

Diagnostic Test
34 The medicine man describes the appearance of the spirit
of the narrator’s sister as
A.

luminous vapor.

B.

a tin ornament.

C.

a baby.

D.

beyond our reach.

35 Why didn’t the girl’s parents send for the medicine man in
the first place?

39 Based on the passage, which choice best describes the
narrator’s relationship with her parents?
A.

The parents seem to treat the narrator as if she
were an adult.

B.

The narrator wishes her parents would give her
more responsibility.

C.

The parents love their youngest child, but not the
narrator.

D.

The narrator receives warmth and validation from
her parents.

A.

He was busy helping another family at the time.

B.

He had to come from a long distance.

C.

They thought the medicine woman would be able
to help their daughter.

A.

A Native American recalls her experience of trying
to save and losing her baby sister.

D.

They preferred a woman to cure their female child.

B.

A Native American child is called upon to make an
offering to the spirits.

C.

A Native American family struggles with illness in
the depths of winter on the Plains.

D.

A Native American family uses their religious beliefs
to try to save their daughter.

36 What evidence from the text shows the girl’s dilemma in
following the medicine woman’s directions?
A.

Lines 7–8 (“the offering . . . naked earth”)

B.

Lines 21–23 (“If I put . . . away”)

C.

Lines 27–29 (“The surface . . . of snow. ”)

D.

Lines 34–35 (“I went on . . . the wind.”)

37 Which title best expresses an important theme of the
passage?

40 What is the central idea in the passage?

41 When the girl says, “bitter remorse was mine” (lines 53–54),
she feels
A.

that she is responsible for her sister’s illness.

B.

badly because she didn’t listen to the medicine
woman.

A.

The Medicine-Man

C.

angry about being given so much responsibility.

B.

Acts of Faith

D.

C.

Native American Culture

guilty because she feels as though she has failed
her sister.

D.

Walking on Snow

38 Which choice provides the best evidence for the answer
to the previous question?

42 Which of the following best describes the meaning of
“thrilled” in line 45?
A.

Inspired

A.

Lines 1–4 (“It was . . . little sister.”)

B.

Refreshed

B.

Lines 12–13 (“Full of . . . spirit”)

C.

Frightened

C.

Lines 44–45 (“No sooner . . . thrilled me.”)

D.

Stimulated

D.

Lines 59–61 (“As soon . . . away.”)

CONTINUE
Peterson's SAT® Prep Guide 2017

  53

Diagnostic Test
Questions 43–52 are based on the following
passages and supplementary material.
Passage 1: Fanny Wright was a reformer, author, and orator, which
were unusual occupations for a woman in the early nineteenth
century.
Passage 2: Young Robert Emmet was condemned to death for
treason after organizing a rebellion against the English in Ireland.
He, too, had achieved fame as an orator, with speeches decrying
tyranny.

35

40

PASSAGE 1
Fanny Wright to a Fourth-of-July Audience
at New Harmony, Indiana (1828)

Line
5

10

15

20

25

		 In continental Europe, of late years, the words patriotism and patriot have been used in a more enlarged
sense than it is usual here to attribute to them, or than
is attached to them in Great Britain. Since the political
struggles of France, Italy, Spain, and Greece, the word
patriotism has been employed, throughout continental
Europe, to express a love of the public good; a preference
for the interests of the many to those of the few; a desire
for the emancipation of the human race from the thrall of
despotism, religious and civil: in short, patriotism there is
used rather to express the interest felt in the human race
in general than that felt for any country, or inhabitants
of a country, in particular. And patriot, in like manner, is
employed to signify a lover of human liberty and human
improvement rather than a mere lover of the country in
which he lives, or the tribe to which he belongs. Used
in this sense, patriotism is a virtue, and a patriot is a
virtuous man. With such an interpretation, a patriot is a
useful member of society capable of enlarging all minds
and bettering all hearts with which he comes in contact;
a useful member of the human family, capable of establishing fundamental principles and of merging his own
interests, those of his associates, and those of his nation
in the interests of the human race. Laurels and statues
are vain things, and mischievous as they are childish; but
could we imagine them of use, on such a patriot alone
could they be with any reason bestowed. . . .

PASSAGE 2
Robert Emmet to the Court That
Condemned Him to Death (1803)

30

  54

		 I am charged with being an emissary of France An
emissary of France! and for what end? It is alleged that
I wish to sell the independence of my country; and for
what end? . . .

www.petersons.com

45

50

		 No; I am no emissary. . . . Sell my country’s independence to France! and for what? Was it a change of masters?
No, but for ambition. Oh, my country! Was it personal
ambition that could influence me? Had it been the soul
of my actions, could I not, by my education and fortune,
by the rank and consideration of my family, have placed
myself amongst the proudest of your oppressors? My
country was my idol! To it I sacrificed every selfish, every
endearing sentiment; and for it I now offer up myself, O
God! No, my lords; I acted as an Irishman, determined
on delivering my country from the yoke of a foreign
and unrelenting tyranny, and the more galling yoke of
a domestic faction, which is its joint partner. . . . It was
the wish of my heart to extricate my country from this
double riveted despotism—I wished to place her independence beyond the reach of any power on earth. I
wished to exalt her to that proud station in the world.
Connection with France was, indeed, intended, but only
as far as mutual interest would sanction or require.
		 Were the French to assume any authority inconsistent with the purest independence, it would be the
signal for their destruction . . .
		

I wished to prove to France and to the world that

55 Irishmen deserved to be assisted . . . I wished to procure

for my country the guarantee which Washington procured for America—to procure an aid, . . . which would
perceive the good, and polish the rough points of our
character . . . These were my objects; not to receive new
60 taskmasters, hilt to expel old tyrants. And it was for these
ends I sought aid from France . . .
		
Let no man dare, when I am dead, to charge me with
dishonor; let no man attaint my memory by believing
that I could have engaged in any cause but that of my
65 country’s liberty and independence . . . The proclamation
of the provisional government speaks for our views;
no inference can be tortured from it to countenance
barbarity or debasement at home, or subjection,
humiliation, or treachery from abroad. I would not have
70 submitted to a foreign oppressor, for the same reason
that I would resist the foreign and domestic oppressor.
In the dignity of freedom, I would have fought upon the
threshold of my country, and its enemy should enter only
by passing over my lifeless corpse. And am I, who lived
75 but for my country, and who have subjected myself to
the dangers of the jealous and watchful oppressor, and
the bondage of the grave, only to give my countrymen
their rights, and my country its independence . . . —no,
God forbid!

Diagnostic Test
43 Which of the following statements from Emmet’s speech
shows that he thinks he is a martyr?

45 Based on the timeline, the historical events of which year
most likely influenced the American concept of patriotism?

A.

Lines 36–38 (“could I not . . . your oppressors”)

A.

1707

B.

Lines 46–47 (“I wished to . . . on earth”)

B.

1776

C.

Lines 54–55 (“I wished to . . . to be assisted”)

C.

1803

D.

Lines 62–63 (“Let no man . . . with dishonor”)

D.

1828

44 Which of the following of Emmet’s statements shows that
he thinks he is a patriot?
A.

Line 32 (“No; . . . emissary”)

B.

Lines 34–35 (“Was it . . . influence me?”)

C.

Lines 54–55 (“I wished . . . assisted”)

D.

Lines 65–66 (“The proclamation . . . for our views”)

Timeline
1707

Acts of Union between Scotland
and England create the Kingdom
of Great Britain

1776–1783

American colonies declare and win
independence

1789

French storm the Bastille (prison),
fight to end French monarchy

1798

Society of United Irishmen rebel
unsuccessfully against British rule

1800

British Parliament passes The Act of
Union, abolishing the Irish parliament

1801

United Kingdom of Great Britain and
Ireland created

1803

United States purchases Louisiana
Territory from France
Robert Emmet leads a rebellion in
Dublin against the union

1803–1815

Napoleonic Wars in Europe
(France vs. European powers)

1808–1833

Spanish wars of independence

1823

France invades Spain to help restore
monarchy

1828

Andrew Jackson elected president of
United States

46 Based on information shown in the timeline, why might
France have turned down Emmet’s request for help?
A.

France was in the midst of trying to restore the
monarchy in Spain.

B.

France was engaged in the drawn-out Napoleonic
Wars.

C.

France had fought its own revolution and didn’t
want to get involved in that of another country.

D.

France was trying to keep the United States from
taking Louisiana.

47 Emmet would not fit Wright’s definition of a patriot
because he
A.

saw no dishonor in his actions.

B.

wanted to free his people.

C.

idolized his own country above all others.

D.

declared the court’s sentence to be unjust.

48 Which one of the following statements is true?
A.

Although years apart, both Wright and Emmet were
advocating to rethink their country’s ideas about
patriotism.

B.

Emmet was focused on freedom and independence
for his own country, while Wright was focused on
freedom and independence for all humankind.

C.

Emmet loved his country more than Wright loved
her country.

D.

Wright didn’t understand tyranny because she lived
in a democracy, while Emmet fought against
tyranny.

CONTINUE
Peterson's SAT® Prep Guide 2017

  55

Diagnostic Test
49 How does the tone of Wright’s speech compare with that
of Emmet’s speech?

51 Which of the following fits Wright’s definition of a patriot?
A.

A person willing to die for their country

A.

Both express anger, although in response to
different causes.

B.

A person who fights for improving the lives of
others

B.

Wright’s expresses a calm plea while Emmet
expresses desperation and anger.

C.

A person who enlists in the armed forces of his
country

C.

Emmet speaks calmly, and Wright speaks
passionately.

D.

A person who loves his country

D.

Both use a tone that prevents their specific
positions from being convincing.

52 When the British government claims that Emmet is an
“emissary” (line 29) of France, they are accusing him of
being

50 Which of the following statements is most analogous to
Emmet’s statement: “In the dignity of freedom, I would
have fought upon the threshold of my country, and its enemy should enter only by passing over my lifeless corpse.”
(lines 72–74)
A.

Never yield to force; never yield to the apparently
overwhelming might of the enemy.

B.

With the enemy at their back, with our bayonets at
their breasts, in the day of their distress, perhaps
the Americans would have submitted . . .

C.

Give me liberty or give me death.

D.

It is sweet and fitting to die for one’s country.

A.

an ambassador.

B.

a spy.

C.

a minister.

D.

a mercenary.

STOP
If you finish before time is called, you may check your work on this section only.
Do not turn to any other section.

  56

www.petersons.com

Diagnostic Test
SECTION 2: WRITING AND LANGUAGE TEST
35 Minutes—44 Questions
TURN TO SECTION 2 OF YOUR ANSWER SHEET TO ANSWER THE QUESTIONS IN THIS SECTION.
DIRECTIONS: Each passage below is accompanied by a number of multiple-choice questions. For some questions, you will
need to consider how the passage might be revised to improve the expression of ideas. Other questions will ask you to
consider how the passage might be edited to correct errors in sentence structure, usage, or punctuation. A passage may
be accompanied by one or more graphics—such as a chart, table, or graph—that you will need to refer to in order to best
answer the question(s).

After reading each passage, select the answer to each question that most effectively improves the quality of writing in
the passage or that makes the passage follow the conventions of Standard Written English. Many questions include a “NO
CHANGE” option. Select that option if you think the best choice is to leave that specific portion of the passage as it is.

Questions 1–11 are based on the following passage.

1

While most American cities must adapt to constant growth, Detroit
is undergoing change as a result of depopulation.

Which choice provides the most logical introduction to
the sentence?
A.

NO CHANGE

B.

Civic growth caused by the depression

C.

The improvement in living conditions

D.

The decrease in pollution

manufacturing, and other factors have wreaked havoc on the

A.

NO CHANGE

once prosperous city, driving away its middle class and 2 it

B.

having left behind vast tracts of urban blight.

left behind vast tracts of urban blight.

C.

to leave behind vast tracts of urban blight.

D.

leaving behind vast tracts of urban blight.

A city of 139 square miles, with a long history of growth and
middle-class success, Detroit now faces an unusual, though
not entirely novel, situation for U.S. cities: depopulation. 1
Economic transformations caused by recessions, the loss of

2

Diagnostic Test — writing and language

Some questions will direct you to an underlined portion of a passage—it could be one word, a portion of a sentence, or the full
sentence itself. Other questions will direct you to a particular paragraph or to certain sentences within a paragraph, or you’ll
be asked to think about the passage as a whole. Each question number refers to the corresponding number in the passage.

CONTINUE
Peterson's SAT® Prep Guide 2017

  57

Diagnostic Test
The statistics 3 are staggering—since 1950, some 60

3

percent of the population has gone elsewhere, leaving the

A.

NO CHANGE

city with 4 20,000 new residents. When the people left,

B.

are staggering since: 1950 some

thousands of businesses went with them.

C.

are staggering since 1950 some

D.

are staggering since; 1950 some

City planners have been responding to the challenge of
depopulation. Over several years, they have studied their

4

urban spaces and used varying and innovative techniques
to 5 confuse the input of some 30,000 of their residents.
Planners have come up with what 6 she calls Detroit
Future City, a vision that takes the long view and is projected
to take some fifty years to implement. Within this plan are
different strands of redevelopment, development, and—most

Which choice provides information that best supports
the claim made by this sentence?
A.

NO CHANGE

B.

100,000 vacant residences or lots

C.

50,000 more middle-class residents

D.

30,000 homeless people

A.

NO CHANGE

B.

belittle

C.

solicit

D.

return

A.

NO CHANGE

B.

they call

C.

he calls

D.

we call

A.

NO CHANGE

B.

In fact,

C.

Nevertheless,

D.

Besides,

5

dramatically—un-development. 7 Similarly, the strategic
plan includes a concept not often seen in U.S. city planning:
downsizing, or what some prefer to call “right sizing.”

6

7

  58

www.petersons.com

Diagnostic Test
[1] One of the boldest suggestions of the plan is a basic

8

conversion of about one third of all Detroit’s urban space. [2]
Making the city more compact, the planners 8

reasoned,

A.

NO CHANGE

B.

insisted

would save money on services and allow them to devote

C.

noted

more resources to a smaller total area. [3] Walking paths,

D.

commented

parks, ponds for rainwater collection and retention (the city’s
sewage system is overburdened), sports fields, meadows,

9

forested areas, campgrounds, and other green space

To improve the flow of this paragraph, sentence 4 should
be placed
A.

where it is now.

B.

before sentence 1.

C.

after sentence 5.

D.

after sentence 6.

for strengthening. [6] The plan also calls for remaining

A.

NO CHANGE

neighborhoods to be 10 transformed but—not by the

B.

transformed—but not by the

traditional models of economic growth. [7] For example, the

C.

transformed but not—by the

city, if organized carefully with viable public transportation

D.

transformed, but not by—the

A.

NO CHANGE

B.

has; namely, healthcare and education; government
and transportation, local businesses

C.

has; namely, healthcare; education; government;
transportation; and local businesses

D.

has; namely, healthcare, education, government,
transportation; and local businesses

initiatives would then gradually transform the shutdown area.
9 [4] The plan contained some creative and bold
suggestions. [5] Controversially, the plan suggests shutting
down services in certain areas to drive current residents
out of them and into neighborhoods being targeted

options, hopes to create jobs right where people live. In
part, the plan is predicated on the idea that within their
own various redevelopment areas, or “natural economic
zones,” people can both live and work in fields that every
city 11 has, namely, healthcare, education, government,

10

11

transportation, and local businesses that meet core needs,
such as grocery stores and eating places. The plan is also
predicated on the idea that the well-planned urban space
generates its own economic success, as well as on the idea
that such areas will eventually draw some outside business
and industry. Debt-ridden Detroit is definitely going to
need the latter. A recent NPR report on Detroit posited that
commercial real estate taxes can make up a substantial 70
percent of the revenue for a city.

CONTINUE
Peterson's SAT® Prep Guide 2017

  59

Diagnostic Test
Questions 12–22 are based on the following passage.

12

In a public square on the Indonesian island of Java, dusk falls.
Families gather; it is a festival day. Children dart around while,
on the edges of the square, vendors 12 hawk snacks and

A.

NO CHANGE

B.

stock

C.

advertise

D.

trade

toys. A large screen, lit from behind, stands prominently in
the square. A twenty-piece percussion orchestra, or gamelan,
prepares to play.

13 At this point, the author is considering deleting the underlined sentence. Should the writer do this?

13 The scene is traditional Java, hundreds of years ago.

A.

Yes, because it inserts an irrelevant opinion.

B.

Yes, because it distracts from the main ideas of the
paragraph.

C.

No, because it provides a transition from the
previous paragraph to this one.

D.

No, because it explains what wayang kulit is.

A.

NO CHANGE

B.

traditions and its origins stretch

C.

traditions, its origins stretch

D.

traditions. Its origins stretch

A.

NO CHANGE

The performance is wayang kulit, or shadow puppetry, one
of the world’s oldest storytelling 14 traditions its origins
stretch back to the ancient spiritual practices of Indonesia’s
original inhabitants, who believed that the spirits of the
ancestors governed the living world. Ceremonial puppet plays

14

addressed the spirits, asking them to help the living.
Over two thousand years ago, islands such as Java, Bali, and
Sumatra saw their first 15 Indian migrants, a nation to which
Indonesia was linked through trade relations. In the centuries
that followed, Indian culture influenced every aspect of
Indonesian life.

15

B.

migrants from India, a nation to which Indonesia

The puppet plays reflected these cultural changes. 16 They

C.

Indian migrants, to which a nation Indonesia

began to depict narratives from Hindu religious texts,

D.

Indian migrants, a nation of people to which
Indonesia

including the Mahabharata, the Ramayana, and the Serat
Menak. Traditional Indonesian stories were blended into Hindu
epics or lost altogether. Later, when Islam began to spread

16 At this point, the writer is considering adding the following sentence.

throughout Indonesia, puppet plays again transformed.

A master of shadow puppetry is called a dalang.
Should the writer make this addition here?

  60

www.petersons.com

A.

Yes, because it provides relevant and new
information about the practice of wayang kulit.

B.

Yes, because it adds an important fact to the
paragraph’s focus on shadow puppetry.

C.

No, because it repeats information that has already
been given.

D.

No, because it distracts from the paragraph’s focus
on cultural changes.

Diagnostic Test
The Islamic religion 17 prohibited the display of gods in

17

human form, so Indonesians adapted their art by making flat,

A.

NO CHANGE

leather puppets that cast shadows on a screen. The puppets

B.

discouraged

18 themselves remain unseen during the performances;

C.

hindered

D.

restricted

shadow puppet tradition. 19 Performances are epic events,

A.

NO CHANGE

lasting all night long from sunset to sunrise with no break at

B.

themselves will remain unseen during the
performances

C.

themselves remained unseen during the
performances

D.

themselves had been remaining unseen during the
performances

A.

NO CHANGE

B.

Performances are epic events, lasting from sunset to
sunrise with no break.

C.

Performances are epic events, lasting all night long
from sunset to sunrise without taking a break.

D.

Performances are epic events, lasting all night.

A.

NO CHANGE

B.

Simultaneously he directs, the gamelan the large,
percussive orchestra which consists of percussive,
instruments some of which are mallets.

C.

Simultaneously, he directs the gamelan—the large
percussive orchestra, which consists of percussive
instruments, some of which are mallets.

D.

Simultaneously, he directs, the gamelan, the large,
percussive orchestra, which consists, of percussive
instruments, some of which, are mallets.

only their shadows were visible. Wayang kulit was born.
Java is particularly well-known for its continuation of the

18

all. They take place in public spaces and are performed on
holidays and at family celebrations. At the center is a large
screen, backlit by a gas or electrical light. Behind this screen
sits the dalang, or shadow master, traditionally a man. He
manipulates the puppets—sometimes more than a hundred
of them in one show—with rods, voicing and singing all of the

19

roles. 20 Simultaneously he directs the gamelan the large
percussive orchestra which consists of percussive instruments
some of which are mallets.

20

CONTINUE
Peterson's SAT® Prep Guide 2017

  61

Diagnostic Test
Each puppet is carefully crafted, a flat figure that is perforated

21

to project a detailed shadow. Artists begin creating a puppet

A.

NO CHANGE

by tracing the outline of a paper model on leather. The leather

B.

to manipulate his parts

figure is painstakingly smoothed and treated before being

C.

to manipulate its parts

passed onto another craftsperson, who paints it. Then, the

D.

to manipulate her parts

These puppets follow an established set of conventions: evil

A.

NO CHANGE

characters have grotesque faces, while noble ones have more

B.

that

C.

than

D.

this

puppet’s moving parts—the arms and hands—are added,
along with the sticks used 21 to manipulate their parts.

22

refined features. They are highly stylized caricatures, rather
22 then realistic figures.

Questions 23–33 are based on the following passage
and supplementary material.
Water issues are hardly unknown to the American Southwest, but
they have recently taken on a new urgency.
23 The arid climate and limited water resources of the
American Southwest 24 has always influenced the peoples

23 Which of the following sentences would make the most
effective introductory sentence to this passage?

of the region. The Anasazi, ancient people of some of the
most inhospitable areas of the Southwest, made a series of
accommodations to 25 they’re hot, arid environment by

A.

Consider a vacation to the American Southwest!

B.

What do you know about the majestic American
Southwest?

C.

The Anasazi are the original people who inhabited
the American Southwest.

D.

There’s a serious problem occurring in the American
Southwest.

A.

NO CHANGE

B.

had always influence

C.

have always influenced

D.

is always influenced

A.

NO CHANGE

B.

their hot, arid environment

C.

there hot, arid environment

D.

its hot, arid environment

means of adaptive agricultural practices, cliff-side residences,
and elaborate catchment systems.

24

25

  62

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Diagnostic Test
Today, the American Southwest, simplistically defined in this
document as encompassing all of Utah, Nevada, New Mexico,

26 Which choice most effectively combines the sentences at
the underlined portion?

Arizona, and California, is the country’s fastest-growing
26 region. It is home to more than 50 million people who
are the source of ever-increasing water demands. Yet, the
region is dependent for its water on just two river systems,
the Colorado and the Rio Grande, of which the former is

region, but it is home

B.

region; home

C.

region it is home

D.

region, home

A.

NO CHANGE

B.

California. However,

C.

California? However,

D.

California, however,

A.

NO CHANGE

27

unequivocally the primary.
The Colorado supplies water to some 38 million users and
irrigates some 300 million acres of farmland, much of it in
27 California! However, the mighty Colorado’s flow was
apportioned almost one hundred years ago to include not just

A.

28

the southwestern United States but also Mexico. It was also
apportioned according to a volume that simply does not exist

B.

hoped

in current years; for example, in the years 2001–2006, river

C.

desired

water that had been 28 projected to flow versus river water

D.

thought

that did flow came up a staggering 34 percent short.
In 2014, the U.S. Department of the Interior warned that the

29 At this point, the writer is considering deleting this
sentence. Should the writer do this?

Colorado River basin area “is in the midst of a fourteen-year

A.

Yes, because it repeats information that has already
been presented in the passage.

B.

Yes, because it blurs the paragraph’s focus by
introducing a new idea.

C.

No, because it illustrates the severity of drought
conditions with a specific example.

D.

No, because it introduces the argument that the
Southwest’s water supply is drying up.

A.

NO CHANGE

B.

Equally dire, and possibly more alarming,

C.

Equally dire and possibly more alarming

D.

Equally dire and, possibly more alarming,

A.

NO CHANGE

B.

For all intentional purposes,

C.

For all intents and purposes,

D.

For all intended purposes,

drought nearly unrivaled in 1,250 years.” 29 It further
noted that the river’s two major reservoirs, Lake Powell and
Lake Mead—the once-massive backup systems for years
in which drought occurs—were, alarmingly, more than 50
percent depleted. 30 Equally dire, and possibly, more
alarming, predictions came out of a recent study, cited in
the Proceedings of the National Academy of Science of the

30

United States, that suggested a 50 percent chance of Lakes
Powell and Mead reaching a level so low that they become
inoperable by the 2020s. 31 For all intensive purposes, the
Southwest’s water supply is drying up.
31

CONTINUE
Peterson's SAT® Prep Guide 2017

  63

Diagnostic Test
32 Compounding the problems of drought, increasing

32

population, and an overly optimistic historical assessment

A.

NO CHANGE

of water resources are problems related to climate change.

B.

Escalating

For example, between 2000 and 2014, the 33 highest air

C.

Inflating

temperatures in much of the Southwest rose as much as 2

D.

Exaggerating

degrees, increasing the negative effects of evapotranspiration,
the evaporation of water from the soil. Finally, climate change
and drought are leading to the greater prevalence and
intensity of fires, including so-called “super fires,” a result,
in part, of the beetle infestations and dying trees that are
weakened by the lack of water.

33 Which choice makes appropriate and effective use of the
data in the accompanying map?
A.

NO CHANGE

B.

lowest air temperatures

C.

hottest water temperature

D.

average air temperatures

Average Temperatures in the Southwestern United States 2000–2014 Versus Long-Term Average

This map shows how the average air temperature from 2000 to 2014 has differed from the long-term average (1895–2014). To provide
more detailed information, each state has been divided into climate divisions, which are zones that share similar climate features.

  64

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Diagnostic Test
Questions 34–44 are based on the following passage.
Women in Film: Troubling Inequalities

34 Which choice most effectively establishes the main topic of
the paragraph?
A.

There are many actresses in Hollywood with
extraordinary talent, but they cannot seem to get the
same roles as men.

B.

Though women land far fewer leading roles than
men, in other categories of filmmaking, they do a little
better.

C.

Women are not adequately represented in Hollywood,
either by the roles they play or by the amount of time
they appear on-screen.

D.

The movie industry needs to pay female actresses
more than their male counterparts, in an effort to
attract new and extraordinary talent.

A.

NO CHANGE

B.

agents

C.

necessities

D.

relationships

A.

NO CHANGE

B.

Women simply can’t expect to play the leading roles
men play or even, in general, while there seems to be
no end of extraordinary acting talent among women
in Hollywood, to be on-screen for as many minutes as
men are in any given film.

C.

Women simply can’t expect to play the leading roles
men play or to be on-screen for as many minutes as
men, and there seems to be no end of extraordinary
acting talent among women in Hollywood in general.

D.

While there seems to be no end of extraordinary
acting talent among women in Hollywood, women
simply can’t expect to play the leading roles men
play or even, in general, to be on-screen for as many
minutes as men are in any given film.

34 In a society in which television and movies have been
well documented as 35 roles of social change, current
data about women in the movies are far from reassuring.
36 Women simply can’t expect to play the leading roles men
play or even, in general, to be on-screen for as many minutes
as men are in any given film, while there seems to be no end
of extraordinary acting talent among women in Hollywood. As
for other categories of filmmaking, at least by Oscar standards,
women seem barely to exist at all.
35

36

CONTINUE
Peterson's SAT® Prep Guide 2017

  65

Diagnostic Test
Indeed, 37 women were the protagonists in only 15

37

percent of the top grossing films of 2013, according to a study

A.

NO CHANGE

conducted at San Diego State University. Other study findings

B.

women were the protagonist

included the fact that when women are on-screen, 38 their

C.

a woman was the protagonists

marriage status is more identifiable than men. Also, males over

D.

the protagonists were a woman

A.

NO CHANGE

B.

their marriage status is more identifiable than of a
man.

C.

their marriage status is more identifiable than that of
men.

D.

their marriage status is more identifiable than men’s
marriage.

age 40 are much more commonly represented on-screen than
women in the same age group.

38

Other inequities have been revealed by Cinemetrics, which
strives to gather objective data on movies, and by other
organizations. 39 For example, in 2013, lead actresses in
full-length films spent 57 minutes on-screen, while lead actors
spent 85 minutes on-screen. Compounding the inequity is
the tendency of the camera to stay on a female actress longer
in a single shot, or stare at 40 them passively, while the

39 Which choice most effectively maintains support for claims
or points in the text?

camera moves more actively when it shows a male character.
In other aspects of films, women 41 are treated even more
outrageously. Since the Oscars began in 1928, only 16 percent
of all nominees have been women. In fact, there were no

A.

NO CHANGE

B.

For example, women direct more documentaries
than narrative films.

C.

For example, the highest paid actress in 2013 made
$33 million dollars.

D.

For example, women buy about half of movie tickets
purchased in the United States.

A.

NO CHANGE

B.

him

C.

her

D.

us

A.

NO CHANGE

B.

are taken advantage of.

C.

are cheated.

D.

fare even worse.

women nominees at all in seven categories of achievement for
the 2014 Oscars. More significantly, Oscar trends do not seem
to be improving over time.
40

41

  66

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Diagnostic Test
Some women, however, have managed to shine despite

42

these inequities. Actress Meryl Streep has been nominated

A.

NO CHANGE

for 19 Oscars as of 2015, easily surpassing both male and

B.

famous

female competitors for the record of most Academy Award

C.

forgotten

nominations. She is 42 confused for her strong, authoritative

D.

lambasted

(2014). Streep has received 43 accolades for such parts, as 15

A.

NO CHANGE

of her 19 Academy Award nominations were in the category of

B.

privileges

C.

recognition

D.

attention

A.

NO CHANGE

B.

character love life

C.

character’s love life

D.

character loves life

roles; she portrayed a powerful—if terrifying—boss in The
Devil Wears Prada (2006) and a formidable leader in The Giver

43

Best Actress in a Leading Role. Even Streep, however, is subject
to the inequities of the film industry: in The Devil Wears Prada
her 44 characters love life was brought to the forefront and

44

depicted as a sacrifice that she, as a woman in power, had to
continually make for the good of her career.

STOP
If you finish before time is called, you may check your work on this section only.
Do not turn to any other section.

Peterson's SAT® Prep Guide 2017

  67

Diagnostic Test
SECTION 3: MATH TEST—NO CALCULATOR
25 Minutes—20 Questions
TURN TO SECTION 3 OF YOUR ANSWER SHEET TO ANSWER THE QUESTIONS IN THIS SECTION.
DIRECTIONS: For Questions 1–15, solve each problem, select the best answer from the choices provided, and fill in the
corresponding circle on your answer sheet. For Questions 16–20, solve the problem and enter your answer in the grid on
the answer sheet. The directions before Question 16 will provide information on how to enter your answers in the grid.

Circle:

Rectangle:

r
C = 2�r
A = �r2

l
A = lw

r

c

a
b

x 2

x

b

1
A = bh
2

Cylinder:

w
V = lwh

h

w

Rectangular
Solid:

l

Triangle:

x

x

Special Right Triangles

a + b2 = c2
2

Sphere:

2x

x 3

Cone:

Rectangular-Based
Pyramid:

h
r

V = �r2h

h

h
r

4
V = �r3
3

The number of degrees of arc in a circle is 360.
The number of radians in the arc of a circle is 2 .
The sum of the measures in degrees of the angles of a triangle is 180.

1
V = �r2h
3

l

w
1
3

V = lwh

Diagnostic Test — math — no calculator

ADDITIONAL INFORMATION:
1.  	 The use of a calculator in this section is not permitted.
2.  	 All variables and expressions used represent real numbers unless otherwise indicated.
3.  	 Figures provided in this test are drawn to scale unless otherwise indicated.
4.  	 All figures lie in a plane unless otherwise indicated.
5.  	 Unless otherwise specified, the domain of a given function f is the set of all real numbers x for which f(x) is a real number.

CONTINUE
Peterson's SAT® Prep Guide 2017

  69

Diagnostic Test
SHOW YOUR WORK HERE

1
6

4

2

–2

0

2

4

6

–2

If the line drawn above is translated 3 units to the left and
6 units down, what is the slope of the new line?

2

  70

A.

−

1
9

B.

−

4
3

C.

−

7
10

D.

–2

Catherine is performing a science experiment on the distance traveled by a snail. She sets the snail on her driveway
and records the time it takes the snail to crawl to the end
of her driveway. She uses the equation D = 0.4t + 12, where
D is the total distance traveled in feet, and t represents the
time in minutes. Which of the following statements best
interprets the meaning of 12 in Catherine’s equation?
A.

It would take the snail 12 minutes to reach the end
of the driveway.

B.

The snail began 12 feet from the beginning of the
driveway.

C.

The snail traveled a total distance of 12 feet.

D.

The snail traveled at a rate of 12 feet per minute.

www.petersons.com

Diagnostic Test
3

If –2x + 5 = 2 – (5 – 2x), what is the value of x?
A.

–2

B.

2

C.

3

D.

5

4

SHOW YOUR WORK HERE

15"
6"

6"

15"
T

The figure shows one square inside another and a rectangle
with diagonal T. The best approximation of the value of T,
in inches, is given by which of the following inequalities?

5

A.

8 40

C.

8 6
+ ≥ 40
m n

D.

8 6
+ > 40
m n

SHOW YOUR WORK HERE

Amy is renting a moving van that charges $19.99 per day,
plus an additional $0.15 per mile. A tax of 7.5% is applied
to both the daily rate and the mileage rate. Which of the
following represents the total charge, y, that Amy will pay
to rent the van for one day and drive it miles?
A.

y = 19.99 + 0.075x + 0.15

B.

y = 1.075(19.99) + 0.15x

C.

y = 1.075(19.99 + 0.15x)

D.

y = 1.075(19.99 + 0.15)x

If nails are bought at 35 cents per dozen and sold at 3 for
1
10 cents, the total profit on 5 dozen is
2
A.

25 cents.

B.

27

C.

1
31 cents.
2

D.

35 cents.

1
cents.
2

A cubic foot of concrete weighs approximately 150 pounds.
How many pounds will a similar block of concrete weigh
if the edges are twice as long?
A.

300 pounds

B.

450 pounds

C.

800 pounds

D.

1,200 pounds

CONTINUE
Peterson's SAT® Prep Guide 2017

  79

Diagnostic Test
9

Which of the following expressions is equivalent to
–2(1 – x)2 + 2(1 – x2)?
A.

–2x

B.

–4x2 + 4x

C.

–4x2 – 4x – 4

D.

0

10 An organization is giving away t-shirts for its 5-kilometer
road race. The cost to produce the t-shirts is defined by the
equation C(x) = 7x + 60, where x is the number of t-shirts
produced. The organization gives away the t-shirts for free
to people who sign up for the race more than one month
in advance and pay the $20 sign-up fee. What is the fewest number of people who must sign up in order for the
organization to profit if the only cost is manufacturing the
t-shirts and the only income is the sign-up fee?
A.

3

B.

5

C.

13

D.

20

11 A small college, which has a population of 2,180 students,
recently held a fundraiser in which each male student
raised $20, and each female student raised $25. Together,
they raised a total of $50,000. If x represents the number of
male students in the college and y represents the number
of female students in the college, which system of equations can be used to represent the scenario?

A.

x + y = 50 , 000
20 x + 25 y = 2,180

B.

x + y = 2,180
20 x + 25 y = 50 , 000

  80

C.

x + y = 2,180
25 x + 20 y = 50 , 000

D.

x + y = 50 , 000
25 x + 20 y = 50 , 000

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SHOW YOUR WORK HERE

Diagnostic Test
12

Which of the following is an expression equivalent to
9 x 3 y 5z 6 ?

SHOW YOUR WORK HERE

3

A.

3y2z3

B.

3xy2z3

C.

9 3 xy 2 z 3

D.

9 3 xy 3 z 2

1

1

5

13
4

2

–4

–2

0

2

4

–2

–4

y = x +1
y = − x2 +1
A system of equations and their graphs are shown above.
Which of the following are solutions to the system?
I. (0, 1)
II. (1, 0)
III. (–1, 0)
IV. (0, –1)

14

A.

I only

B.

II only

C.

I and III only

D.

II and IV only

If

x y
− = 5 , what is the value of 8x – 6y?
3 4

A.

–120

B.

–60

C.

60

D.

120

CONTINUE
Peterson's SAT® Prep Guide 2017

  81

Diagnostic Test
15 At a restaurant, the rates for meals are $7.50 for a lunch
and $12.00 for a dinner. One weekend, the restaurant sold
a total of 241 meals for $2,523.00. Which of the following
systems of equations can be used to determine the number of lunches, x, and the number of dinners, y, that the
restaurant sold?

A.

7.5 x + 12 y = 241
x + y = 2, 523

B.

12 x + 7.5 y = 241
x + y = 2, 523

C.

7.5 x + 12 y = 2, 523
x + y = 241

D.

12 x + 7.5 y = 2, 523
x + y = 241

16
United States Population

Number of people (in millions)

400M

300M

200M

100M

0

1920

1940

1960

1980

2000

Years

The graph shows the relationship between the population
of the United States (in millions) and the year the population
was recorded. Which of the following statements is true
about the data shown on the graph?

  82

A.

There is a weak correlation between the variables.

B.

There is a strong correlation between the variables.

C.

There is no clear correlation between the variables.

D.

There is an exponential correlation between the
variables.

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SHOW YOUR WORK HERE

Diagnostic Test
17

SHOW YOUR WORK HERE

GDP of China

10T

D olla r s in t r illions

8T

6T

4T

2T

0

1970

1980

1990

2000

2010

2020

Years

The graph shows data representing the gross domestic product (GDP), in trillions of dollars, of China from
1970 through 2013. Which of the following function
types would best represent the data?

B.

13
( x − 1970 )
8
y = ln (2015 – x)

C.

y = (x – 1970)3 + 2

D.

y = 1970e0.0003x

A.

y=

18 A college graduate goes to work for x dollars per week.
After several months, the company gives all the employees
a 10% pay cut. A few months later, the company gives all
the employees a 10% raise. Which expression is equal to
the college graduate’s weekly salary resulting from these
changes?
A.

0.90x

B.

0.99x

C.

x

D.

1.01x

CONTINUE
Peterson's SAT® Prep Guide 2017

  83

Diagnostic Test
19 Which of the following systems of inequalities has a solution set that intersects the first quadrant of the xy-plane?

A.

 y ≤ −2 x + 4

 y > −2 x + 2

B.

 x ≤ −2

y ≥ 5

C.

y ≥ x

 y < −3 − x

D.

 y > 5 + 3x

 y ≤ −3 + 3 x

20 Suppose two legs of a right triangle are such that the
length of one is 3 units shorter than the other. If the hypotenuse is 4 units long, what is the length of the longer
leg of the triangle?
A.
B.
C.
D.

3.5 units

(3 + 7 ) units

3 + 23 units
2
3 units

21 What is the original price of an item if it costs $12.60 after
a 10% discount is applied to the selling price?

22

A.

$11.34

B.

$12.48

C.

$13.86

D.

$14.00

1
A recipe for a homemade weed killer calls for 1 gallons of
3
white vinegar and 2 cups of table salt. Miguel made a large
batch of the weed killer and used 7 cups of table salt. If he
followed the recipe correctly, how many gallons of white
vinegar did he use?
A.
B.
C.
D.

  84

4
2
3
1
5
3
6
4

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SHOW YOUR WORK HERE

Diagnostic Test
23

A circle has a circumference that is equal to the perimeter
of a hexagon. The sides of the hexagon are each 22 inches
long. Which of the following is closest to the length of the
radius of the circle?
A.

7

B.

14

C.

21

D.

28

24 If (x – 4) and (x + 2) are factors of f(x), which of the following
graphs could represent the function f(x)?
A.

C.

–4

–2

4

4

2

2

0

2

–4

4

–2

0

–2

–2

–4

–4

2

4

2

4

D.

B.

–4

–2

4

4

2

2

0

2

4

–4

–2

0

–2

–2

–4

–4

CONTINUE
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  85

Diagnostic Test
SHOW YOUR WORK HERE

25

None

1 to 3

4 or more

Group A

8

23

19

Group B

14

21

5

Total

22

44

24

The table above shows data from demographic researchers studying the number of living siblings people have. If
a person is chosen at random from Group A, what is the
probability that the person has no living siblings?
A.

4
25

B.

4
11

C.

7
11

D.

22
25

26 During the Apollo 14 mission, astronaut Alan Shepard hit
a golf ball on the moon. The height of the ball in meters is
modeled by the function f(t) = –0.81t2 + 55t + 0.02, where
t is the time in seconds after the ball was hit. What does
0.02 stand for in this equation?

  86

A.

Acceleration of the ball due to gravity

B.

Vertical velocity of the ball

C.

Horizontal velocity of the ball

D.

Height of the ball before it is hit

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Diagnostic Test
27 If k is a positive constant other than 1, which of the following could be the graph of kx + y = c?
A.

C.

–4

–2

4

4

2

2

0

2

–4

4

–2

0

–2

–2

–4

–4

2

4

2

4

D.

B.

–4

–2

4

4

2

2

0

2

4

–4

–2

0

–2

–2

–4

–4

CONTINUE
Peterson's SAT® Prep Guide 2017

  87

Diagnostic Test
28 The Cyber Corporation buys a new machine for $80,000.
If the machine loses 15% of its value each year, what is its
value after 4 years?
A.

$41,760.50

B.

$42,750.50

C.

$48,000.00

D.

$49,130.00

29 The table below shows the total number of medals won
by the United States in the last six Winter Olympics.

Number of
Medals

Year

12

1994

13

1998

31

2002

25

2006

37

2010

28

2014

How many medals will the United States have to win in the
2018 Olympics in order for the average number of medals
for the years 1994 to 2018 to be one more than the average
number of medals won during the years 1994 to 2014?

30

  88

A.

29

B.

31

C.

32

D.

36

A
6x
is written in the form 3 +
,
x
+2
2x + 4
what is the value of A?
If the expression

A.

–12

B.

–6

C.

6

D.

12

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SHOW YOUR WORK HERE

Diagnostic Test
DIRECTIONS: For Questions 31–38, solve the problem and enter your answer in the grid, as described below, on the answer
sheet.
1.  	 Although not required, it is suggested that you write your answer in the boxes at the top of the columns to help you fill in
the circles accurately. You will receive credit only if the circles are filled in correctly.
2.  	 Mark no more than one circle in any column.
3.  	 No question has a negative answer.
4.  	 Some problems may have more than one correct answer. In such cases, grid only one answer.
5.  	 Mixed numbers such as 3
		

If 3

7
1
must be gridded as 3.5 or .
2
2

1
is entered into the grid as
2

, it will be interpreted as

31
1
, not 3 .
2
2

6.  	 Decimal answers: If you obtain a decimal answer with more digits than the grid can accommodate, it may be either rounded
or truncated, but it must fill the entire grid.
7
12

Answer:

Answer: 2.5

Write answer
in boxes.

.

Fraction
line
0

Grid in
result.

0

0

0

0
1

1

2

2

3

3

3

4

4

4

5

5

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6

6

6

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9

9

1

2

Decimal
point

0

Answer: 201
Either position is correct.

0
1

1

0
2

2

0

0

1

1

1

1

2

2

2

2

2

1

3

3

3

3

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4

4

Acceptable ways to grid

2
are:
3

.

1

0

0

0

1

1

1

2

2

2

0

1

.

0

0

0

0

0

0

1

1

1

1

1

1

1

2

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2

2

2

2

2

3

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3

3

3

3

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3

4

4

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5

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3

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8

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9

9

9

9

9

9

9

6

6

CONTINUE
Peterson's SAT® Prep Guide 2017

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Diagnostic Test
31 If (ax – 1)(2x + b) = 4x2 + 4x – 3, what is the value of a + b?

32 Derek has $50 to spend on organic produce at the local
farmer’s market. A pint of berries costs $4.00, a 1-pound
bag of peaches costs $3.75, and a head of lettuce costs
$1.50. If he buys at least two of each item, what is the
maximum number of pints of berries he could purchase?

33 In a 3-hour examination of 350 questions, there are 50
mathematical problems. If twice as much time should be
allowed for each mathematical problem as for each of the
other questions, how many minutes should be spent on
the mathematical problems?

34 In the 1924–25 season of the National Hockey League
(NHL), the Montreal Canadiens won 57% of their games.
During the 1947–48 season, they won 33% of their games.
If there were twice as many games played in the 1947–48
season as in the1924–25 season, what percentage of the
games did the Montreal Canadiens win in these two seasons of the league? (Do not grid the percentage sign.)

35 A polling company surveys 625 randomly selected registered voters to determine whether a proposed ballot
measure might pass. Of those surveyed, 400 voters were in
favor of the ballot measure. The polling company reports
that the poll results have a conservative margin of error
of 4%. If 9,000 people actually vote, what is the minimum
number of people likely to vote for the ballot measure?

  90

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SHOW YOUR WORK HERE

Diagnostic Test
36 The average weight of a medium-sized bottlenose dolphin
is 400 pounds. If a particular medium-sized bottlenose
dolphin weighs 110% of the average, how many pounds
does the dolphin weigh?

37

SHOW YOUR WORK HERE

3 x + y = −4
x + y = 13
If (x, y) is a solution for the system of equations above, what
is the value of y?

38

−3 x + 2 y = −1
6 x − by = 8
What is the value for b that will make the system above
have no solution?

STOP
If you finish before time is called, you may check your work on this section only.
Do not turn to any other section.

Peterson's SAT® Prep Guide 2017

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Diagnostic Test
SECTION 5: ESSAY
50 Minutes—1 Essay

Directions: The essay gives you an opportunity to show how effectively you can read and comprehend
a passage and write an essay analyzing the passage. In your essay, you should demonstrate that you
have read the passage carefully, present a clear and logical analysis, and use language precisely.
Your essay will need to be written on the lines provided in your answer booklet. You will have enough
space if you write on every line and keep your handwriting to an average size. Try to print or write
clearly so that your writing will be legible to the readers scoring your essay.

• Evidence, such as facts, statistics, or examples, to support claims
• Reasoning to develop ideas and to connect claims and evidence
• Stylistic or persuasive elements, such as word choice or appeals to emotion, to add power
to the ideas expressed

Adapted from “Penn State Hack Exposes Theft Risk of Student Personal Data” by Peter Krapp, originally published in The
Conversation on May 20, 2015. Peter Krapp is a professor of film & media studies at University of California, Irvine.
(This passage was edited for length.)
1

Pennsylvania State University’s College of Engineering took its computer network offline on May 15 after disclosing two
cyberattacks. The perpetrators were able to access information on 18,000 students, who are being contacted this week with
the news that their personal identifying information is in hackers’ hands.

2

Three days later, the computer network is back online, with new protections for its users. One of the two attacks is ascribed
by a forensic cybersecurity corporation retained by Penn State to computers apparently based in China.

3

As a researcher who has published on hacking and hacktivism and serves on the board of the UC Irvine data science
initiative, I believe two aspects of this news story deserve particular attention.
Compromising student data

4

Penn State announced last week that the FBI alerted it on November 21, 2014, about an attack with custom malware that
started as early as September 2012.

5

Why did it take so long for Penn State to disclose the breach, despite the fact that the experience of large-scale hacks
in 2013 and 2014 (against Target, Home Depot, and others) clearly demonstrated an urgent need for quick and full
disclosure—both to help the victims and to preserve a modicum of trust?

6

Penn State stated only that any disclosure would have tipped off the perpetrators before their access to the College of
Engineering computers could be cut off. Meanwhile, student data may have been compromised for at least six months,
maybe longer.

7

Another conspicuous problem with public discussion of events like this is, in fact, the lack of distinction often made in the
media between actual appropriation of data (as at Penn State) and mere temporary disabling or defacement of websites
(as happened to Rutgers University last month). That is like being unable to make a difference between a grand theft auto
and keying a car.

8

The question is, what can universities do to limit the risk to their students?

Diagnostic Test — essay

As you read the passage below, consider how Peter Krapp uses the following:

CONTINUE
Peterson's SAT® Prep Guide 2017

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Diagnostic Test
9

The exposure of student data in higher education is not limited to Social Security numbers or email passwords. Information
collected and retained by educational institutions includes full name, address, phone number, credit and debit card
information, workplace information, date of birth, personal interests and of course academic performance and grade
information.

Concern with data practices
Data Storage and Security
Very much

75%

Transparency About Data Use
Very much

80%

Legal Standards & Oversight
Very much

Somewhat

Somewhat

Somewhat

Only a little

Only a little

Only a little

Not at all

Not at all

Not at all

No response

No response

No response

Collection of Location Data
Very much

58%

Collection of Video/Audio Data
Very much

59%

81%

Collection of Telecom Data
Very much

Somewhat

Somewhat

Somewhat

Only a little

Only a little

Only a little

Not at all

Not at all

Not at all

No response

No response

No response

64%

A survey conducted by the Obama administration collected responses from 24,092 individuals on how much they
trusted various institutions to keep their data safe. There was a high level of concern around transparency and legal
standards.

...
10

President Obama only recently called for laws covering data hacking and student privacy. “We’re saying that data collected
on students in the classroom should only be used for educational purposes,” he stated in his speech to the Federal Trade
Commission (FTC) earlier this year.
Data privacy concerns

  94

11

If students’ right to privacy needs to be protected from the specter of foreign intelligence agencies poking around the Penn
State Engineering School, then by the same logic it should be protected also against data-mining by for-profit actors right
here in the US.

12

Until May 2014, Google, for instance, routinely mined its apps for education services for advertising and monetizing
purposes. When Education Week reported that Google was mining student emails, it quickly led not only to lawsuits but also
to landmark legislation. The California Senate Bill 1177 was enacted to prevent educational services from selling student
information or mining it for advertising purposes.

13

Yet, almost a year later, students in California remain just as concerned about their data privacy as before—since the new
state law was watered down to apply only to K–12 and not to higher education. And when it was disclosed earlier this spring
that education publisher Pearson secretly monitored social media to discern references to their content, the legislative
response was one that, according to the Electronic Privacy Information Center (EPIC) in Washington, DC, “fails to uphold
President Obama’s promise that the data collected in an educational context can be used only for educational purposes.”

14

Students in higher education nationwide are still in a position where they cannot opt out of the computer services of their
learning institutions, and so they have no expectation of privacy.

15

Despite President Obama’s promises for safeguarding the privacy of consumers and families, and despite the fact that a
number of technology companies concerned with growing consumer distrust recently signed a pledge to safeguard student
privacy, neither Google nor Apple signed on.

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Diagnostic Test
16

The President’s Council of Advisors on Science and Technology (PCAST) was tasked to examine current and likely future
capabilities of key technologies, both those associated with the collection, analysis, and use of big data and those that can
help to preserve privacy, resulting in a direct recommendation to strengthen US research in privacy-related technologies.

17

And overwhelmingly, respondents to a White House survey recently expressed severe reservations about the collection,
storage, and security and use of private information.

18

Maybe it is time for higher education to heed those signals.
Write an essay in which you explain how Peter Krapp builds an argument to persuade his audience that
the use of college students’ personal information for anything other than educational purposes is a
serious violation of privacy and a major breach of computer security. In your essay, analyze how Peter
Krapp uses one or more of the features listed above (or features of your own choice) to strengthen
the logic and persuasiveness of his argument. Be sure that your analysis focuses on the most relevant
features of the passage. Your essay should not explain whether you agree with the writer’s claims, but
rather explain how he builds an argument to persuade his audience.

STOP
If you finish before time is called, you may check your work on this section only.
Do not turn to any other section.

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  95

Answer Keys and Explanations
Section 1: Reading Test

1. C

12. A

23. B

33. C

43. D

2. A

13. A

24. A

34. A

44. D

3. A

14. D

25. B

35. C

45. B

4. C

15. A

26. B

36. B

46. B

5. D

16. D

27. B

37. B

47. C

6. B

17. A

28. A

38. B

48. B

7. D

18. D

29. D

39. A

49. B

8. B

19. C

30. B

40. A

50. C

9. D

20. C

31. A

41. D

51. B

10. B

21. A

32. C

42. C

52. B

11. A

22. B

READING TEST RAW SCORE
(Number of correct answers)

1.

2.

The correct answer is C. Address appears in the author’s
description of the black bear, including qualities that
the bear possesses. So the context here tells us that
address is not being used to mean direction, choice D,
because one can possess a great sense of direction but
not direction. Habitat, choice A, can also be eliminated
because the correct answer choice should refer to a trait
of the black bear, and one’s habitat is not a personal
trait. Anxiety is a personal trait, but it contradicts the
author’s description of the black bear as possessing
great courage, so choice B can be eliminated. Therefore,
the correct answer is choice C, since skill is a positive
personality trait that would be listed among traits such
as strength and courage.
The correct answer is A. Choice A is correct because
Audubon clearly wanted to inform readers about black
bears even though he was mainly known as a studier of
birds. Had this not been his goal, he would not have
written this particular passage, nor would he address
the reader directly with phrases such as, “You are
probably not aware . . . ” Choice B can be eliminated
because its conclusion is too general; Audubon seems
as though he was likely fascinated by the black bear, but
there is no strong evidence that this man, who mostly
studied birds, was fascinated by all mammals. Choice C

is incorrect because it assumes the reader is more
interested in what Audubon knows than the topic
he is actually discussing, shifting the focus from
bears and placing it onto the author, himself.
Choice D is incorrect because Audubon never
compared the black bear to any species of bird in
the passage.
3.

The correct answer is A. In lines 5–6, the author
compares the bear’s migratory behavior to that of
the deer, so even though the deer and the bear are
very dissimilar mammals, they do share some
significant traits. Choice B is incorrect because the
author is not making a direct comparison here even
if it is likely that other mammals eat roots. Choice C
is incorrect because the comparison between the
bear and the hog ended with the discussion of how
both animals wallow in mud, and it is unlikely a hog
would eat its own species, so the author clearly is
not comparing the bear to a hog when stating that
the bear might eat a young pig. Choice D is
incorrect because the author is not making a
comparison between bees and bears in these lines;
a bee would not rob its own tree.

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Answer Keys and Explanations
4.

5.

6.

7.

8.

  98

The correct answer is C. The author explicitly states
that “many of our woodsmen and hunters who have
seen the Bear performing the singular operation just
described, imagine that it does so for the purpose of
leaving behind an indication of its size and power,”
which supports choice C. Choice A represents
Audubon’s claim, which begins with the phrase “It
seems to me,” not that of the huntsmen and woodsmen.
There’s no indication that the tree will be used for
hibernation, choice B, nor that such behavior is a way of
obtaining food, choice D.
The correct answer is D. The author explains how bears
forage for food by using descriptive phrases that create
a placid, or peaceful, image. The author mentions that
bears may eat livestock, but that is not the main focus,
so choices A and B are incorrect. While the lines in
choice C do show that the bear eats a wide range of
foods, the author’s use of descriptive phrases is
evidence that he is creating an image as opposed to
making a scientific list of what the bear eats.
The correct answer is B. The previous question
indicated that the author was trying to create a placid,
or peaceful, effect with his description of how bears
behave in swamps in lines 17–18, and the words “passes
much of its time” and “wallowing” help achieve that
effect. The wording in lines 16–17 creates a sense of
discomfort by using phrases such as “During the
summer heat” and “gloomy,” so choice A is not the best
answer. Focusing on the bear’s hunger, as choice C does,
or how it seizes prey, as choice D does, does not
contribute to the placid atmosphere the author is trying
to achieve.
The correct answer is D. Audubon explains that the
bear sharpens its claws and the wild boar sharpens its
tusks for the same purpose: to prepare themselves for
competition during the mating season, choice D—not
for defense from human predators, choice C, which are
only mentioned very briefly in the passage. A predator
would not be very effective if it went out of its way to
warn potential prey that it was a predator, so choice A
does not make much sense. The author only mentions
that the black bear forages for crayfish and roots and
does not indicate what the wild boar eats, so choice B is
incorrect.
The correct answer is B. The fact that Audubon calls
man the bear’s “most dangerous” enemy shows that he
has some sympathy for hunted bears, making choice B
the correct answer. Nothing indicates that Audubon
hunts, so choice A is incorrect. He never denies that
bears are dangerous nor does he state that bears are

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more dangerous than people, so choices C and D are
also incorrect.
9.

The correct answer is D. By referring to man as “the
most dangerous of its enemies,” Audubon creates an
image of man as almost villainous in his treatment of
hunted bears, which indicates that he has sympathy for
the creatures. Choice A refers to the bear’s migratory
habits in a neutral way, indicating nothing about the
author’s feelings for the bear. Neither does choice B,
which merely refers to a basic instinct of the bear.
Choice C only indicates how the bear protects itself
from its most dangerous enemies; it does not comment
on those enemies in a way that expresses the author’s
sympathy for bears.

10.

The correct answer is B. The author addresses the
reader directly with such phrases as “as you well know”
and “you also know that” when stating information
about bears. This shows that he believes the reader
already has some knowledge about bears and contradicts the assumption in choice A. While it is entirely
possible that some readers may have a fear of bears,
there is no evidence that the author believes this in the
passage, so choice C is not the best answer. Although
the author assumes that the reader knows that bears
are “good climbers,” he never compares their climbing
abilities to that of apes, so choice D is incorrect.

11.

The correct answer is A. The first paragraph of the
letter indicates that Wythe and Jefferson have an
established relationship and Jefferson believes Wythe
had a positive influence on Jefferson’s education, thus
we can infer that Wythe was a teacher. The letter does
not imply any familial relationship or that Wythe and
Jefferson are friends—choices B and C—rather, it shows
a more formal connection. There is no indication of how
Wythe is employed—choice D. While he could be a
public servant of some kind, that conclusion could not
be drawn from the information provided.

12.

The correct answer is A. Jefferson’s overall concern is
that his nephew should receive a solid education,
choice A. This letter has a greater purpose than merely
staying in touch, choice D. His advice about studying
languages, choice C, is part of the overall advice about
education, and morality is also a part of his course of
study, choice B.

13.

The correct answer is A. Jefferson indicates that
learning Italian might actually make it harder to learn
Spanish, which he views as the more useful language.
This contradicts the conclusion in choice C. Jefferson
only briefly mentions French to indicate that it is a

Answer Keys and Explanations
similar language to Italian and Spanish and never
indicates he believes that it is more important to learn
French than Spanish or Italian, so choice B is incorrect.
Although he mentions that Spanish and Italian are
similar, Jefferson never suggests that they are so similar
that it makes sense to speak Italian to Spanish people,
which eliminates choice D.
14.

15.

The correct answer is D. The word “prosecute” in this
context is used to mean “pursue,” so in lines 23–25,
Jefferson is suggesting that Peter lay Italian aside to
learn Spanish instead. Choices A and B introduce the
problem of learning Italian and Spanish at the same
time, but neither choice suggests how Peter should deal
with this problem. Choice C is incorrect because it
merely indicates Jefferson’s own observations about
people who speak Italian, French, and Spanish.
The correct answer is A. Jefferson compares conscience to a physical limb of the body to show that it is
natural and present in all human beings, choice A.
Choices B, C, and D may be true, but Jefferson only
covers the first point: that morality is as natural as an
arm or leg and is given to all “in a stronger or weaker
degree” (lines 43–44).

16.

The correct answer is D. In advising Peter about
language studies, Jefferson tells him to learn Spanish
(lines 26–29). He predicts that it will be valuable in the
future because of “connections with Spain and Spanish
America,” choice D. He dismisses Italian, choice C, as
unworthy of Peter’s attention and doesn’t discuss
America’s future with France, choice B, or England,
choice A.

17.

The correct answer is A. By “lost time,” Jefferson means
“wasted time,” choice A. In his third point, Jefferson tells
Peter that he shouldn’t bother to study moral philosophy because it is something everyone knows
intuitively, stating that morality “is as much a part of his
nature as the sense of hearing, seeing, feeling.” “Lost
time” could refer to the past, choice B, or mean “missing
time,” choice C, but Jefferson is not discussing the past
or a gap in time in this context. He says nothing about
youth either, so choice D does not make much sense.

18.

The correct answer is D. Jefferson claims that he has
never seen anyone who spoke French, Spanish, and
Italian who didn’t get them confused, choice D. All three
languages are derived from Latin, choice A, as Jefferson
notes (“all of them degenerated dialects of the Latin”),
but that is not the reason he gives to avoid the study of
Italian. Jefferson does think it is necessary to learn a
language other than French, choice B, because he

recommends learning Spanish. Jefferson states that it is
useful to be multilingual, as he advocates the study of
Spanish; he only suggests that one should be careful
when choosing which languages to study, so choice C is
incorrect.
19.

The correct answer is C. Jefferson says morality is
innate, therefore it doesn’t need to be taught. In fact, he
argues, teaching it can have a negative effect of leading
“astray” (lines 47–48).

20.

The correct answer is C. The family tree shows that
Peter’s father died when Peter was only 3 years old, and
that his mother had 5 other children to care for, so
choice C is the best conclusion to reach. The fact that
Peter’s father was already dead when Thomas wrote this
letter eliminates choice A, since a comparison cannot be
fairly made between someone currently living and
someone who is dead. Choice B is too speculative and is
not as logical a conclusion as choice C. Choice D
assumes that Peter was responsible for caring for his 5
siblings, and there is no evidence of this in the family
tree or the passage.

21.

The correct answer is A. The letter shows Jefferson to
be very involved in ensuring that his nephew gets the
best education since it refers to a number of important
study topics and how Peter should approach them.
Expressing that he felt “very happy” to receive letters
from Peter also helps create a warm tone. Although
Jefferson clearly cares for his nephew, he does not seem
particularly worried about the boy since there is no
anxious language in this letter, so choice B can be
eliminated. The letter’s warmth also eliminates choices
C and D since an “objective and matter-of-fact” tone
would be contrary to warmth, as would a “distant and
preoccupied” tone.

22.

The correct answer is B. In the first paragraph, the
author provides a number of statistics that support
choice B. However, the author never compares plastic
waste to other kinds of waste, so choice A cannot be
concluded based on information in this particular
passage. The author writes that “World plastics production has experienced almost constant growth for
more than half a century,” which contradicts choice C.
While the entire passage discusses the impact of plastic
waste on the planet, the author never compares that to
the impact other forms of waste have, so choice D is not
the best answer.

23.

The correct answer is B. Lines 4–6 provide specific
statistics about how plastic accounts for 10 percent of
the 1.4 billion tons of trash the world generates each

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  99

Answer Keys and Explanations
year. By any standards, that is a massive amount of
waste. Choice A indicates that plastic production
continues to grow, but, without specific numbers, there
is no way to use this information to conclude that
plastic is responsible for a massive amount of waste.
That the International Maritime Organization has had to
ban dumping of plastic waste indicates it is a big
enough problem to warrant such action, but choice C is
simply not as specific as choice B is, so it is not the best
answer choice. Choice D also refers to a problem but
fails to supply specific numbers.
24.

The correct answer is A. The overall tone is neutral,
choice A. The author provides facts and data using
scientific terminology and explanations. Nowhere does
he attempt to persuade the reader to a particular view,
choice B, inject a personal story, choice C, or offer his
own opinion, choice D, although he does offer differing
viewpoints on possible solutions.

25.

The correct answer is B. In lines 72–74, Halden’s quote
(“We need to design the next generation of plastics to
make them more biodegradable…”) shows that he
believes science can find a way to make plastics
biodegradable, choice B, which he says will eliminate
the problem. The author states that new laws may help
minimize the problem (“New laws … could require
handling plastics more responsibly at the end of their
useful life through recycling, proper disposal, and
extended producer responsibility”), not Halden, so
choice A is incorrect. The author describes how some
sea creatures have ingested plastic, but Halden never
advocates for the invention of edible plastic in this
passage, so choice C is incorrect. Banning all new plastic
production is an extreme and unrealistic solution that
Halden never suggests, so choice D is incorrect.

26.

27.

  100

The correct answer is B. The article includes scientific
data and explanations of the effects of plastics in the
ocean and waterways, offering information to the public,
choice B. The specific data in this passage contradict the
idea that the author is merely theorizing that the
environment could be in danger, and this conclusion is
too general in any event, so choice A is not the best
answer. Choices C and D are too extreme and unrealistic;
it is unlikely a movement will halt all plastic production,
which is not even something the author advocates, and
cleaning up the oceans is too huge a job for readers to
personally take on. Choice B remains the most logical
and realistic answer.
The correct answer is B. Lines 49–51: “Research has
shown that harmful and persistent substances can both
bioaccumulate … and biomagnify …” indicate that

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plastic pollution in the ocean is a problem by specifying
that they are “harmful.” Choice A merely states what
happens when animals eat plastics without suggesting
that this is or is not a problem, so choice A is not the
best answer. Choice C describes how certain problems
associated with plastic pollution are still unknown, so it
fails to support the previous question’s conclusion that
plastic pollution is a problem. Choice D merely sets up
questions to follow without providing evidence that
plastic pollution in the ocean is a problem.
28.

The correct answer is A. Choice A is the only choice
that refers specifically to the food chain as detailed in
the diagram. Choices B and D may have some bearing
on how plastics impact the ocean, but they don’t
explain the disruption to the food chain. Although the
diagram depicts the emptying of wastewater into the
ocean, it makes no statement about the legality of this
practice, so choice C is incorrect.

29.

The correct answer is D. The author explains that a law
was passed in Illinois to prevent infiltration of these tiny
microbeads into the water system (lines 33–37), choice
D. The microbeads were ending up in the Great Lakes
and Chicago’s waterways. None of the other options are
offered as a reason for the law.

30.

The correct answer is B. Knowing the meaning of
absorb, choice C, does not assist in understanding the
meaning of adsorbed. The words share the same suffix,
but it’s the prefix that makes the word less familiar.
Absolve, choice A, has the same prefix as absorb but
does not help in figuring out the meaning of adsorb.
The “sorb” in “sorbet” (a frozen dessert) has nothing to
do with the word “adsorbed,” so choice D is incorrect.
Adhere, choice B, is when something sticks to something
else; so adsorb must mean when one substance sticks to
another.

31.

The correct answer is A. The author indicates that
there are ways humans may be ingesting plastic waste
in lines 57–65. The author had already discussed how
sea creatures ingest such waste in the previous
paragraphs, so choice B is not the best answer. Choice C
had already been discussed earlier, too. Choice D is
incorrect because the author never theorizes about why
people dump plastic waste into the oceans in this
passage.

32.

The correct answer is C. These lines (“And, finally, what
proportion of humans’ exposure to plastic ingredients
and environmental pollutants occurs through
seafood?”) deal with how much plastic waste from the
oceans people might be ingesting by eating seafood.

Answer Keys and Explanations
Choice A has to do with the transference of plastic
pollutants, not the amount of such pollutants people
are ingesting. Choice B deals with the effects of plastic
pollutants. Choice D is a general statement about the
efforts to answer the questions the author raises.
33.

34.

35.

36.

The correct answer is C. As she places the offering, the
narrator wonders whether or not her sister will die, so it
is logical to conclude that she wants to place her
offering correctly to prevent that from happening. No
one at home asks her about the offering once she
returns, so choices A and D can be eliminated. By
worrying about her sister’s potential death, the narrator
indicates that she has more important things on her
mind than whether or not she’ll be trusted to make
offerings again in the future, so choice B is not the best
answer.
The correct answer is A. In line 70, the medicine man
describes seeing a “luminous vapor above the house,”
and clarifies that “It was the spirit of the little child who
is sick” in line 72. Choice B describes objects that were in
his medicine bag, not the appearance of the sister’s
spirit. The sister was a baby, but the medicine man
never describes her as a baby, so choice C is not the best
answer. In line 73, the medicine man says the “spirit is
beyond our reach,” but this describes the spirit’s
location, not its appearance.
The correct answer is C. The family didn’t send for the
medicine man until the medicine woman had given up
hope, which makes choice C the most logical answer.
What the medicine man had been doing before arriving
at the family’s house is never indicated, so there is no
evidence for choice A. Although we are told that the
medicine man had to travel quite a distance, choice B,
this is not offered as a reason why he was not contacted
in the first place. Although the family did have a
medicine woman attempt to cure their child before
calling the medicine man, the narrator never implies
that the healers’ genders were of any significance.
The correct answer is B. These lines show the girl
having trouble analyzing how nature will affect the
delivery of her offering and unsure if she is making the
right decision. Choice A describes the instructions given
by the medicine woman; it doesn’t explain why they
were difficult to follow. Choice C describes the variance
in the terrain where she was searching for the right
spot, but it does not describe any particular problem.
Choice D describes the narrator’s surroundings without
indicating that they were causing her any particular
problem.

37.

The correct answer is B. The narrator has faith that her
offering will save her sister, and the medicine man has
faith that the girl’s spirit still exists. Choice B emphasizes
the theme of faith in this passage. Choice A is not the
best answer because it places the medicine man at the
center of the story when it is mainly about the narrator’s
faith that her actions will save her sister. Choice C
implies that the passage is mostly concerned with
Native American culture when the rituals in the story
are not as important as how the characters feel about
them. Choice D describes one passing incident in the
passage that says nothing about the story’s overall
theme.

38.

The correct answer is B. Choice B most effectively
indicates the strength of the narrator’s faith by
describing how she believes her offering will save her
sister. Choice A merely describes the offering without
indicating the narrator’s faith that it has any power to
save her sister. Choice C describes a moment in the
story in which the narrator’s faith is shaken, so it
contradicts the conclusion in the previous question.
Choice D mistakes the correct answer to the previous
question.

39.

The correct answer is A. The parents seem oblivious to
the natural emotions of a child. Though they give the
narrator responsibility and take care of her physical
needs (lines 48–49), they don’t seem to offer comfort or
consolation. The narrator is probably given more
responsibility than she wants, which makes choice B
incorrect. The parents naturally focus on the child likely
to die, but there is no indication they prefer the younger
child, so choice C is incorrect. The narrator seems very
alone and her parents display little warmth or affection,
making choice D incorrect.

40.

The correct answer is A. Choice A covers the central
idea of the passage best by including its most important
details: the sister’s death and the narrator’s efforts to
save her. Choice B narrowly focuses on the girl following
the instructions of the medicine woman without
indicating why she was following those instructions.
Choices C and D do not capture the concept of the
narrative and the point of view because they both shift
focus from the narrator to the entire family.

41.

The correct answer is D. The narrator feels remorseful
because she feels as if she failed since her sister dies
despite her own efforts to save the baby. The narrator
may feel responsible for her sister’s death, but there is
no evidence that she has any reason to feel as though
she caused her sister’s illness, so choice A is not the best

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Answer Keys and Explanations
answer. The narrator follows the instructions of the
medicine woman meticulously, so choice B is not the
best conclusion. The narrator may have felt burdened
by the amount of responsibility she took on, but her
remorse is more directly related to her sister’s death, so
choice C is not the best answer.
42.

The correct answer is C. The modern meaning of
“thrilled” is usually “excited,” but in line 45, the author is
talking about fear and doubt. Choices A, B, and D all
have positive connotations that fail to capture the
feelings of heightened fear the narrator was experiencing when she began thinking that her offering had
failed to save her sister’s life.

43.

The correct answer is D. In choice D, Emmet acknowledges his death sentence and declares that he is dying
for a cause. Choice C explains why he asked France for
assistance, and choice B explains that he was fighting
for his country’s independence and freedom from
tyranny. In choice A, he defends his actions by saying
that he could have stood by and done nothing, given
his family’s privileged position, but chose instead to
defend liberty.

44.

The correct answer is D. Choice D shows Emmet’s
pride in his country by saying that his government
speaks for him. In choice A, he merely states that he is
not working for France, but this alone does not prove
that he feels pride in his own country. In choice B, he
theorizes about the different influences that could have
caused him to betray his country. Choice C makes a
statement about how he feels the people in his country
deserve assistance from France, but that alone does not
make a strong case for the author’s patriotism.

45.

46.

  102

The correct answer is B. According to the timeline,
“American colonies declare” independence in 1776, and
the fact that the country needed to define itself outside
of the rule of another country probably caused
America’s view of patriotism to focus more on the
country, itself, rather than a greater concern for all
people. Choice A refers to events that did not concern
America. Choices C and D do refer to events that
concerned America, but neither had as significant an
impact on the country and its views as declaring
independence in 1776 did.
The correct answer is B. Noting the date of Emmet’s
speech in his defense of his actions prior to his execution, choice A is not possible. The Louisiana Purchase
by the United States, choice D, would have given France
more resources to help. However, the Napoleonic Wars
were draining the French treasury, choice B, leaving the

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country overburdened and uninterested in becoming
involved elsewhere. The French Revolution had started
in 1789, triggering the rise of Napoleon, but this was
not directly related to its refusal to aid Emmet.
47.

The correct answer is C. Emmet loved his country and
declared his patriotism toward Ireland as his reason for
his actions, choice C. Wright’s view was more inclusive,
defining it as freeing all people, whatever the country.
Emmet’s primary focus was on his own people, choice B,
not all of humankind. He defended his actions, choice A,
but this action is unrelated to Wright’s concept of a
patriot, as is a court that renders an unjust sentence,
choice D.

48.

The correct answer is B. Although both Emmet and
Wright wanted freedom from tyranny, Emmet’s focus
was on Ireland, and Wright had a broader objective of
freedom and independence for all humankind, choice B.
Choice A is incorrect because only Wright states that her
country’s concept of patriotism needs to be rethought.
Wright was focused on advancing an expansive concept
of patriotism that included other countries, which does
not mean that she did not love her country, so choice C
is not the best answer. Choice D is incorrect because
Wright indicates that she does understand tyranny by
advocating for its opposite: freedom and concern for all
people.

49.

The correct answer is B. Wright’s speech reads as a
sincere yet calm plea for understanding and concern for
all people, while Emmet’s feels more motivated by his
desperation to save himself and his anger regarding the
charges of treason against him, which is clear through
his regular use of exclamations throughout his speech.
Wright does not express anger, so choice A is incorrect.
It is Emmet who is passionate and angry and Wright
who is calm, not the other way around, so choice C is
incorrect. Someone can speak calmly yet sincerely, as
Wright does, or desperately and angrily, as Emmet does,
and still be convincing, so choice D is not the best
answer.

50.

The correct answer is C. All the options refer to patriotism, but choice A is mostly about perseverance and the
importance of freedom. Choice B speculates on how
Americans might react if attacked and suggests submission as one possibility. Choice C sets up the either-or
scenario in which there can only be one of two outcomes:
liberty or death. Emmet’s statement is analogous because
there are only two choices: if the enemy wishes to come
into his homeland (symbolized by the word “threshold”),
he will defend it to the death. Choice D expresses too

Answer Keys and Explanations
much pleasure in the act of fighting for and dying for one’s
country while Emmet’s statements show that he is
motivated more by his belief in the righteousness of
freedom than more romantic ideas about dying for one’s
country.
51.

The correct answer is B. Wright says Americans’ idea of
a patriot is someone who loves his/her country, choice
D, but that idea is too narrow. She explains that
Europeans see patriotism as a more expansive concept
that extends to freedom for all humans. This idea, she
explains, includes working toward the best interests of
all human lives, wherever they are, so that they are free
from despotism. This statement suggests that she
herself holds these ideas, choice B, and wants others to
consider them. Physically fighting for one’s country,
choices A and C, is an example of vanity.

52.

The correct answer is B. Emmet speaks of being
accused of secretly selling his country out to France,
which would be the actions of a spy. In this context, an
emissary is a spy, choice B. The government wouldn’t
accuse someone of being an ambassador, an official
position as a representative, choice A. Similarly, a
minister is an official head of a government department,
which eliminates choice C. Choice D is incorrect
because a mercenary is a paid soldier, and there is
nothing that suggests the government thinks Emmet is
a “hired gun.”

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Answer Keys and Explanations
Section 2: Writing and Language Test
1. A

11. A

21. C

31. C

41. D

2. D

12. A

22. C

32. A

42. B

3. A

13. C

23. D

33. D

43. A

4. B

14. D

24. C

34. C

44. C

5. C

15. B

25. B

35. B

6. B

16. D

26. D

36. D

7. B

17. A

27. B

37. A

8. A

18. C

28. A

38. C

9. B

19. B

29. C

39. A

10. B

20. C

30. B

40. C

WRITING TEST RAW SCORE
(Number of correct answers)

1.

The correct answer is A. Choice A correctly sets up the
sentence’s focus on the various causes of depopulation
and their negative impact. Choices B, C, and D are
incorrect because they present potentially positive
developments.

2.

The correct answer is D. Choice D follows the same
parallel construction as the preceding gerund phrase,
“driving away its middle class.” Choices A, B, and C are
incorrect because they do not demonstrate parallel
construction.

3.

4.

  104

The correct answer is A. Choice A is correct because
the dash is being used to indicate the following
explanation of what is staggering. Choice C is incorrect
because it creates a run-on sentence. Choices B and D
are incorrect because they move the punctuation to the
wrong part of the sentence.
The correct answer is B. Choice B correctly maintains
the sentence’s focus on the effects of depopulation.
Choices A and C are incorrect because they suggest a
growth in residency, which contradicts the main
argument of the passage. Choice D is incorrect because
homelessness is unrelated to the main focus of the
passage, which is Detroit’s depopulation.

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5.

The correct answer is C. Choice C, solicit, correctly
establishes the implied intent of the city planners within
the context of the passage—to solicit, or ask for, the
input of city residents while studying their urban
spaces. Choices A, B, and D do not make sense within
the context of the passage, as all of these terms imply
the opposite of the city planner’s intent.

6.

The correct answer is B. The plural pronoun they is
required here to agree with the plural noun planners, so
they call (choice B) is the correct answer. Choices A and
C are incorrect because they incorrectly use singular
pronouns. Choice D is incorrect because “we call” is
first-person plural, not third-person plural, which is what
is required here.

7.

The correct answer is B. Choice B is correct because “in
fact” properly suggests an elaboration of the previous
sentence. Choice A is incorrect because it suggests the
second sentence is an equal example, rather than an
elaboration of the first. Choice C is incorrect because it
implies the author is drawing a conclusion despite a
contradiction. Choice D is also incorrect because it
implies that the second sentence is a second example
supporting a contrast scenario, which is not the case.

Answer Keys and Explanations
8.

The correct answer is A. In choice A, the sentence
explains the planners’ logic. Choice B implies that there
was resistance to their thinking or that their thinking
was absolute, which is incorrect. Choices C and D do not
denote the careful calculation that the rest of the
sentence implies.

15.

The correct answer is B. Choice B is correct because
the phrase, “a nation to which Indonesia was linked
through trade relations” should refer to the country of
India, not “migrants.” Choices A and D are incorrect
because the modifier is misplaced. Choice C is incorrect
because it is awkward and doesn’t make sense.

9.

The correct answer is B. This sentence would best
serve as a transition between the previous and current
paragraphs. The previous paragraph refers to the city
planner’s strategic plan, and the current paragraph
begins with an example of one of the plan’s boldest
suggestions. A sentence that mentions that the plan
contained bold suggestions, choice B, would work well
here. The other choices would place this notion after
providing examples of the plan’s bold suggestions,
rendering it ineffective and unnecessary.

16.

The correct answer is D. Choice D is correct because
this sentence is tangential to the paragraph’s focus on
the cultural influences of Hindu culture on the
Indonesian way of life. Choice A is incorrect because this
information is not relevant. Choice B is incorrect
because this paragraph is not focused on the actual
practice of shadow puppetry. Choice C is incorrect
because this information has not been presented before
this sentence.

17.

The correct answer is A. Choice A reflects the paragraph’s emphasis that depicting gods in human form
was not allowed under any circumstance. Choices B, C,
and D are incorrect because these words do not suggest
complete proscription the way choice A does.

10.

The correct answer is B. Choice B correctly places the
dash so that it separates the two clauses. Choices A, C,
and D are incorrect because they place the dash in the
wrong place.

11.

The correct answer is A. Choice A is correct because
items in a series should be separated by commas.
Choice B is incorrect because it is unnecessarily wordy
and misuses semicolons. Choice C is incorrect because it
misuses semicolons. Choice D is also incorrect because
it incorrectly mixes a semicolon with commas.

18.

The correct answer is C. Choice C is correct because it
places the verb in the past tense. Choice A is incorrect
because the present tense is an inappropriate shift from
the past tense that the rest of the paragraph uses.
Choices B and D are incorrect because the sentence
requires the simple past tense.

12.

The correct answer is A. Choice A is correct because
“hawk” means to call out and sell, which is what vendors
at festivals do. Choice B is incorrect because, while the
vendors may stock merchandise, the author’s intent
here is to describe how they sell that merchandise.
Similarly, choices C and D are incorrect because the
vendors are selling—not advertising or trading—goods.

19.

13.

The correct answer is C. Choice C correctly suggests
that this sentence provides a transition between the
scene depicted in the first paragraph and the ancient
and modern tradition of puppetry, and the writer
should not delete it. Choice A is incorrect because this
sentence is not an opinion. Choice B is incorrect
because this sentence does not distract from but rather
helps in contextualizing the main ideas of the paragraph. Choice D is incorrect because it is the following
sentence that explains what wayang kulit is.

The correct answer is B. Choice B conveys both the
duration of the events and the lack of interruptions
without being redundant or wordy. Choice A is incorrect
because “all night long” is redundant. Choice C is
incorrect because it is redundant and wordy; while it is
constructed better by omitting “at all,” “all night long
from sunset to sunrise” is still redundant. Choice D is
incorrect because just noting that the performance
lasted all night isn’t as informative as noting that, from
dusk to dawn, the performance had no interruptions.

20.

The correct answer is C. As written, the sentence is a
run-on and needs some added punctuation to reduce
confusion. Choice C fixes the run on by adding the
appropriate commas and an em-dash. Choices B and D
add commas in inappropriate places, creating additional confusion.

21.

The correct answer is C. Choice C is correct because
the pronoun “its” should refer back to the antecedent
“the puppet.” Choice A is incorrect because “their” is
plural and “puppet” is not. Choices B and D are incorrect
because “the puppet” is neither male nor female.

14.

The correct answer is D. Choice D correctly uses end
punctuation to correct this run-on sentence. Choices A,
B, and C are incorrect because they do not employ the
correct punctuation to denote two independent
clauses.

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Answer Keys and Explanations
22.

The correct answer is C. Choice C is correct because
comparisons require “than,” not “then Choice A is
incorrect because “then” is an adverb that refers to time.
Choices B and D are incorrect because they don’t make
sense in the sentence.

23.

The correct answer is D. The intent of this passage is to
bring attention to the serious water issues occurring in
the American Southwest, so choice D would be the
most effective introductory sentence. The passage is not
an enticement to vacation in the American Southwest,
so choice A is incorrect. It is also not focused on
pointing out the majestic aspects of the American
Southwest, so choice B is incorrect. The Anasazi people
are a supporting detail about the American Southwest
in the context of this passage, so choice C is also
incorrect.

24.

25.

26.

  106

The correct answer is C. Choice C is correct because
the sentence subject that includes “arid climate and
limited water resources,” is a compound subject, which
means the verb must also be plural. Choice A is
incorrect because “has” is singular. Choice B is incorrect
because it is the past perfect instead of the present
perfect tense. Choice D is incorrect because it is present
tense and changes the meaning of the sentence.
The correct answer is B. Choice B is correct because
the sentence requires a possessive pronoun in order to
refer back to the Anasazi people. Choice A is incorrect
because “they’re” is a contraction, not a possessive.
Choice C is incorrect because “there” is an adverb, not a
possessive. Choice D is incorrect because “its” is singular
possessive, while the antecedent, “people,” is plural.
The correct answer is D. Choice D combines the
sentences in a way that helps emphasize the connection between the two. Choice A is incorrect because
“but” implies that the two ideas are contradictory.
Choice B is incorrect because it misuses a semicolon to
connect an independent clause and what is now a
dependent clause. Choice C is incorrect because it
creates a run-on sentence.

27.

The correct answer is B. Choice B is correct because
the sentence is a statement, and thus it should end in a
period. Choice A is incorrect because the sentence does
not warrant the excitement or surprise that an exclamation point conveys. Choice C is incorrect because the
sentence is not a question. Choice D is incorrect
because it creates a run-on sentence.

28.

The correct answer is A. Choice A is correct because
the context of the sentence suggests that a scientific

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forecast or prediction about water flow has been made.
Choices B and C are incorrect because they reflect
subjective impulses and do not maintain the neutral
and scientific tone of the passage. Choice D is incorrect
because it is vague and fails to suggest a basis in
research.
29.

The correct answer is C. This sentence adds new and
relevant information by providing an example that
shows the reader how severe the drought really is.
Choice A is incorrect because this information has not
been presented previously in the passage. Choice B is
incorrect because this sentence does not blur the
paragraph’s focus; rather, it is relevant to the paragraph’s
main argument. Choice D is incorrect because this
sentence does not introduce this argument; rather, it
supports the argument.

30.

The correct answer is B. Choice B is correct because
the dependent clause “and possibly more alarming”
must be set off by commas. Choice A is incorrect
because it contains one too many commas. Choice C is
incorrect because it contains no commas. Choice D is
incorrect because the comma should be before “and.”

31.

The correct answer is C. Although it may sound
otherwise, the conventional expression is “for all intents
and purposes,” choice C. Choices A, B, and D are
incorrect.

32.

The correct answer is A. Choice A is correct because
the author’s intent is to suggest that there are other
factors that make the problems of drought worse and
more complicated. Choices B, C, and D are incorrect
because each word emphasizes an increase in severity
rather than an increase in the number of factors
contributing to problems in the area.

33.

The correct answer is D. The accompanying map
measures average air temperatures, so choice D is
correct. The map does not measure the highest or
lowest air temperatures, so choices A and B are
incorrect. It also doesn’t measure water temperature, so
choice C is incorrect.

34.

The correct answer is C. Choice C is correct because
this first paragraph establishes that women do not get
as many leading roles as men nor do they spend as
much time on-screen. Choice A is incorrect because it
does not stress the number of roles nor the amount of
on-screen time. Choice B is incorrect because it does not
cite the amount of on-screen time, alluding to it only
vaguely. Choice D is incorrect because the main idea of
the passage is not a plea to pay female actresses more
than their male counterparts to attract new talent.

Answer Keys and Explanations
35.

The correct answer is B. Choice B is correct because
only “agents” expresses the idea that television and
movies play an active role in causing social change.
Choices A, C, and D are incorrect because they don’t
suggest a causal relationship to social change.

40.

The correct answer is C. Choice C is correct because
the pronoun should refer to the female actress using
“her.” Choice A is incorrect because it is plural. Choice B
is incorrect because it is masculine. Choice D is incorrect
because it is plural and first person.

36.

The correct answer is D. Choice D is correct because
the subordinate clause that begins with “while” is tied to
the central premise of the sentence and needs to be at
the beginning. Choice A is incorrect because at the end
of the sentence, the relationship of the clause to the
central idea is lost. Choices B and C are incorrect
because they obscure the relationship of the clause to
the central idea of the sentence.

41.

The correct answer is D. Choice D maintains the
objective and neutral tone of the passage while noting
unfavorable comparisons. Choices A, B, and C are
incorrect because the tone of each choice is emotionally
charged, which is inconsistent with the objective tone
of the passage.

42.

The correct answer is B. Given the context of the
sentence and paragraph, famous is the most appropriate word choice. Streep is offered as an example of a
woman who has achieved great acclaim as an actress,
despite gender inequalities in the film industry. The
previous sentence mentions her many award nominations, so referring to her fame here makes the most
sense. The other answer choices either don’t make
sense given the context, or we are not given enough
information to determine if they are appropriate
choices, so they are incorrect.

43.

The correct answer is A. Choice A is correct because
“accolades” connotes the achievements, honor, and
respect that the passage goes on to describe. Choice B
is incorrect because the awards were not privileges.
Choices C and D are incorrect because they do not
connote the honor and respect that “accolades” does.

44.

The correct answer is C. The word “characters” should
be the possessive “character’s,” so choice C is correct.
Choice A is incorrect because “characters” is plural
instead of possessive. Choice B is incorrect because
“character” is not possessive. Choice D is incorrect
because it does not convey the appropriate meaning.

37.

The correct answer is A. Choice A is correct because
the noun “women” should agree in number with
“protagonists.” Choice B is incorrect because “protagonists” should be plural. Choice C is incorrect because
“protagonist” should be singular if it is to agree with the
subject. Choice D is incorrect because “a woman” is not
plural.

38.

The correct answer is C. Choice C clarifies that the
comparison is between women’s marital status and the
marital status of men. Choice A is incorrect because it
makes an illogical comparison between women’s
marital status and men in general. Choice B is incorrect
because it uses the singular “man,” meaning that only
one man’s marriage status is being compared. Choice D
is incorrect because it compares women’s marital status
to the grammatically incorrect phrase “men’s marriage.”

39.

The correct answer is A. Choice A supports the
paragraph’s claim that women are underrepresented in
the film industry. Choices B, C, and D are incorrect
because they don’t support the paragraph’s main focus
that women are underrepresented in the film industry.

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Answer Keys and Explanations
Section 3: Math Test—No Calculator

1. B

6. A

10. A

14. A

2. B

7. D

11. C

15. D

3. B

8. C

12. C

16. 80

4. C

9. B

13. D

17. 6

18. 0.55
or 5/9
19. 15
20. 4

5. C

MATH TEST—NO CALCULATOR RAW SCORE
(Number of correct answers)
1.

The correct answer is B. The slope of a line can be
determined by finding the difference in the y-coordinates divided by the difference in the x-coordinates for
any two points on the line. Using the points indicated,
5 −1
4
= − . Translating the line moves all the
the slope is
3
0−3
points on the line the same distance in the same

5.

(2 x ) 2 + (3 x ) 2 = 13 2
4 x 2 + 9 x 2 = 169
13 x 2 = 169
x 2 = 13

direction, and the image will be a parallel line.
4
Therefore, the slope of the line is − .
3
2.

The correct answer is B. The constant 12 represents the
starting distance on the driveway. In other words,
before the snail even moved, it was already 12 feet from
the beginning of the driveway. Therefore, Catherine
must have placed the snail 12 feet from the start of her
driveway before she began recording the time it took
for the snail to get to the end of her driveway.

3.

The correct answer is B. Solve for x:
−2 x + 5 = 2 − (5 − 2 x )
−2 x + 5 = 2 − 5 + 2 x
−2 x + 5 = −3 + 2 x
Add 2x to bo
oth sides.
5 = −3 + 4 x
Add +3 to both sides.
8 = 4x
Divide by 4.
2= x

4.

The correct answer is C. The right triangle, of which T is
the hypotenuse, has legs that are 6 inches and 9 inches.
Hence,
T 2 = 62 + 92
T 2 = 36 + 81 = 117
T = 117
Therefore, 10 < T < 11.

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The correct answer is C. Let the legs equal 2x and 3x.
By the Pythagorean theorem,

x = 13
The shorter leg measures 2x, or 2 13 .
6.

The correct answer is A. The amount of grain added
each week is 1,000 bushels. Divide 1,000 bushels by 0.8
bushels per cubic foot to obtain 1,250 cubic feet per
week. So, the total amount of grain in the silo is 32,500
(what is already there) plus 1,250w (the amount added
each week times the number of weeks), which must be
less than or equal to the volume of the silo, 45,000 cubic
feet.

7.

The correct answer is D. The vertex of a parabola is
found when the equation is written in the form
y = a(x – h)2 + k. You don’t need to perform any
calculations, because only choice D is written in this
form.

8.

The correct answer is C.
− x 2 + 2 y − (− x − y ) = − x 2 + 2 y + x + y
= − x 2 + x + 3y

Answer Keys and Explanations
9.

The correct answer is B. Since Jorge’s truck is traveling
at an average speed of 55 miles per hour and the truck
gets 6 miles per gallon, the number of gallons of diesel
used each hour can be found by the equation
55 miles 1 gallon 55 . The truck uses 55 gallons of
×
=
6
1 hour
6 miles 6
55
diesel per hour, so it uses
h gallons of diesel in h
6
hours. The truck’s fuel tank has 125 gallons of diesel at

14.

650 − 500 150
=
= 0.3
500
500

the beginning of the trip. Therefore, the function that

Since the population is growing, add the percent of
change, in decimal format, to 1 to get 1.3.
Therefore, the expression representing the population
growth is 500 × 1.3x.

models the number of gallons of diesel remaining in the
tank h hours after the trip begins is d (h) = 125 −
10.

55h
.
6

The correct answer is A.
M
M−N
A(M − N) = M
A=

AM − AN = M
AM − M = AN
M( A − 1) = AN
AN
M=
A −1
11.

12.

The correct answer is C. The down payment is only
made once, and the sum of the down payment plus the
monthly payments is equal to the cost of the sofa.
The correct answer is C. The sum of the values that
satisfy the equation is the sum of the solution
2.5 + 3 = 5.5.
2n 2 − 11n + 15 = 0

(2n − 5) (n − 3) = 0

The correct answer is A. The initial population of the
town is 500. The rate of change between consecutive x
values (1 year, 2 years, 3 years) is not constant. As a
result, the expression cannot be linear, but exponential.
Determine the percent of increase of the population
from the initial population to that of one year:

15.

The correct answer is D. You can see right away that
4 +1
choice A is not correct, because
= 2.5, which is not
2
equivalent to the fraction for all values of x. The three
other answer choices all start with a whole number, so
to find out which of them is correct, rewrite the given
fraction as a mixed number. Since the denominator is
x + 2 and the numerator includes a 4x, rewrite it so that
it is the sum of two fractions where one has a 4(x + 2), or
4x + 8, in the numerator. Then, simplify the expression
until you have a whole number and check the result
against the three answer choices to find the correct one.
4x +1 4x
1
=
+
x +2 x +2 x +2
4x
8
1
8
=
+
+
−
x +2 x +2 x +2 x +2
4 x + 8 1− 8
=
+
x +2 x +2
7
=4−
x +2

2n − 5 = 0 , n − 3 = 0
n = 2.5, n = 3
13.

The correct answer is D. The graph will cross the x-axis
at the point where the function (that is, the y-coordinate) has a value of 0. As a result, the following
equation needs to be solved:
480 − 60t = 0
−60t = −480
t =8
Since t represents the independent variable, the point
is (8, 0).

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Answer Keys and Explanations
16.

The correct answer is 80. If AC = BC, then m–A =
m–B = 50°.
A

50

˚

50

˚

19.

B

The correct answer is 15. To begin,
f(p + 3) = 5(p + 3)+ 12 = 5p + 15 + 12 = 5p + 27.
Similarly, f(p) = 5p + 12. Thus,
f ( p + 3) − f ( p) = 5p + 27 − (5p + 12)
= 5p + 27 − 5p − 12
= 15

80

20.

˚

C

80

The correct answer is 4. Solve for x:
9
16
+
=5
x −2 x +3

˚

16 
 9
+
( x − 2) ( x + 3) = 5 ( x − 2) ( x + 3)
 x − 2 x + 3 

E

9 x + 27 + 16 x − 32 = 5 x 2 + 5 x − 30
D

In ∆ABC, m – ACB = 180° – (m – A + m – B). So,
m – ACB = 80°. Further, m – ACB = m – ECD because
they are vertical angles. Therefore, m – ECD = 80°.
17.

The correct answer is 6. First, solve for x by multiplying
the second equation and adding it to the first one:

5 x − 4 y = 13
2[ x + 2 y = 4 ]
7 x = 21
x =3
Then, substitute x = 3 into the second equation to get
the value of y:
3 + 2y = 4
2y = 1
1
y=
2
x
1
Finally, substitute x = 3 and y =
into
to get the
y
2
value of the ratio:
x 3
= =6
y 1
2
18.

The correct answer is

5
= 0.55 . Solve for b:
9

−9 x + 5 = −9 ( x − b)
−9 x + 5 = −9 x + 9b
5 = 9b
5
=b
9

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0 = 5 x 2 − 20 x − 25
0 = 5( x + 1)( x − 5)
x = −1 or x = 5
To get the sum of the solutions, simply add –1 and 5:
–1 + 5 = 4.

Answer Keys and Explanations
Section 4: Math Test—Calculator

1. C

9. B

17. D

25. A

33. 45

2. D

10. B

18. B

26. D

34. 41

3. D

11. B

19. A

27. D

35. 5400

4. A

12. D

20. C

28. A

36. 440

5. A

13. C

21. D

29. A

37. 21.5

6. C

14. D

22. B

30. B

38. 4

7. B

15. C

23. C

31. 5

8. D

16. B

24. A

32. 9

MATH TEST—CALCULATOR RAW SCORE
(Number of correct answers)

1.

The correct answer is C. 15% of $12.50 is (0.15)($12.50)
= $1.875. So, the discount for every 10 audio singles
purchased is $1.875. Multiply this by 4 to get the savings
when purchasing 40 audio singles: 4($1.875) = $7.50.

2.

The correct answer is D. 20 square yards = 180 square
feet. At $1.30 per square foot, it will cost 180 × $1.30 =
$234.00.

3.

The correct answer is D. Let x represent the score of
one of the two games for which he scored identically.
Then, the score of the third game is x + 20. Since the
average of all six games is 182, solve the following
equation for x:
212 + 181+ 160 + x + x + ( x + 20)
= 182
6
573 + 3 x
= 182
6
573 + 3 x = 1, 092
3 x = 519
x = 173
So, his six scores were 160, 173, 173, 181, 193, and
212. Therefore, the second highest score is 193.

4.

The correct answer is A. The volume of the fish tank is
11(14)(9) = 1,386 cubic inches. The amount needed to
fill the tank is 1,386 ÷ 231 = 6 gallons.

5.

The correct answer is A. The amount of protein in m
cups of milk is 8m grams, and the amount of protein in
n eggs is 6n grams. The problem asks for the amount to
meet or exceed the recommended daily intake, which
sets up a greater-than-or-equal-to scenario.

6.

The correct answer is C. The total charge that Amy will
pay is the daily rate, the mileage rate, and the 7.5% tax
on both. If Amy drove x miles, then the total charge is
(19.99 + 0.15x) + 0.075(19.99 + 0.15x), which can be
rewritten as 1.075(19.99 + 0.15x).

7.

The correct answer is B. 5

1
dozen nails are bought for
2

1
5 dozen × 35 cents per dozen = 192.5 cents. There are
2
1
66 nails in 5 dozen and 66 ÷ 3 = 22 sets sold at 10
2
cents per set, so 22 sets × 10 cents per set = 220 cents.
1
The profit is 220 – 192.5 = 27 cents.
2
8.

The correct answer is D. The weights are proportional
to the volumes, and the volumes vary as the cubes of
their dimensions. If the edges are doubled, the volume
becomes 23 = 8 times as large. Therefore, the weight is
8 × 150 = 1,200 pounds.

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  111

Answer Keys and Explanations
9.

The correct answer is B.

16.

The correct answer is B. The scatterplot shows a strong
correlation between the variables. As the years increase,
the population also increases.

17.

The correct answer is D. The points on the graph
display a pattern of exponential growth, as a curve
could be used to connect them; therefore, an exponential curve would best represent the data.

18.

The correct answer is B. The graduate starts at x dollars
per week. After the pay cut, the graduate receives 90%
of the original salary. The 10% raise adds 9% to the
salary (10% of 90%), so the new salary is 0.99x.

19.

The correct answer is A. The solution set is as follows.
Note that it extends into all quadrants except the third.

−2(1− x ) 2 + 2(1− x 2 ) = −2(1− 2 x + x 2 ) + 2(1− x 2 )
= −2 + 4 x − 2 x 2 + 2 − 2 x 2
= −4 x 2 + 4 x

10.

11.

12.

The correct answer is B. One method to find the
correct answer is to create an inequality. The income
from the sign-up fees for x people is 20x. For the
organization to profit, 20x must be greater than the cost
of x t-shirts Therefore, 20x > 7x + 60 can be used to
model the situation. Solving this inequality yields x >
4.6. Since there can’t be 4.6 people, round the answer
up to 5.
The correct answer is B. The total number of students,
x + y, is equal to 2,180, so the answer must be choice B
or choice C. The male students raised $20 each, and the
female students raised $25 each. Since x represents the
number of male students, then the amount the male
students raised is represented by 20x, and the amount
the female students raised is represented by 25y. The
total amount raised is $50,000, so the sum is 20x + 25y =
50,000. That leaves choice B as the only correct answer.

(

9 x 3 y 5z 6 = 9 x 3 y 5z 6

)

15.

  112

2°
1

2

x

y = –2y+ 4

20.

The correct answer is C. The two intersections of the
graphs of the equations are at the points (0, 1) and
(–1, 0). Substituting 0 for x and 1 for y makes both
equations true.Also, substituting –1 for x and 0 for y
makes both equations true.

The correct answer is C. The legs have lengths x and
x – 3, and the hypotenuse has length 4. Using the
Pythagorean theorem yields:
x 2 + ( x − 3)2 = 4 2
x 2 + x 2 − 6 x + 9 = 16

The correct answer is D. Find the solution to this
problem by using the structure of the given equation.
x y
Multiplying both sides of the equation − = 5 by 24
3 4
will clear fractions from the equation and yield
8x – 6y = 120.
The correct answer is C. If x is the number of lunches
sold and y is the number of dinners sold, then x + y
represents the number of meals sold during the
weekend. The equation 7.5x + 12y represents the total
amount collected in the weekend. Therefore, the correct
system of equations is x + y = 241 and 7.5x + 12y =
2,523.

www.petersons.com

y = –2x+ 4

1
3

1
5
= 9 3 xy 3 z 2

14.

4°

The correct answer is D. The cube root of an expression
1
is equal to that expression raised to the
power, so
3
3

13.

y

2x2 − 6x − 7 = 0
x=

−( −6 ) ± ( −6 )2 − 4(2) i ( −7)
2(2)

x=

6 ± 92 6 ± 2 23 3 + 23
=
=
2
4
4

A length cannot be negative, so the hypotenuse has
length
21.

3 + 23
units.
2

The correct answer is D. If the discount is 10%, then
$12.60 is 90% of $14.00.

Answer Keys and Explanations
22.

The correct answer is B. To solve for the number of
gallons of white vinegar, set up a proportion.

29.

12 + 13 + 31+ 25 + 37 + 28 146
=
6
6
= 24.3
≈ 24

1
3=x
2 7
28
2x =
3
2
4
x =4 =4
6
3

1

23.

24.

25.

26.

27.

28.

The new average will have to be 25, but it will be
spread over 7 Winter Olympics:

The correct answer is C. A hexagon with 22-inch sides
has a perimeter of 6 × 22, or 132 inches.
C = 2pr
132 = 2pr
132 = 2(3.14 )r
132 = 6.28r
21.02 = r
The correct answer is A. Because the function f has
(x – 4) and (x + 2) as factors, the function should have
zeros when x – 4 = 0 and x + 2 = 0. The only graph that
shows a curve that has x-intercepts at –2 and 4 is
choice A.

12 + 13 + 31+ 25 + 37 + 28 + x 146 + x
=
= 25
7
7
146 + x = 175
x = 29
30.

31.

The correct answer is D. If k is a positive constant other
than 1, then the equation kx + y = c can be rewritten as
y = –kx + c. If k is positive, then –k is negative, and it
must be something other than –1. The only graph with
a negative slope other than –1 is answer choice D.
The correct answer is A. If the machine loses 15% of its
value each year, then each year its value is 85% of what
it was the year before. Therefore, we are looking for the
fifth term of a geometric sequence with first term
$80,000 at time zero and common ratio 0.85. Using the
formula a n = a1 × r n −1, we can compute a5 = $80,000 ×
(0.85)4 = $41,760.50.

6x
by
2x + 4
factoring out the greatest common factor (2).
A
3x
3x
= 3+
The result is
. Set
and solve.
x +2
x +2
x +2
A = –6.
The correct answer is B. Simplify the ratio

The correct answer is 5. Expand the left side and then
equate corresponding coefficients:
(ax − 1) ⋅ (2 x + b ) = 4 x 2 + 4 x − 3

The correct answer is A. There are a total of 8 + 23 + 19
= 50 people in Group A. Then, reduce the fraction to
8
4
find that
.
=
50 25
The correct answer is D. When t = 0, the height of the
ball is 0.02 m, so 0.02 represents the height of the ball
before it is hit.

The correct answer is A. To find the current average,
add all of the medals and divide by 6:

2ax 2 − 2 x + abx − b = 4 x 2 + 4 x − 3
2ax 2 + (ab − 2) x − b = 4 x 2 + 4 x − 3
So, 2a = 4, ab – 2 = 4, and –b = –3. So, a = 2 and b = 3.
Therefore, a + b = 5.
32.

The correct answer is 9. Since he must buy at least two
of each item, first determine the cost of buying exactly 2
of each:
2($4.00 + $3.75 + $1.50) = $18.50
Now, subtract that from $50: $50 – $18.50 = $31.50.
Divide this difference by the cost of a pint of berries
($4.00) to get $7.875.
So, Derek would be able to buy at most 7 more pints
of berries. This, together with the 2 pints we accounted for at the start of the solution, gives a maximum of
9 pints of berries that he could purchase.

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Answer Keys and Explanations
33.

The correct answer is 45. Letting m be the time per
regular question, 2m is the time per math problem. The
total time for all the regular questions is 300m, and the
total time for all the math problems is 50(2m). Since the
exam is 3 hours, or 180 minutes, 300m + 100m = 180

37.

3 x + y = −4
−( x + y = 13)

180 9
=
. The time to
400 20
 9 9
. All 50 math
do a math problem is 2   =
 20  10
9
problems can be done in 50   = 45 minutes.
 10 

2 x = −17
x = −8.5

minutes, 400m = 180, and m =

34.

35.

36.

  114

Then solve for x and substitute into one of the equations to solve for y:

The correct answer is 41. The number of games that
the Montreal Canadiens played is not provided, but it is
given that the ratio of the number of games played in
the 1947–48 season to that in the 1924–25 season is 2:1.
Problems like this can be solved by plugging in real
numbers. Let’s say that there were 100 games in the
1924–25 season, so they won 57 of those games. In the
1947–48 season, there were twice as many games, or
200 games, so they won 33% of 200 games, or 66
games. Altogether they won 123 out of 300 games, and
this fraction can be simplified to 41 out of 100, or 41%.
The correct answer is 5400. Of the number of voters
polled, 400 of 625, or 64%, were in favor of the measure.
If the margin of error is 4%, the likely population
proportion will be between 60% and 68%: 60% of 9,000
total voters is (0.6)(9,000) = 5,400.
The correct answer is 440. If the dolphin weighs 110%
of the average, it weighs 10% more than the average
weight of 400 pounds, or 0.10 × 400 = 40 pounds. The
dolphin weighs 400 + 40 = 440 pounds.

www.petersons.com

The correct answer is 21.5. First, combine the
equations by subtracting (x + y = 13) from (3x + y= –4):

−8.5 + y = 13
y = 21.5
38.

The correct answer is 4. To find the value that will
make the system have no solution, you must show that
the two equations are parallel lines. In standard form,
parallel lines are equal on the left side of their equations, but not the right. In standard form, the first
equation would need to be equal to the second on the
left side. Find a factor that you could multiply the first
equation by:
−3 x + 2 y = −1
6 x − by = 8
( −3 x + 2 y = −1)( −2)
6x − 4y = 2
so b = 4
If b = 4, the left side of both equations are equal, but
not the right. Therefore, they are parallel.

Answer Keys and Explanations
Section 5: Essay
Analysis of Passage
The following is an analysis of the passage by Peter Krapp, noting how the writer used evidence, reasoning, and stylistic or persuasive
elements to support his claims, connect the claims and evidence, and add power to the ideas he expressed. Check to see if you
evaluated the passage in a similar way.

1

Pennsylvania State University’s College of Engineering took
its computer network offline on May 15 after disclosing
two cyberattacks. The perpetrators were able to access
information on 18,000 students, who are being contacted
this week with the news that their personal identifying
information is in hackers’ hands.

1

The writer cites a specific example of a computer security
breach and uses facts and statistics to show the seriousness
of the problem.

Three days later, the computer network is back online, with
new protections for its users.
2

One of the two attacks is ascribed by a forensic
cybersecurity corporation retained by Penn State to
computers apparently based in China.

2

By mentioning that the computer hackers appear to be
located in China, the writer underscores the global risks of
the security breach.

3

As a researcher who has published on hacking and
hacktivism and serves on the board of the UC Irvine data
science initiative, I believe two aspects of this news story
deserve particular attention.

3

The writer establishes his credentials to write about and offer
an argument on this topic.

Compromising student data
4

Penn State announced last week that the FBI alerted it on
November 21, 2014, about an attack with custom malware
that started as early as September 2012.

4

The writer uses facts and dates to lay the groundwork for the
point he is about to make.

5

Why did it take so long for Penn State to disclose the breach,
despite the fact that the experience of large-scale hacks in
2013 and 2014 (against Target, Home Depot, and others)
clearly demonstrated an urgent need for quick and full
disclosure—both to help the victims and to preserve a
modicum of trust?

5

The writer poses a rhetorical question to address one
aspect of this story that he earlier said deserves “particular
attention.” His first point is that Penn State took too long to
disclose the security breach. He cites two past breaches to
support his argument about the need for quick disclosure
and uses evocative words (“help the victims and to preserve
a modicum of trust”) to emphasize how important early
disclosure is.

6

Penn State stated only that any disclosure would have
tipped off the perpetrators before their access to the
College of Engineering computers could be cut off.

6

The writer states Penn State’s justification for delaying
disclosure. He uses the word only to imply that Penn State
should have been more forthcoming, thus strengthening
his own position. At the same time, he presents himself as
reasonable because he offers Penn State’s argument before
going on with his own.

7

Meanwhile, student data may have been compromised
for at least six months, maybe longer.

7

The writer shows why, in his opinion, Penn State’s position is
weak.

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Answer Keys and Explanations
8

Another conspicuous problem with public discussion
of events like this is, in fact, the lack of distinction often
made in the media between actual appropriation of
data (as at Penn State) and mere temporary disabling
or defacement of websites (as happened to Rutgers
University last month). That is like being unable to make a
difference between a grand theft auto and keying a car.

8

The writer now addresses the second point that he feels
“deserve[s] particular attention” — The media do not treat
breaches like the one at Penn State seriously, often viewing
them more like nuisance problems. He supports his view that
the media should be able to make a distinction between
serious and non-serious security breaches by juxtaposing a
significant crime (grand theft auto) with a minor infraction
(keying a car).

The question is, what can universities do to limit the risk
to their students?
9

The exposure of student data in higher education is not
limited to Social Security numbers or email passwords.
Information collected and retained by educational
institutions includes full name, address, phone number,
credit and debit card information, workplace information,
date of birth, personal interests and of course academic
performance and grade information.

9

The writer shows how dangerous these kinds of security
breaches are, using specific examples of the kinds of
information that can fall into hackers’ hands.

10

A survey conducted by the Obama administration
collected responses from 24,092 individuals on how
much they trusted various institutions to keep their
data safe. There was a high level of concern around
transparency and legal standards. (https://www.
whitehouse.gov/issues/technology/big-data-review)

10

The writer strengthens his viewpoint by providing statistical
evidence in the form of a survey that reveals how concerned
people are about the safety of their personal data.

11

The writer supports his argument by quoting an authority
no less than the president of the United States. (He also
makes the point that he believes the president’s concern is
overdue, by saying this statement was made “only recently.”)

12

The writer underscores the depth of the problem by pointing
out that there are many entities that might access students’
computer data: not only foreign intelligence agencies (as was
the case with the Penn State breach), but US companies that
mine computer data for profit. The writer argues that just as
foreign breaches have been recognized as serious, so should
domestic for-profit breaches.

13

The writer offers a specific example of a US company mining
personal data for profit to strengthen his case.

The California Senate Bill 1177 was enacted to prevent
educational services from selling student information or
mining it for advertising purposes.

14

The writer underscores the validity of his argument by citing
legislation enacted to address the problem.

15 Yet, almost a year later, students in California remain just
as concerned about their data privacy as before—since
the new state law was watered down to apply only to
K–12 and not to higher education.

15

The writer points out that the final form of the legislation to
protect students’ data does not extend to college students.

...
11

President Obama only recently called for laws covering
data hacking and student privacy. “We’re saying that data
collected on students in the classroom should only be
used for educational purposes,” he stated in his speech to
the Federal Trade Commission (FTC) earlier this year.
Data privacy concerns

12

If students’ right to privacy needs to be protected from
the specter of foreign intelligence agencies poking
around the Penn State Engineering School, then by the
same logic it should be protected also against datamining by for-profit actors right here in the US.

13

Until May 2014, Google, for instance, routinely mined
its apps for education services for advertising and
monetizing purposes. When Education Week reported
that Google was mining student emails, it quickly led not
only to lawsuits but also to landmark legislation.

14

  116

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Answer Keys and Explanations
16

And when it was disclosed earlier this spring that
education publisher Pearson secretly monitored social
media to discern references to their content, the legislative
response was one that, according to the Electronic Privacy
Information Center (EPIC) in Washington, DC, “fails to
uphold President Obama’s promise that the data collected
in an educational context can be used only for educational
purposes.”

16

The writer reinforces his argument that not enough is being
done to protect college students’ personal data by citing
another example of a US company data-mining for profit. He
also quotes an authoritative source to support this claim.

17

Students in higher education nationwide are still in a
position where they cannot opt out of the computer
services of their learning institutions, and so they have no
expectation of privacy.

17

The writer points out a reality that makes and keeps college
students vulnerable: “they cannot opt out of the computer
services of their learning institutions.”

18

Despite President Obama’s promises for safeguarding
the privacy of consumers and families, and despite the
fact that a number of technology companies concerned
with growing consumer distrust recently signed a pledge
to safeguard student privacy, neither Google nor Apple
signed on.

18

The writer underscores college students’ vulnerability by
pointing out that two major companies (Google and Apple)
have refused to sign a pledge to safeguard student privacy.

19 The President’s Council of Advisors on Science and
Technology (PCAST) was tasked to examine current
and likely future capabilities of key technologies, both
those associated with the collection, analysis, and use
of big data and those that can help to preserve privacy,
resulting in a direct recommendation to strengthen US
research in privacy-related technologies.

19

The writer points out that an advisory board to the US
president has recommended strengthening research on the
problem.

20

And overwhelmingly, respondents to a White House
survey recently expressed severe reservations about
the collection, storage, and security and use of private
information.

20

21

Maybe it is time for higher education to heed those
signals.

The writer cites a White House survey that reflected “severe
reservations about the collection, storage, and security and
use of private information.” The survey shows that despite
the measures that have been put in place, the public still
feels their data is at risk.

21

The writer concludes his argument by saying it’s time for
colleges (where people are particularly vulnerable) to pay
attention to—and act on—the information they have, in
order to remedy the situation.

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Answer Keys and Explanations
Sample Essays
The following are examples of a high-scoring and low-scoring essay, based on the passage by Peter Krapp.

High-Scoring Essay
Peter Krapp builds a well-constructed argument designed to persuade readers that computer hacking is a danger that steals
our most valuable possession—personal information—from our most vulnerable people—students. The article focuses on
personal information about college students stolen from college computer networks. The writer builds his argument brick by
brick. He starts by giving an example of student data stolen from a college by hackers in China, relating his topic to an event
that really happened. The writer quickly establishes himself as an expert in the topic so readers can trust his information.
Krapp asks and answers questions about the theft of data, goes into more depth about the type of information stolen from
college networks, and then briefly touches on the laws, or the lack of laws, to protect students. He points out that data is
stolen from students by not only foreign hackers, but also by American companies. The writer finishes with a call for colleges
to protect student data, clearly stating his purpose.
The theft of 18,000 students’ data from Pennsylvania State University’s College of Engineering is a strong example to start the
article. Naming the college and identifying the large number of students tells the reader immediately that a single incident
affected many people. The location of the hackers shows that protecting the physical location of the data is useless, and it
adds a bit of foreign intrigue.
The second paragraph is a single sentence, but accomplishes two important things. It gives the reader his credentials,
proving that he is qualified to speak as an authority on the topic. It also provides structure for the next section by telling the
reader that the writer will focus on two aspects of the news story.
Krapp reveals the first aspect to be discussed right away—Penn State knew that data was being stolen for six months before
doing anything about it. To emphasize just how astonishing Penn State’s behavior is, Krapp asks the same question that any
reader might ask: Why? He asks the question in words that urge a fast response such as “urgent need” and “help the victims.”
He demonstrates that Penn State’s response is weak by describing the response as “stated only.”
The next paragraph starts with “Another conspicuous problem.” This phrase presents the second aspect he wants to discuss,
which is the media’s lack of understanding about the difference between the major problem of stealing data and the minor
problem of disabling a website for a short period of time. He helps readers to understand the difference by comparing
stealing a car to causing only minor damage to the car.
Krapp follows the reader’s natural thought process by asking the next question: If it’s a problem, how do we fix it? He presents
information in two ways, listing the type of information at risk and displaying bar charts that show the results of a presidential
survey on the public’s perception of data security.
The section title, “Data privacy concerns,” informs readers that Krapp is addressing another concern, the theft of student data
by US companies that want to mine data for profit. He gives two examples that are recognized and used by college students:
Google and Pearson. He points out that students are particularly vulnerable to theft by the corporations that provide the
apps and class materials that students must use for school. The school dictates which apps are to be used, so students “cannot
opt out.” He touches again on the law, but reveals that it doesn’t protect college students. This information reinforces his
arguments that college students are vulnerable to data theft. Presenting the facts this way says that college students are not
only vulnerable, they are betrayed by the universities, US companies, and laws that should protect them.
The argument Krapp has built leads to a solid conclusion that Krapp states in a single, final sentence that is brief but
summarizes the conclusion readers should draw. The brevity of the statement ensures that readers will get the point of the
article and remember it in the future. The information that Krapp presents in examples, lists, survey results and bar charts
provides a solid foundation for the conclusion that becomes obvious to any reader. Protect student data.

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Answer Keys and Explanations
Low-Scoring Essay
Peter Krapp tries to persuade us that students should be allowed to opt out of giving their data to colleges and companies.
This would protect them from having their full name, address, phone number, credit and debit card information, workplace
information, date of birth, personal interests and of course academic performance and grade information stolen by people
like Chinese hackers and companies like Google and Pearson.
Peter Krapp persuades us by giving lots of examples of data being stolen, like Pennsylvania State University’s College of
Engineering, Target, Home Depot, Google, and Pearson. One of the examples includes a chart that shows how much 24,092
individuals trusted various institutions to keep their data safe. Most of them didn’t think their data was safe.
All of this information, charts, and examples that Peter Krapp included in his essay is very convincing. I think we should pass
some laws like Peter Krapp suggested. The laws would keep American companies from stealing our data and punish hackers
from other countries who steal our data.
All college students should try to opt out of giving away all of their private information because we know the colleges won’t
protect it. The colleges actually tell students to use apps that they know companies steal data from. This makes college
students more vulnerable than anyone else to losing their data.
If companies are going to use the data they take from students, they should pay the students for the data. Then, students
might not mind so much if companies take their data because the students would make money from it and college is very
expensive.
When I go to college, I will opt out of giving my data to colleges and companies. My information is important to me and I
don’t want it stolen from me.

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Computing Your Scores
COMPUTING YOUR SCORES
Now that you've completed this diagnostic test, it's time to compute your scores. Simply follow the instructions on the following
pages, and use the conversion tables provided to calculate your scores. The formulas provided will give you as close an approximation
as possible on how you might score on the actual SAT® exam.

To Determine Your Practice Test Scores
1. After you go through each of the test sections (Reading, Writing and Language, Math—No Calculator, and Math—Calculator)
and determine which answers you got right, be sure to enter the number of correct answers in the box below the answer
key for each of the sections.
2. Your total score on the practice test is the sum of your Evidence-Based Reading and Writing Section score and your Math
Section score. To get your total score, convert the raw score—the number of questions you got right in a particular section—
into the “scaled score” for that section, and then calculate the total score. It sounds a little confusing, but we’ll take you
through the steps.

To Calculate Your Evidence-Based Reading and Writing Section Score
Your Evidence-Based Reading and Writing Section score is on a scale of 200–800. First determine your Reading Test score, and then
determind your score on the Writing and Language Test.
1. Count the number of correct answers you got on the Section 1: Reading Test. Remember that there is no penalty for wrong
answers. The number of correct answers is your raw score.
2. Go to Raw Score Conversion Table 1: Section and Test Scores on page123. Look in the “Raw Score” column for your raw
score, and match it to the number in the “Reading Test Score” column.
3. Do the same with Section 2: Writing and Language Test to determine that score.
4. Add your Reading Test score to your Writing and Language Test score.
5. Multiply that number by 10. This is your Evidence-Based Reading and Writing Section score.

To Calculate Your Math Section Score
Your Math score is also on a scale of 200–800.
1. Count the number of correct answers you got on the Section 3: Math Test—No Calculator and the Section 4: Math
Test—Calculator. Again, there is no penalty for wrong answers. The number of correct answers is your raw score.
2. Add the number of correct answers on the Section 3: Math Test—No Calculator and the Section 4: Math Test—Calculator.
3. Use the Raw Score Conversion Table 1: Section and Test Scores on page 123 and convert your raw score into your Math
Section score.

To Obtain Your Total Score
Add your score on the Evidence-Based Reading and Writing Section to the Math Section score. This is your total score on this SAT®
Practice Test, on a scale of 400–1600.

Subscores Provide Additional Information
Subscores offer you greater details about your strengths in certain areas within literacy and math. The subscores are reported on
a scale of 1–15 and include Heart of Algebra, Problem Solving and Data Analysis, Passport to Advanced Math, Expression of Ideas,
Standard English Conventions, Words in Context, and Command of Evidence.

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Computing Your Scores
Heart of Algebra
The Heart of Algebra subscore is based on questions from the Math Test that focus on linear equations and inequalities.

• Add up your total correct answers from these questions:
•
•

ºº Math Test—No Calculator: Questions 1–3, 6, 9, 11, 17, 18
ºº Math Test—Calculator: Questions 5, 6, 10, 11, 14, 15, 19, 27, 29, 37, 38
Your Raw Score = the total number of correct answers from all of these questions
Use the Raw Score Conversion Table 2: Subscores on page 124 to determine your Heart of Algebra subscore.

Problem Solving and Data Analysis
The Problem Solving and Data Analysis subscore is based on questions from the Math Test that focus on quantitative reasoning,
the interpretation and synthesis of data, and solving problems in rich and varied contexts.

• Add up your total correct answers from these questions:
•
•

ºº Math Test—No Calculator: None
ºº Math Test—Calculator: Questions 1–3, 7, 8, 16–18, 21, 22, 25, 28, 32, 33–36
Your Raw Score = the total number of correct answers from all of these questions
Use the Raw Score Conversion Table 2: Subscores on page 124 to determine your Problem Solving and Data Analysis
subscore.

Passport to Advanced Math
The Passport to Advanced Math subscore is based on questions from the Math Test that focus on topics central to your ability
to progress to more advanced math, such as understanding the structure of expressions, reasoning with more complex equations,
and interpreting and building functions.

• Add up your total correct answers from these questions:
•
•

ºº Math Test—No Calculator: Questions 7, 8, 10, 12–15, 19, 20
ºº Math Test—Calculator: Questions 9, 12, 13, 24, 26, 30, 31
Your Raw Score = the total number of correct answers from all of these questions
Use the Raw Score Conversion Table 2: Subscores on page 124 to determine your Passport to Advanced Math subscore.

Expression of Ideas
The Expression of Ideas subscore is based on questions from the Writing and Language Test that focus on topic development,
organization, and rhetorically effective use of language.

• Add up your total correct answers from these questions in Section 2: Writing and Language Test:
ºº Questions 1, 4, 5, 7–9, 12, 13, 16, 17, 19, 20, 23, 26, 28, 29, 32–35, 39, 41–43

• Your Raw Score = the total number of correct answers from all of these questions
• Use the Raw Score Conversion Table 2: Subscores on page 124 to determine your Expression of Ideas subscore.

Standard English Conventions
The Standard English Conventions subscore is based on questions from the Writing and Language Test that focus on sentence
structure, usage, and punctuation.

• Add up your total correct answers from these questions in Section 2: Writing and Language Test:
ºº Questions 2, 3, 6, 10, 11, 14, 15, 18, 21, 22, 24, 25, 27, 30, 31, 36–38, 40, 44

• Your Raw Score = the total number of correct answers from all of these questions
• Use the Raw Score Conversion Table 2: Subscores on page 124 to determine your Standard English Conventions
subscore.

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Computing Your Scores
Words in Context
The Words in Context subscore is based on questions from the Reading Test and the Writing and Language Test that address
word/phrase meaning in context and rhetorical word choice.

• Add up your total correct answers from these questions in Sections 1 and 2:
•
•

ºº Reading Test: Questions 1, 5, 12, 16, 27, 30, 41, 42, 51, 52
ºº Writing and Language Test: Questions 5, 8, 12, 17, 28, 32, 35, 43
Your Raw Score = the total number of correct answers from all of these questions
Use the Raw Score Conversion Table 2: Subscores on page 124 to determine your Words in Context subscore.

Command of Evidence
The Command of Evidence subscore is based on questions from the Reading Test and the Writing and Language Test that ask
you to interpret and use evidence found in a wide range of passages and informational graphics, such as graphs, tables, and charts.

• Add up your total correct answers from these questions in Sections 1 and 2:
•
•

ºº Reading Test: Questions 3, 6, 9, 14, 19, 23, 28, 32, 36, 38, 42, 43, 44
ºº Writing and Language Test: Questions 1, 4, 13, 16, 29, 33, 34, 42
Your Raw Score = the total number of correct answers from all of these questions
Use the Raw Score Conversion Table 2: Subscores on page 124 to determine your Command of Evidence subscore.

Cross-Test Scores
The SAT® exam also reports two cross-test scores: Analysis in History/Social Studies and Analysis in Science. These scores are based
on questions in the Reading Test, Writing and Language Test, and both Math Tests that ask you to think analytically about texts and
questions in these subject areas. Cross-test scores are reported on a scale of 10–40.

Analysis in History/Social Studies
• Add up your total correct answers from these questions:

•
•

ºº Reading Test: Questions 11–21, 43–52
ºº Writing and Language Test: Questions 12, 13, 16, 17, 19, 20
ºº Math Test—No Calculator: Question 14
ºº Math Test—Calculator: Questions 11, 16, 17, 25, 29, 34, 35
Your Raw Score = the total number of correct answers from all of these questions
Use the Raw Score Conversion Table 3: Cross-Test Scores on page 126 to determine your Analysis in History/Social
Studies cross-test score.

Analysis in Science
• Add up your total correct answers from these sections:
• Reading Test: Questions 1–10, 22–32
• Writing and Language Test: Questions 25, 26, 28, 29, 32, 33
• Math Test—No Calculator: Questions 2, 13
• Math Test—Calculator: Questions 4, 5, 8, 22, 26, 36
• Your Raw Score = the total number of correct answers from all of these questions
• Use the Raw Score Conversion Table 3: Cross-Test Scores on page 126 to determine your Analysis in Science cross-test
score.

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Computing Your Scores

Raw Score

Math Section Score

Reading Test Score

Writing and Language
Test Score

10
10
10
10
11
12
13
13
14
15
16
16
17
18
19
19
20
21
21
22

20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39

450
460
470
480
480
490
500
510
520
520
530
540
550
560
560
570
580
590
600
600

22
23
23
24
24
25
25
26
26
27
28
28
29
29
30
30
31
31
32
32

23
23
24
25
25
26
26
27
28
28
29
30
30
31
32
32
33
34
34
35

40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58

610
620
630
640
650
660
670
670
680
690
700
710
730
740
750
760
780
790
800

33
33
34
35
35
36
37
37
38
38
39
40
40

Writing and Language
Test Score

Writing and Language
Test Score

10
10
10
11
12
13
14
15
15
16
17
17
18
19
19
20
20
21
21
22

Reading Test Score

Reading Test Score

200
200
210
230
240
260
280
290
310
320
330
340
360
370
380
390
410
420
430
440

Math Section Score

Math Section Score

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19

Raw Score

Raw Score

Raw Score Conversion Table 1: Section and Test Scores

36
37
38
39
40

Conversion Equation 1 Section and Test Scores
READING TEST
RAW SCORE (0–52)

WRITING AND LANGUAGE TEST
RAW SCORE (0–44)

CONVERT

CONVERT

10
READING
TEST SCORE (10–40)

WRITING AND LANGUAGE
TEST SCORE (10–40)

READING AND WRITING
TEST SCORE (20–80)
MATH TEST
RAW SCORE
(0–58)

MATH TEST—NO CALCULATOR
RAW SCORE (0–20)

MATH TEST—CALCULATOR
RAW SCORE (0–38)

EVIDENCE-BASED
READING AND WRITING
SECTION SCORE (200–800)

EVIDENCE-BASED
READING AND WRITING
SECTION SCORE (200–800)

CONVERT
MATH SECTION
SCORE (200–800)
MATH SECTION
SCORE (200–800)

TOTAL SAT® SCORE
(400–1600)

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Computing Your Scores

  124

Raw Score
(# of correct answers)

Expression of Ideas

Standard English Conventions

Heart of Algebra

Problem Solving
and Data Analysis

Passport to Advanced Math

Words in Context

Command of Evidence

Raw Score Conversion Table 2: Subscores

0

1

1

1

1

1

1

1

1

1

1

1

1

3

1

1

2

1

1

2

2

5

2

2

3

2

2

3

3

6

3

3

4

3

2

4

4

7

4

4

5

4

3

5

5

8

5

5

6

5

4

6

6

9

6

6

7

6

5

6

7

10

6

7

8

6

6

7

8

11

7

8

9

7

6

8

8

11

8

8

10

7

7

8

9

12

8

9

11

8

7

9

10

12

9

10

12

8

8

9

10

13

9

10

13

9

8

9

11

13

10

11

14

9

9

10

12

14

11

12

15

10

10

10

13

14

12

13

16

10

10

11

14

15

13

14

17

11

11

12

15

14

15

18

11

12

13

15

15

19

12

13

15

20

12

15

21

13

22

14

23

14

24

15

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Computing Your Scores
Conversion Equation 2 Subscores
HEART OF ALGEBRA
RAW SCORE (0–19)

EXPRESSION OF IDEAS
RAW SCORE (0–24)

COMMAND OF EVIDENCE
RAW SCORE (0–18)

PROBLEM SOLVING AND DATA
ANALYSIS RAW SCORE (0–17)

CONVERT

CONVERT

CONVERT

CONVERT

HEART OF ALGEBRA
SUBSCORE (1–15)

EXPRESSION OF IDEAS
SUBSCORE (1–15)

COMMAND OF EVIDENCE
SUBSCORE (1–15)

PROBLEM SOLVING AND DATA
ANALYSIS SUBSCORE (1–15)

STANDARD ENGLISH CONVENTIONS
RAW SCORE (0–20)

WORDS IN CONTEXT
RAW SCORE (0–18)

PASSPORT TO ADVANCED
MATH RAW SCORE (0–16)

CONVERT

CONVERT

CONVERT

STANDARD ENGLISH CONVENTIONS
SUBSCORE (1–15)

WORDS IN CONTEXT
SUBSCORE (1–15)

PASSPORT TO ADVANCED
MATH SUBSCORE (1–15)

Analysis in History/Social
Studies Cross-Test Score

Analysis in Science
Cross-Test Score

10

10

18

28

26

1

10

11

19

29

27

2

11

12

20

30

27

3

12

13

21

30

28

4

14

14

22

31

29

5

15

15

23

32

30

6

16

16

24

32

30

7

17

17

25

33

31

8

18

18

26

34

32

9

20

19

27

35

33

10

21

20

28

35

33

11

22

20

29

36

34

12

23

21

30

37

35

13

24

22

31

38

36

14

25

23

32

38

37

15

26

24

33

39

38

16

27

24

34

40

39

17

28

25

35

40

40

Raw Score
(# of correct answers)

Analysis in Science
Cross-Test Score

0

Raw Score
(# of correct answers)

Analysis in History/Social
Studies Cross-Test Score

Raw Score Conversion Table 3: Cross-Test Scores

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Computing Your Scores
Conversion Equation 3: Cross-Test Scores
ANALYSIS IN
HISTORY/SOCIAL STUDIES
TEST

QUESTIONS

RAW SCORE

ANALYSIS IN SCIENCE
QUESTIONS

Reading Test

11–21, 43–52

1–10, 22–32

Writing and
Language Test

12, 13, 16, 17, 19, 20

25, 26, 28, 29, 32, 33

Math Test—No
Calculator

14

2, 13

Math
Test—Calculator

11, 16, 17, 25, 29,
34, 35

4, 5, 8, 22, 26, 36

TOTAL

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ANALYSIS IN HISTORY/
SOCIAL STUDIES
RAW SCORE (0–35)

ANALYSIS IN SCIENCE
RAW SCORE (0–35)

CONVERT

CONVERT

ANALYSIS IN HISTORY/
SOCIAL STUDIES
CROSS-TEST SCORE (10–40)

ANALYSIS IN SCIENCE
CROSS-TEST SCORE (10–40)

RAW SCORE

part iii: reading
strategies for the sat ®

Chapter 3: Evidence-Based Reading Test Strategies

Chapter 3:
Evidence-Based Reading
Test Strategies
OVERVIEW
A Closer Look at the Evidence-Based Reading Test
Basic Steps for Answering Evidence-Based
Reading Questions
Tips for Taking the Reading Test
Strategies for Answering Specific Question Types
Exercise: Evidence-Based Reading Test
Answer Key and Explanations
Summing It Up

A CLOSER LOOK AT THE EVIDENCE-BASED READING TEST
The SAT® Evidence-Based Reading Test is focused on demonstrating comprehension and reasoning skills through responses to
a variety of reading passages. The passages are chosen to reflect the complexity and reading levels appropriate to college and
career readiness. The questions will require you to analyze the text and use textual evidence to assess meaning and to support
ideas. The emphasis is on using the text to support your answers and on understanding the overall concepts and how they are
developed through the course of the passage.
Passage topics are drawn from U.S. and world literature, historical and social science documents, and scientific writing. There will
be four single passages and one paired passage, with a total of 52 questions. The single passages’ lengths are between 500–750
words; the paired passage has the same word length between the two passages.

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EvidenceBased
Reading Test
Strategies

The reading comprehension questions are designed to assess how well you read and understand information. The questions
don’t test the specifics you have learned in your course work. They are based solely on explicit and implicit information contained
in the passage. At least one passage will be accompanied by an informational graphic from which one or more questions will be
drawn. These questions require you to analyze the data presented in such formats as tables, graphs, and charts.

NOTE: Evidence-based reading questions are not arranged in order of difficulty. The questions
for each passage will generally begin with broader questions about the overall ideas in the
text and will then focus on specific portions of the passage.

Peterson’s SAT® Prep Guide 2017

Question Format
On the SAT® exam, each evidence-based reading passage and question set starts with a direction line that looks like this:

Questions 1–10 are based on the following passage.
OR

Questions 1–10 are based on the following passage[s] and supplementary material.
OR

Questions 1–10 are based on the following two passages.
The direction line is followed by a brief introduction to the text. The introduction describes the origin of the passage. Sometimes,
the introduction will include additional background information.
The questions for each passage are in standard multiple-choice format with four answer choices each. Most often, these questions
ask you to do one of the following:

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Chapter 3
EvidenceBased
Reading Test
Strategies

•
•
•
•
•
•

Determine central themes and ideas as presented in the passage.
Determine the author’s purpose or point of view of the text.
Cite textual evidence to support inferences, conclusions, or arguments.
Cite evidence to illustrate or support interpretations of meaning, mood, or tone of the passage.
Analyze words and phrases in the context of the passage.
Analyze information in an accompanying table, graph, chart, etc.

BASIC STEPS FOR ANSWERING EVIDENCE-BASED READING QUESTIONS
To answer the reading questions, follow these five steps:
1. Read the introduction.
2. Read the questions.
3. Read the passage with the questions in mind.
4. Answer the questions.
5. For any question you’re not sure of, eliminate obviously wrong answers and take your best guess. Answer all the questions.
Let’s look at the five steps in more detail.
1. You don’t want to blow past the introductory paragraph because it can be very helpful to you. It might provide some
important background information about the passage, or it might set the stage so you know what you’re reading about.
2. Read the questions so you know what to look for in the passage. For example, if there’s a question about the theme,
consider that as you read. If there’s a question that references a particular word or phrase, or perhaps a quotation in
the text, look for it as you read the passage.
3. Now read the passage as quickly as you can without getting lost. Don’t fret over details; focus on the larger ideas and
try to follow the sequence, argument, or plot.

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4. Answer the questions that are easiest for you. Then tackle the others in order, referring to the text to find and confirm
answers as time permits.
5. Eliminate answer choices that you know are incorrect and guess at the remaining choices. Remember, there is no
penalty for incorrect answers, so be sure not to leave any answer circles blank.
Now that you’re familiar with how to approach evidence-based reading passages and questions, let’s try a few. Note that in the
SAT® and all of the practice tests in this book, the reading passages will appear in two columns. For instructional purposes, we
have placed the passages full page in this chapter.

NOTE: Never skip the introduction as it is likely to contain some important information about both the passage and the
types of questions that accompany it. The introduction will identify the type of passage being presented, the source or
author of the passage, the era in which the passage was written, or the event that the passage describes. All of this information will help you focus your reading and find the correct answers to the questions.

Sample Reading Passage 1
Questions 1–4 are based on the following passage.
José Martí was a Cuban teacher, organizer, writer, and poet. The lyrics to the popular folksong “Guantanamera” were adapted from
one of Martí’s poems. Although Cuban by birth, he is considered one of the most influential writers in all of Latin America. He died in
the battle for Cuba’s independence from Spain, a cause to which he had devoted much of his life.
The following excerpt comes from an article by Martí that was published in El Partido Liberal (Mexico City), March 5, 1892.
Our America
The prideful villager thinks his hometown contains the whole world, and as long as he can stay on as mayor or
humiliate the rival who stole his sweetheart or watch his nest egg accumulating in its strongbox he believes the universe
to be in good order, unaware of the giants in seven-league boots who can crush him underfoot or the battling comets
Line in the heavens that go through the air devouring the sleeping worlds. … It is the hour of reckoning and of marching in
5 unison, and we must move in lines as compact as the veins of silver that lie at the roots of the Andes. …
Our youth go out into the world wearing Yankee- or French-colored glasses and aspire to rule by guesswork a
country they do not know. … To know is to solve. To know the country and govern it in accordance with that knowledge
is the only way of freeing it from tyranny. The European university must yield to the American university. The history of
America from the Incas to the present must be taught in its smallest detail, even if the Greek Archons go untaught. Our
10 own Greece is preferable to the Greece that is not ours; we need it more. Statesmen who arise from the nation must
replace statesmen who are alien to it. …

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What a vision we were: the chest of an athlete, the hands of a dandy, and the forehead of a child. We were a whole
fancy dress ball, in English trousers, a Parisian waistcoat, a North American overcoat, and a Spanish bullfighter’s hat.
The Indian circled about us, mute, and went to the mountaintop to christen his children. The black, pursued from afar,
15 alone and unknown, sang his heart’s music in the night, between waves and wild beasts. The campesinos, the men of
the land, the creators, rose up in blind indignation against the disdainful city, their own creation. We wore epaulets
and judge’s robes, in countries that came into the world wearing rope sandals and Indian headbands. The wise thing
would have been to pair, with charitable hearts and the audacity of our founders, the Indian headband and the judicial
robe, to undam the Indian, make a place for the able black, and tailor liberty to the bodies of those who rose up and
20 triumphed in its name. … No Yankee or European book could furnish the key to the Hispanoamerican enigma. So the
people tried hatred instead, and our countries amounted to less and less each year. Weary of useless hatred … we are
beginning, almost unknowingly, to try love. The nations arise and salute one another. “What are we like?” they ask, and

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begin telling each other what they are like. … The young men of America are rolling up their sleeves and plunging their
hands into the dough, and making it rise with the leavening of their sweat. They understand that there is too much
25 imitation, and that salvation lies in creating. Create is this generation’s password. Make wine from plantains; it may be
sour, but it is our wine! …
Anyone who promotes and disseminates opposition or hatred among races is committing a sin against humanity. …
To think is to serve. We must not, out of a villager’s antipathy, impute some lethal congenital wickedness to the continent’s
light-skinned nation simply because it does not speak our language or share our view of what home life should be or
30 resemble us in its political failings, which are different from ours, or because it does not think highly of quick-tempered,
swarthy men or look with charity, from its still uncertain eminence, upon those less favored by history who, in heroic
stages, are climbing the road that republics travel. But neither should we seek to conceal the obvious facts of the problem,
which can, for the peace of the centuries, be resolved by timely study and the urgent, wordless union of the continental
soul. For the unanimous hymn is already ringing forth, and the present generation is bearing industrious America along
35 the road sanctioned by our sublime forefathers. From the Rio Bravo to the Straits of Magellan, the Great Cemi,* seated
on a condor’s back, has scattered the seeds of the new America across the romantic nations of the continent and the
suffering islands of the sea!
*Cemi is a deity or ancestral spirit of the Taíno people—one of the indigenous groups of the Caribbean.

1.

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What is the most likely reason Martí wrote this article?
A.

To appeal to fellow Latin Americans to be more like the Europeans

B.

To gain support from fellow Latin Americans to join in a fight against European invaders

C.

To rally support for himself as a leader of a revolution to free his country from tyranny

D.

To encourage fellow Latin Americans to educate themselves about their own mixed heritage

Review the introduction and note that Martí was a teacher, writer, and organizer and that he spent a good part of his life fighting
for Cuban independence. You can infer from that information that he would try to persuade people to educate themselves. Martí
talks about how little his fellow countrymen know of their own heritage and says that they cannot expect to govern themselves
unless they understand their own heritage (lines 6–8). Choice D pinpoints one rationale Martí gives for education. Overall, the
excerpt does not address revolution, nor does Martí present himself as revolutionary. He appeals to pride in one’s heritage. These
ideas are connected, because Martí believes that his fellow Latin Americans can free themselves through education (lines 7–8:
“To know the country and govern it in accordance with that knowledge is the only way of freeing it from tyranny.”). The correct
answer is choice D.
2.

As used in line 12, “vision” refers to
A.

how others perceive the peoples of Latin America.

B.

a picture of a unified culture.

C.

Martí’s ideals about freedom.

D.

the opportunity to create a new beginning.

When a question refers to a specific part of the text, quickly reread the line(s) and the surrounding text. In paragraph 3, Martí
offers a detailed, mocking description of the disparate parts blended to compose the peoples of Latin America. The picture he
paints is of a person comprising various elements of other cultures—none his or her own. This, he says, is how Europeans saw
the people of Latin America. The vision is the image others have of them and how it is misunderstood and even hated by others.
So, the correct answer is choice A.

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3.

Which statement represents Martí’s attitude toward education?
A.

Universal education is of the utmost importance.

B.

Everyone should learn about Latin American history.

C.

Education is necessary for new leadership.

D.

Education should include the study of Greek democracy as a model for freedom.

This question is asking for an overview, so you need to think about the points Martí makes and his overall message. You can
eliminate choices A and B because they are general statements that anyone could make—they are not ideas that are represented
in the passage (though they are related). Choice D is an incorrect interpretation of the text. Choice C is the best answer because
Martí says that education is important for new leaders, and these leaders must be well-versed in the history of Latin America.
People who are educated about their own country and heritage will rise as leaders and take over for “statesmen who are alien to
it” (line 11). The correct answer is choice C.
4.

Which of the following best describes of the overall tone of the article?
A.

Passionate and resolute

B.

Angry and bitter

C.

Scolding and arrogant

D.

Inflammatory and rebellious

This type of question also requires you to think about the point of view and presentation, which determine the tone. Martí tries
to persuade his readers, advising them to wake up to threats from those who don’t understand them or hate them (lines 20–21:
“No Yankee or European book could furnish the key to the Hispanoamerican enigma. So the people tried hatred instead, and
our countries amounted to less and less each year.”). His passion comes through in the use of flowery and emotional language:
“the hour of reckoning and of marching in unison” (lines 4–5); similes: “move in lines as compact as the veins of silver” (line 5);
and metaphors: “sang his heart’s music in the night, between waves and wild beasts” (line 15). He expresses a resolve, not anger
or bitterness: “Make wine from plantains; it may be sour, but it is our wine!” (lines 25–26). Although Martí admonishes his fellow
countrymen for being naïve and for ignoring dangers (line 3): they are “unaware of the giants in seven-league boots who can
crush him underfoot”), he does not take the position of preacher. Rather, he speaks as one of the people (lines 4–5: “It is the
hour of reckoning and of marching in unison, and we must move in lines as compact as the veins of silver that lie at the roots of
the Andes.”). Martí does not use inflammatory language; nor does he call for rebellion, choice D. Rather, he tries to persuade his
fellow Latin Americans to unite in spite of their differences and to defend their lands from foreigners who do not understand
them. The correct answer is choice A.

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Sample Reading Passage 2
Questions 1–4 are based on the following two passages.
The following passages discuss theories of how culture and language spread across Europe and Central Asia to form modern European
and Asian peoples. Two teams of scientists—one based at the University of Copenhagen and one at the University of Adelaide—presented studies about the DNA of ancient Europeans, based on 170 skeletons found in countries from Spain to Russia.
Passage 1 is excerpted from “When modern Eurasia was born,” originally published by the University of Copenhagen on June 10, 2015,
by the Center for GeoGenetics and the Natural History Museum of Denmark.

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PASSAGE 1
When Modern Eurasia Was Born
With this new investigation, the researchers confirm that the changes came about as a result of migrations. The
researchers think that this is interesting also because later developments in the Bronze Age are a continuation of this new
social perception. Things add up because the migrations can also explain the origin of the northern European language
Line families. Both language and genetics have been with us all the way up to the present. Kristian Kristiansen [professor of
5 archaeology at the University of Gothenburg, Sweden] even thinks that it was crucial that events happened during these
few centuries, as crucial as the colonization of the Americas.
One of the main findings from the study is how these migrations resulted in huge changes to the European gene-pool,
in particular conferring a large degree of admixture on the present populations. Genetically speaking, ancient Europeans
from the time post these migrations are much more similar to modern Europeans than those prior to the Bronze Age.
Mobile warrior people

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The re-writing of the genetic map began in the early Bronze Age, about 5,000 years ago. From the steppes in the
Caucasus, the Yamnaya Culture migrated principally westward into North and Central Europe, and to a lesser degree, into
western Siberia. Yamnaya was characterized by a new system of family and property. In northern Europe the Yamnaya
mixed with the Stone Age people who inhabited this region and along the way established the Corded Ware Culture,
which genetically speaking resembles present-day Europeans living north of the Alps today.

15

Later, about 4,000 years ago, the Sintashta Culture evolved in the Caucasus. This culture’s sophisticated new weapons
and chariots were rapidly expanding across Europe. The area east of the Urals and far into Central Asia was colonized
around 3,800 years ago by the Andronovo Culture. The researchers’ investigation shows that this culture had a European
DNA background.
During the last part of the Bronze Age, and at the beginning of the Iron Age, East Asian peoples arrived in Central Asia.

20 Here it is not genetic admixture we see, but rather a replacement of genes. The European genes in the area disappear.

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These new results derive from DNA analyses of skeletons excavated across large areas of Europe and Central Asia, thus
enabling these crucial glimpses into the dynamics of the Bronze Age. In addition to the population movement insights, the
data also held other surprises. For example, contrary to the research team’s expectations, the data revealed that lactose tolerance rose to high frequency in Europeans, in comparison to prior belief that it evolved earlier in time (5,000–7,000 years ago).

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Passage 2 is excerpted from “European invasion: DNA reveals the origins of modern Europeans,” published in March 2015, by Alan
Cooper, a director at the Australian Centre for Ancient DNA at the University of Adelaide, and Wolfgang Haak, a senior research fellow
at the University of Adelaide.
PASSAGE 2
European Invasion: DNA Reveals the Origins of Modern Europeans
25

What we have found is that, in addition to the original European hunter-gatherers and a heavy dose of Near Eastern
farmers, we can now add a third major population: steppe pastoralists. These nomads appear to have “invaded” central
Europe in a previously unknown wave during the early Bronze Age (about 4,500 years ago).
This event saw the introduction of two very significant new technologies to western Europe: domestic horses and
the wheel. It also reveals the mysterious source for the Indo-European languages.

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30

The genetic results have answered a number of contentious and long-standing questions in European history. The
first big issue was whether the first farmers in Europe were hunter-gatherers who had learnt farming techniques from
neighbours in southeast Europe, or did they instead come from the Near East, where farming was invented.
The genetic results are clear: farming was introduced widely across Europe in one or two rapid waves around 8,000
years ago by populations from the Near East—effectively the very first skilled migrants.

35

At first the original hunter-gatherer populations appear to have retreated to the fringes of Europe: to Britain, Scandinavia and Finland. But the genetics show that within a few thousand years they had returned, and significant amounts
of hunter-gatherer genomic DNA was mixed in with the farmers 7,000 to 5,000 years ago across many parts of Europe.
Wheeling across Europe

But there was still a major outstanding mystery. Apart from these two groups, the genomic signals clearly showed
that a third—previously unsuspected—large contribution had been made sometime before the Iron Age, around 2,000
40 years ago. But by whom?
We have finally been able to identify the mystery culprit, using a clever new system invented by our colleagues at
Harvard University.
Instead of sequencing the entire genome from a very small number of well-preserved skeletons, we analysed 400,000
small genetic markers right across the genome. This made it possible to rapidly survey large numbers of skeletons from
45 all across Europe and Eurasia.
This process revealed the solution to the mystery. Our survey showed that skeletons of the Yamnaya culture from
the Russian/Ukrainian grasslands north of the Black Sea, buried in large mounds known as kurgans, turned out to be the
genetic source we were missing.
1.

How do the passages illustrate the contributions of DNA evidence to scientific inquiry?
A.

Both provide examples of how DNA evidence enabled scientists to fill in gaps in their knowledge about human
migrations.

B.

Both describe how DNA analysis is used in scientific investigations.

C.

The passages imply that DNA evidence can solve evolutionary questions.

D.

The passages show how scientists solved the mysteries of DNA evidence.

Chapter 3

Both passages illustrate how DNA has been used to answer questions about human migration patterns—questions that had
been unresolved before the ability to use DNA as evidence for such studies, choice A. Neither passage gives details about the
actual scientific methodology as both are focused on the results. Choices C and D are not correct interpretations of the passages.
The correct answer is choice A.
2.

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Which lines from the text support the idea that migration of human populations can be tracked through DNA testing?
A.

Line 1: (“With this … migrations.”)

B.

Lines 8–9: (“Genetically … the Bronze Age.”)

C.

Lines 26–27: (“These nomads … 4,500 years ago).”)

D.

Lines 28–29: (“This event … Indo-European languages.”)

This type of question requires that you find specific text to support an answer to a question. Although the concept given—that
DNA testing was used to track population migrations—can be found in both passages, you only need to review the specific lines
given in the answer choices to find the correct answer. Review only the options to find the one that that best supports the concept.

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Choices C and D are not directly related to DNA evidence. Choice A confirms the idea of migration, but the text does not refer
to DNA as the process that tracked the migration. Choice B refers to the genetic evidence, another term for DNA, which shows
a similarity between ancient and modern Europeans. This evidence illustrates that scientists have been tracking the migration
patterns. The correct answer is choice B.
3.

What is the best description of the two passages?
A.

They show conflicting claims about the migrations.

B.

They describe different scientific methodologies.

C.

They provide supplementary information.

D.

They are written from a different perspective.

When reading paired passages, compare the passages as you read. This question asks you to compare the information. On your
first reading, you probably noticed that they do not contradict one another, choice A; nor do they show different points of view,
choice D. Both passages describe scientific studies and what scientists were able to learn from them, and both describe using
DNA as the methodology, so you can eliminate choice B. But the second passage adds to the information in the first, making
them supplements, choice C, to one another. The correct answer is choice C.
4.

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Based on information in the two passages, which of the following statements could be made about scientific inquiry?
A.

DNA evidence showed why many Europeans are lactose intolerant.

B.

Research can provide historical information.

C.

Evidence showed that the steppe pastoralists introduced horses and the wheel to Western Europe.

D.

Scientific investigation can provide evidence about human history not obtainable through other means.

Like the preceding question, this one asks you to compare the two passages, except this question looks at a topic not directly
discussed in the passages but one that is implicit within it—the nature of scientific inquiry. So while choices A and C are true,
neither one answers the question. Choice B is also true, but it doesn’t address the nature of scientific investigations; it is simply
a general statement that could apply to many texts. Choice D, however, states a fact about scientific inquiry that can be gleaned
from the text: Both passages describe how scientists were able to use DNA data to answer questions they were unable to address
before the use of DNA testing was available. The correct answer is choice D.

NOTE: The paired passages have the same types of questions as the single passages. Some
of the questions, however, may ask you to synthesize the information presented in the two
passages by noting commonalities or comparing them in some way.

TIPS FOR TAKING THE READING TEST
You will be allotted 65 minutes to answer the 52 questions in the Reading section. That’s a lot of questions in a short period of
time. However, you can use some specific strategies and techniques to move through this portion of the SAT® efficiently. Check
out these strategies for answering the evidence-based reading questions quickly and accurately.

Answer All of the Questions Before You Start the Next Passage
There won’t be time to go back to reread the passages and recheck your answers, so answer every question that you can about
the passage. If you don’t know an answer, skip the question and return to it when you have answered the other questions in

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the passage. Check your time, and if you think you can answer one of the skipped questions with a quick reread of part of the
passage, go ahead. If not, or if you find it is taking too long, just give it your best guess and then move on. Remember, wrong
answers are not held against you, so don’t leave anything blank. Make sure you have answered every question for each passage
before you move on to the next one.

Remember That the Questions Get More Specific
The question order often holds a key to understanding a passage. The SAT® Reading Test organizes the questions from broader
questions about themes, purpose, point of view, and main ideas to more specific questions about explicit and implicit meaning,
specific language, and structure of the text. Review the questions and take note of the information they ask for before reading
the passage. As you read, underline or make notes to highlight text that may answer a question. Remember, answers will be in
the text—either stated or implied.

Paired Passages
For the paired passages, look for the characteristics in each passage that tie them together. Skim the questions so you know
what to look for and underline parts of the text that you may want to refer to. Try to form an overview of the two passages. Ask
yourself why they are paired: what do they have in common; how are they different. In answering the questions for this type of
passage, follow the same strategy: first answer all the questions you are fairly sure about. Then fill in the others based on best
guesses, and, if time allows, review the evidence in the text to support your guess and answer the question accordingly. Make
sure you have answered all the questions, and then go to the next passage.

Don’t Panic When You Read an Unfamiliar Passage
The passages can be unfamiliar. In their attempt to be fair, the test-makers purposely choose a variety of passages. This helps
make sure that each test-taker can demonstrate his or her reading and analysis skills. Remember, you’re not being tested on your
knowledge of the topic but on how well you do the following:

•
•
•
•
•

Understand the author’s assumptions, point of view, and theme

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Determine how the author supports the main ideas in the text

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Determine how the author uses specific language to create mood or tone
Analyze the logical structure of the text
Analyze overall meaning as well as specific words and phrases in context of the passage

Remember That Everything You Need to Know Is Right There in Front of You
The introductory paragraph and the passage have all the information you’ll need to answer the questions. Even if the passage is
about the price of beans in Bulgaria or the genetic makeup of a wombat, don’t worry. It’s all right there on the page.

Start with the Passages That Interest You
A point is a point. It doesn’t matter if the point comes from answering correctly a question about a piece of fiction or a scientific
experiment. If the style and subject matter appeal to you, you will probably go through a passage more quickly and answer the
questions more easily. So before you start, quickly check the topics by skimming the titles and introductions. Start with the topics
that are most familiar or most interesting. Then work your way down to the ones that you think will be hardest. Make a notation
so you know which passages you have completed, and double check that your answers on the answer sheet correspond to the
correct question numbers.

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Highlight Important Information as You Read the Passages
It pays to be an active reader, and making quick notes is a part of this process. Let the questions guide your notations. Based on
the questions, actively use your pencil to look for and bracket the text evidence that addresses questions about the big ideas,
themes, and purpose as you read. Use other markings to indicate places in the text that may provide evidence for other questions,
for example, a specific word or phrase referred to in a question. The questions will be related to the most important information in
the passage. If you’ve highlighted those pieces of information, you’ll be able to find them more easily to help answer the questions.

Don’t Get Bogged Down in the Details
Remember, you don’t have to understand every bit of information. You just have to find the information you need to answer the
questions. Don’t waste your time trying to analyze technical details or information not related to a question.

Don’t Confuse a “True” Answer with a “Correct” Answer
The fact that an answer choice is true doesn’t mean it’s right. What does that mean? It means that a certain answer choice may be
perfectly true—in fact, all of the answer choices may be true. But the right answer must be the correct answer to the question that’s
being asked. Only one of the answer choices will be correct and, therefore, the right choice. Read carefully—and don’t be fooled!

STRATEGIES FOR ANSWERING SPECIFIC QUESTION TYPES
As you learned earlier, each reading question asks you to do one of the following:

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•
•
•
•
•
•

Determine central themes and ideas as presented in the passage.
Determine the author’s purpose or point of view of the text.
Cite textual evidence to support inferences, conclusions, or arguments.
Cite evidence to illustrate or support interpretation of meaning, mood, or tone of the passage.
Analyze words and phrases in the context of the passage.
Analyze quantitative information in an accompanying graphic, such as a table, graph, or chart.

The following tips present strategies for dealing with specific types of SAT® reading questions. You should use these strategies
in combination with the basic steps for approaching the reading test. These strategies don’t take the place of the basic steps but
are extra tools to help you with certain types of questions. These tools work with both single and paired passages on any topic.
1. A broad question about the passage requires you to think about the passage as a whole. Keep this in mind as you read
the passage, noting any references to an overview in the introduction and/or conclusion of the passage.
2. Before you begin, note the subject of the passage and the voice/perspective. These items can often be found in the
introduction. Ask yourself if the author is providing information about a specific topic, making an argument for or
against something, or telling a story. Whether fiction or nonfiction, ask yourself who is telling the story and why.
3. Look for evidence of inferences in the text and evaluate how they support the author’s ideas.
4. As you read, notice the overall tone and mood of the passage and find textual evidence—specific words or phrases—
that contribute to these elements.
5. If a question asks about a specific word, read the surrounding text to verify the context in which the word is used.
6. Examine the graphic illustration specifically in response to the question associated with it. If it asks to apply the illustration to the text, read that portion of the text in conjunction with the graphic illustration.
To help you see how these tips work, read the passage in the exercise on the next page. Then read each tip and try to use it to help
you answer each question before you read the answer explanation. Note, again, that on the real exam and in the practice tests
in this book, reading passages will appear in two columns. For instructional purposes, the following passages appear full page.

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EXERCISE: EVIDENCE-BASED READING TEST
30 Minutes—23 Questions
DIRECTIONS: Each passage or pair of passages below is followed by a number of questions. After reading each passage
or pair, choose the best answer to each question based on what is stated or implied in the passage or passages and in any
accompanying graphics (such as a table or graph).

Questions 1–6 are based on the following passage.
The passage below discusses how Alaska Native cultural practices and heritage are being preserved in the twenty-first century.
(For the full article, please visit http://nps.gov.)
Alaska Native cultural practices continue to be a central force in virtually all villages throughout Alaska. In order to
maintain cultural knowledge and ensure its survival, Alaska Native people need to learn the best methods of recording
and archiving music, dance, and oral history. Along with the expansion of Europeans and Americans into Alaska were
Line accompanying hardships for the indigenous people: epidemic diseases, strong Christian missionary activities, and western
5 educational policies such as English language-only rules. These resulted in decimated populations throughout the entire
territory of Alaska, a decline in indigenous languages, and, in many cases, the abolishment of traditional religion and
associated music and dance repertoires.
Native people are deeply spiritual people; historically, they had a rich ceremonial life that was profoundly expressed
through music and dance—core means by which people communicate their identities and beliefs. With the introduction
10 of Christianity, traditional cultures, including aspects such as music and dance, were not viewed favorably by the missionaries. Sadly, most of the missionaries did not tolerate masked dancing and other forms of religious expressions. Dance,
language, and ceremonial practices either had to be practiced in secret, or were lost.
In the 1960s, during the Native Solidarity Movement, as Alaska Native people became more politically active, their
re-identification with their cultures, languages, music, and dance became a banner of their newfound political and social
15 strength. One of the major outcomes of that movement has been a renaissance in traditional music and dance practices,
resulting in multiple dance festivals and younger people becoming actively involved in their village dance groups. …
The Fifth Annual Kingikmiut Dance Festival featured a large Russian dance group, as well as the Tikigaq Traditional
Dancers of Point Hope and dance groups from Brevig Mission and other villages on the Seward Peninsula. Kingikmiut,
or Wales, was once known as the dance capital of the Seward Peninsula. Captain Henry Trollope visited Wales in 1853–54
20 and wrote “… the place is sort of a capital in these parts and has four dancing houses, which is a very expressive manner
of estimating the extent and population for a place.” (Ray 1975) Because of its strategic location, Kingikmiut flourished.
Before the 1900 and 1917 epidemics, it consisted of two related villages and consolidated into one village once the populations had been decimated by disease. After these terrible epidemics, western educators’ English-only policies forced
music, dance, and other expressions of traditional Native culture to go underground.

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Repression of Native culture by western educators and missionaries was common all over Alaska and is a major
reason why many Alaska Native languages are threatened today. In the first part of the twentieth century, traditional
dance and music became associated with the old ways and were looked down upon. After the 1960s, a strong revitalization movement arose. Today there is a renaissance in traditional music and dance practices. In Wales and other Seward
Peninsula communities, the younger people, who make up a large percentage of the population, have a great thirst for
30 learning to sing and dance their traditional songs.
25

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1.

2.

3.

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4.

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5.

6.

Which of the following best reflects the main idea of the passage?
A.

There has been a recent surge in interest in Native cultures everywhere.

B.

Native Alaskans have always tried to preserve their history and culture.

C.

Western expansion into Alaska resulted in loss of native cultures.

D.

Native Alaskan traditions were revived in the 1960s.

What can you infer about how the U.S. civil rights movement of the 1960s might have influenced the Native Solidarity
Movement?
A.

It was a catalyst for the Native Solidarity Movement, which provided an avenue for political activity and renewed
cultural identity.

B.

It led to new laws that give the Native Alaskan people the right to practice their religion and express their native
culture.

C.

It allowed the Native Alaskan people to rebel against the forced repression of their culture.

D.

It revived Native Alaskan cultural practices that had been lost.

What was the relationship between the missionaries in Alaska and the Native Alaskans?
A.

The Native Alaskans and missionaries lived in harmony because they had the same religious beliefs.

B.

The missionaries embraced the Native Alaskans and rescued them from epidemics of disease.

C.

The Native Alaskans resented the missionaries and tried to make them leave.

D.

The missionaries repressed native culture and the Native Alaskans rebelled by holding on to their culture.

What evidence does the passage provide to suggest that Alaskan culture was influenced by Russia?
A.

Lines 3–4: (“Along with … indigenous people”)

B.

Line 17: (“The Fifth Annual … dance group”)

C.

Lines 18–19: (“Kingikmiut, or Wales, … Seward Peninsula.”)

D.

Line 21: (“Because of … Kingikmiut flourished.”)

What conclusion could you infer from the passage?
A.

Native Alaskan cultures are still thriving.

B.

Cultural ties are very strong and can’t be easily extinguished.

C.

Many minor native languages are being lost.

D.

Music and dance is a common way to express one’s culture.

As used in line 24, “go underground” most nearly means
A.

hide in a hut.

B.

be practiced in an ice hut.

C.

practice in secret.

D.

be wiped out.

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Questions 7–11 are based on the following passage.
The following is an excerpt from “The Purloined Letter,” a short story by Edgar Allan Poe. Poe is best known for writing poems and stories
in the horror and mystery genre. However, he is also considered the inventor of detective fiction. The Mystery Writers of America call
their awards for excellence in the genre “Edgars,” in honor of Poe.
The Purloined Letter
… “Well, then; I have received personal information, from a very high quarter, that a certain document of the last
importance, has been purloined from the royal apartments. The individual who purloined it is known; this beyond a
doubt; he was seen to take it. It is known, also, that it still remains in his possession.”
Line
5

“How is this known?” asked Dupin.
“It is clearly inferred,” replied the Prefect, “from the nature of the document, and from the non-appearance of certain
results which would at once arise from its passing out of the robber’s possession; that is to say, from his employing it as
he must design in the end to employ it.”
“Be a little more explicit,” I said.

“Well, I may venture so far as to say that the paper gives its holder a certain power in a certain quarter where such
10 power is immensely valuable.” The Prefect was fond of the cant of diplomacy.
“Still I do not quite understand,” said Dupin.
“No? Well; the disclosure of the document to a third person, who shall be nameless, would bring in question the
honor of a personage of most exalted station; and this fact gives the holder of the document an ascendancy over the
illustrious personage whose honor and peace are so jeopardized.”
15

“But this ascendancy,” I interposed, “would depend upon the robber’s knowledge of the loser’s knowledge of the
robber. Who would dare—”

“The thief,” said G., “is the Minister D—, who dares all things, those unbecoming as well as those becoming a man.
The method of the theft was not less ingenious than bold. The document in question—a letter, to be frank—had been
received by the personage robbed while alone in the royal boudoir. During its perusal she was suddenly interrupted
20 by the entrance of the other exalted personage from whom especially it was her wish to conceal it. After a hurried and
vain endeavor to thrust it in a drawer, she was forced to place it, open as it was, upon a table. The address, however, was
uppermost, and, the contents thus unexposed, the letter escaped notice. At this juncture enters the Minister D—. His lynx
eye immediately perceives the paper, recognises the handwriting of the address, observes the confusion of the personage
addressed, and fathoms her secret. After some business transactions, hurried through in his ordinary manner, he produces
25 a letter somewhat similar to the one in question, opens it, pretends to read it, and then places it in close juxtaposition to
the other. Again he converses, for some fifteen minutes, upon the public affairs. At length, in taking leave, he takes also
from the table the letter to which he had no claim. Its rightful owner saw, but, of course, dared not call attention to the
act, in the presence of the third personage who stood at her elbow. The minister decamped; leaving his own letter—one
of no importance—upon the table.” …
30

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“You looked among D—’s papers, of course, and into the books of the library?”

“Certainly; we opened every package and parcel; we not only opened every book, but we turned over every leaf in
each volume, not contenting ourselves with a mere shake, according to the fashion of some of our police officers. We
also measured the thickness of every book-cover, with the most accurate ad measurement, and applied to each the most
jealous scrutiny of the microscope. Had any of the bindings been recently meddled with, it would have been utterly
35 impossible that the fact should have escaped observation. Some five or six volumes, just from the hands of the binder,
we carefully probed, longitudinally, with the needles.”
“You explored the floors beneath the carpets?”

Peterson’s SAT® Prep Guide 2017

“Beyond doubt. We removed every carpet, and examined the boards with the microscope.”
“And the paper on the walls?”
“Yes.”

40

“You looked into the cellars?”
“We did.”
“Then,” I said, “you have been making a miscalculation, and the letter is not upon the premises, as you suppose.”
“I fear you are right there,” said the Prefect. “And now, Dupin, what would you advise me to do?”
“To make a thorough re-search of the premises.”

45

“That is absolutely needless,” replied G—. “I am not more sure that I breathe than I am that the letter is not at the Hotel.”

7.

8.

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9.

Which of the following best describes the mystery to be solved in “The Purloined Letter”?
A.

What information is in the letter?

B.

What will happen if the letter is exposed?

C.

Where is the letter?

D.

Who has the letter?

Who is being blackmailed, and by whom?
A.

The royal lady is being blackmailed by the Minister D.

B.

The Minister D is being blackmailed by the Prefect G.

C.

The Prefect G is being blackmailed by Dupin.

D.

The royal lady is being blackmailed by the Prefect G.

What is the most likely reason that Dupin asks the Prefect to describe the search in such detail?
A.

To make sure the police haven’t missed anything

B.

To look for other clues

C.

To hear a good story

D.

To verify his own ideas

10. As used in line 10, “cant” most nearly means
A.

inclination.

B.

jargon.

C.

music.

D.

inability.

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11. Which of the following is a line that could be in the letter?
A.

It would be such an honor to have you in attendance at this surprise gala.

B.

You know how much I would like to be in your employ again.

C.

I’m so sorry that this is late notice, but can I meet with you and the prince next week?

D.

I beg of you to meet me on Monday when the prince is in Monaco.

Questions 12–17 are based on the following passage and supplementary material.
This passage is the introduction to a report from the Office of the Chief Technologist at the U.S. space agency NASA, entitled “Emerging
Space: The Evolving Landscape of the 21st Century American Spaceflight”
(http://www.nasa.gov/sites/default/files/files/Emerging_Space_Report.pdf).
America stands today at the opening of a second Space Age. Innovative NASA programs and American entrepreneurs
together are transforming the space industry. These initiatives—both at NASA and in the private sector—are expanding
the nation’s opportunities for exploration and for the economic development of the solar system.
Line

Today’s space economy extends some 36,000 kilometers (22,369 miles) from the surface of the Earth and includes an
5 array of evidence-based technologies—satellite communications, global positioning satellites, and imaging satellites—on
which our economy depends. These technologies are now an integral part of our economy, and they would not exist
if not for the over 50 years of research, development, and investment in the enabling technologies by NASA and other
government agencies that seeded these efforts and allowed them to bloom. As we expand our activities in the solar
system over the next decades, NASA programs and investments will provide the seed and soil that encourage economic
10 development increasingly farther from Earth. The first signs of this are already visible.
The next era of space exploration will see governments pushing technological development and the American
private sector using these technologies as they expand their economic activities to new worlds. NASA’s next objectives
for exploration—visits to asteroids and Mars— are more complex than any previous space mission attempted. They will
happen in the context of relatively smaller NASA budgets and an expanding commercial space economy. Teaming with
15 private-sector partners to develop keystone markets like low Earth orbit (LEO) transportation and technological capabilities like asteroid mining will help NASA achieve its mission goals, help the space economy evolve to embrace new
ambitions, and provide large economic returns to the taxpayer through the stimulation and growth of new businesses
and 21st-century American jobs.

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Motivated by an intrinsic desire to explore space, successful American entrepreneurs have pledged and spent
20 hundreds of millions of dollars to develop technologies aimed at fundamentally improving space access. Since 2003,

commercial human spaceflight has received $2.5 billion in private investment.1 At the same time, a new generation of
space enthusiasts are engaging directly though small-scale projects. Through cubesats, suborbital and orbital adventures,
and citizen science opportunities, the United States is transitioning from a spacefaring nation to a nation of spacefarers.
In addition to executing its scientific and human spaceflight programs, NASA also has a legislated responsibility to
25 “encourage, to the maximum extent possible, the fullest commercial use of space.” As part of fulfilling this responsibility,

this report examines how NASA has collaborated with American private-sector individuals and companies investing in
space exploration, collectively known as “emerging space.” Today, more than fifty years after the creation of NASA, our
goal is no longer just to reach a destination. Our goal is to develop the capabilities that will allow the American people to
explore and expand our economic sphere into the solar system. Although when NASA was founded only a government
30 program could undertake a voyage from the Earth to the Moon, this may not be true in the future. By taking full advantage
of the combined talents of government and the American private sector, our next journeys beyond Earth will come sooner
and we will catalyze new industries and economic growth in the process.

1. 2013 Commercial Spaceflight Industry Indicators, Commercial Spaceflight Federation

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NASA provides a number of resources for people willing to contribute as citizen scientists. All are available online, meaning all you need
is access to the Internet. Software tools are also provided. More than 1.2 million people from 80 countries have participated in NASA’s
citizen science projects. This table captures just a few of the projects.

Project

Citizen Scientist Role

Number of Participants

Be a Martian

Tag rover images and map craters
from satellite pictures

1,230,000

HiTranslate

Help translate NASA’s HiRISE project
captions into different languages

1,021 new in 2012

International Space
Apps Challenge

Develop mobile applications,
software, hardware, data
visualization, and platforms to
address current challenges relevant
to space exploration and social
need

2,083 from 17 countries in
2012

Lunar Impacts

Independent observers can
monitor the rates and sizes of large
meteoroids striking the far side of
the Moon

26 impact candidates

Rock Around the World

Help Mars scientists better
understand the red planet by
sending rocks to NASA for analysis

12,461 rocks received

Stardust at Home

Search for the first samples of
solid matter from outside the solar
system

30,649 from 2006 to 2012

Target Asteroids!

Observe asteroids, to help scientists
refine orbits and determine the
composition of near-Earth objects
(NEOs) in support of the OSIRIS-Rex
mission

104 registered users from 23
countries

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12. How did the government investment in space boost the economy?
A.

It expanded the knowledge about space beyond our solar system.

B.

It led to the development of popular commercial products.

C.

It led to human spaceflight programs, which required specialized products.

D.

It expanded space exploration farther from the Earth.

13. How will the next era of space exploration be different from the past?
A.

It will include private and public investment.

B.

It will require citizen participation.

C.

It will send people to explore and perhaps live on other planets.

D.

It will be much more expensive.

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14. What is meant by: “the United States is transitioning from a spacefaring nation to a nation of spacefarers.” (line 23)?
A.

The United States will no longer be able to use space for warfare.

B.

People will be more interested in space exploration.

C.

The United States will be able to dominate space exploration.

D.

People will be able to travel in space.

15. Which of the following is the best summary of the passage?
A.

NASA is ushering in a new chapter in the space age that will allow the commercialization of space.

B.

In the next decades, ordinary people will be able to travel in space.

C.

NASA is the U.S. agency that is charged with developing space programs.

D.

The Second Space Age will expand the economy and increase complex technologies in space exploration.

Questions 16 and 17 refer to the table.

16. What is the most likely reason that the “Be a Martian” project has so many more participants than the other projects?
A.

More people are interested in exploring Mars than in other aspects of the program.

B.

Other projects require more sophisticated/complex technology skills.

C.

It is the only project that is designed for young people.

D.

The other projects all require more time as a volunteer.

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17. What skills are needed to participate in the NASA projects as a citizen-scientist?
A.

The ability to identify different kinds of rocks

B.

The ability to use a telescope

C.

The ability to use software technology and the Internet

D.

The ability to locate and recognize the planets and other celestial bodies

Chapter 3
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Questions 18–23 are based on the following two passages.

The following passages are excerpted from narratives written by two explorers. Passage 1 is by Sir Earnest Shackleton from an account
he wrote entitled: South! The Story of Shackleton’s Last Expedition (1914–1917). Passage 2 is part of an account by Hiram Bingham
III, from “The Discovery of Machu Picchu,” first published in Harper’s Monthly magazine in 1913.
PASSAGE 1
Some intangible feeling of uneasiness made me leave my tent about 11 p.m. that night and glance around the
quiet camp. The stars between the snow-flurries showed that the floe had swung round and was end on to the swell, a
position exposing it to sudden strains. I started to walk across the floe in order to warn the watchman to look carefully
Line for cracks, and as I was passing the men’s tent the floe lifted on the crest of a swell and cracked right under my feet. The
5 men were in one of the dome-shaped tents, and it began to stretch apart as the ice opened. A muffled sound, suggestive
of suffocation, came from beneath the stretching tent. I rushed forward, helped some emerging men from under the
canvas, and called out, “Are you all right?”

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“There are two in the water,” somebody answered. The crack had widened to about four feet, and as I threw myself
down at the edge, I saw a whitish object floating in the water. It was a sleeping-bag with a man inside. I was able to
10 grasp it, and with a heave lifted man and bag on to the floe. A few seconds later the ice-edges came together again with
tremendous force. Fortunately, there had been but one man in the water, or the incident might have been a tragedy. The
rescued bag contained Holness, who was wet down to the waist but otherwise unscathed. The crack was now opening
again. The James Caird and my tent were on one side of the opening and the remaining two boats and the rest of the
camp on the other side. With two or three men to help me I struck my tent; then all hands manned the painter and rushed
15 the James Caird across the opening crack. We held to the rope while, one by one, the men left on our side of the floe
jumped the channel or scrambled over by means of the boat. Finally I was left alone. The night had swallowed all the
others and the rapid movement of the ice forced me to let go the painter. For a moment I felt that my piece of rocking
floe was the loneliest place in the world. Peering into the darkness; I could just see the dark figures on the other floe.
PASSAGE 2
Nor was I in a great hurry to move. The water was cool, the wooden bench, covered with a woolen poncho, seemed
20 most comfortable, and the view was marvelous. On both sides tremendous precipices fell away to the white rapids of

the Urubamba River below. In front was the solitary peak of Huay-na Picchu, seemingly inaccessible on all sides. Behind
us were rocky heights and impassable cliffs. Down the face of one precipice the Indians had made a perilous path, which
was their only means of egress in the wet season, when the bridge over which we had come would be washed away. Of
the other precipice we had already had a taste. We were not surprised to hear the Indians say they only went away from
25 home about once a month.

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Leaving the huts, we climbed still farther up the ridge. Around a slight promontory the character of the stone-faced
andenes began to improve, and suddenly we found ourselves in the midst of a jungle-covered maze of small and large
walls, the ruins of buildings made of blocks of white granite, most carefully cut and beautifully fitted together without
cement. Surprise followed surprise until there came the realization that we were in the midst of as wonderful ruins as any
30 ever found in Peru. It seemed almost incredible that this city, only five days’ journey from Cuzco, should have remained
so long undescribed and comparatively unknown. Yet so far as I have been able to discover, there is no reference in the
Spanish chronicles to Machu Picchu. It is possible that not even the conquistadors ever saw this wonderful place. From
some rude scrawls on the stones of a temple we learned that it was visited in 1902 by one Lizarraga, a local muleteer.
It must have been known long before that, because, as we said above, Wiener [an Austrian-French explorer], who was
35 in Ollantaytambo in the 70’s, speaks of having heard of ruins at a place named “Matcho Picchu,” which he did not find.
18. How do the two passages differ in their tone?
A.

Passage 1 has a tone of surprise, and Passage 2 has a tone of fear.

B.

Passage 1 has a threatening tone, and Passage 2 has an adventurous tone.

C.

Passage 1 exhibits a tone of danger, and Passage 2, a tone of awe.

D.

Passage 1 features a gloomy tone, and Passage 2, a festive tone.

19. What does the passage reveal about Shackleton’s character?
A.

He is fearful for himself and others.

B.

He is a risk-taker who likes adventure.

C.

He is a heroic man who likes attention.

D.

He is a leader who puts others first.

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20. Which of the following best describes a purpose common to both passages?
A.

They want to tell the world about the dangers of undeveloped places.

B.

They want to tell the world about new and undiscovered places.

C.

They want to persuade others to visit undeveloped places.

D.

They want to persuade people to help develop the locales described.

21. As used in line 12, “unscathed” most nearly means
A.

fearless.

B.

not whole.

C.

not harmed .

D.

harmless.

22. What idea in the text leads to the conclusion that Bingham thought he discovered Machu Pichu?
A.

It would take five days to get there from the closest city of Cuzco.

B.

The Native Indians didn’t seem to know about it.

C.

He dismisses the others who may have been there.

D.

It was hidden behind thick jungle growth.

23. Based on the two passages, what can be inferred about the nature of expeditions?
A.

They are often dangerous and can’t be undertaken alone.

B.

They usually add to the knowledge about a place.

C.

They are good travel destinations for people who enjoy going to little-known places.

D.

They represent undiscovered parts of the world.

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ANSWER KEY AND EXPLANATIONS
1. C

6. C

11. D

16. B

20. B

2. A

7. C

12. B

17. C

21. C

3. D

8. A

13. A

18. C

22. C

4. B

9. D

14. D

19. D

23. B

5. B

10. B

15. D

1. The correct answer is C. The first paragraph in the passage provides a summary. It explicitly states the reasons
that the Native Alaskan culture was almost wiped out and
what needs to be done to ensure that it is not lost forever.
The rest of the passage supports the concepts introduced
here.

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2. The correct answer is A. Lines 13–15 explain that the
Native Solidarity Movement led to increased political
activity. Although the passage doesn’t mention the civil
rights movement by name, the question identifies its era,
and the passage states that the Native Solidarity Movement
occurred during the same time period. A logical inference,
therefore, could be that the civil rights movement had some
impact on the Native American Solidarity Movement. There
is no information about laws that might have affected
Native Alaskans, choice B. The passage does not indicate
that the Native Alaskans rebelled, choice C. The passage
does indicate that the cultural practices have been revived,
but the text does not connect the revival, even indirectly,
to the civil rights movement.
3. The correct answer is D. The passage does not directly
state the relationship of the Native Alaskan people to the
missionaries, but it does describe how the missionaries
imposed Christianity on them and tried to force the Native
Alaskans to abandon their own belief systems. There is no
indication of living in harmony, choice A, nor of the missionaries rescuing them in the face of epidemic disease, choice
B. Although the passage suggests that Native Alaskans
may have had reason to resent the missionaries, there is
no indication that they tried to make them leave, choice C.
4. The correct answer is B. The presence of a Russian dance
group among other native groups indicates that the Russian
culture had been mixed with that of the Native Alaskan
groups. Choice A does not explain any influences on culture.
There is no information that explains the proximity of the
Seward Peninsula to Russia, choice C. Kingikmuit is located
on the Seward Peninsula, as described in the passage, but,

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as in choice C, the passage does not explain its geographic
relationship to Russia, choice D.
5. The correct answer is B. The main idea of the text shows
that although missionaries tried to impose their beliefs on
the Native Alaskans, they were not completely successful.
The Native Alaskans maintained their belief systems and
passed on their traditions in secret when it was not possible to be open about them. Today, these traditions are no
longer repressed and are thriving. Overall, this suggests
the conclusion in choice B. Although choice A is true, it is
neither a conclusion nor an inference; that idea is stated in
the text. Both answer choices C and D are general concepts
that could be applied to the Native Alaskan culture, but
nothing in the text suggests such a generality.
6. The correct answer is C. The idiom “go underground”
means doing something in secret. In this case, the western educators who had come to Alaska forced their own
culture onto the Native Alaskans, who then practiced their
traditions in secret rather than give them up.
7. The correct answer is C. The conversation in the excerpt
reveals, in general terms, the responses listed in the other
choices A, B, and D. The mystery to be solved by either the
Prefect or Dupin is in locating the letter, choice C.
8. The correct answer is A. It is the royal lady who has told
the Prefect the story, which he retells to Dupin and the
narrator (line 1: “I have received personal information,
from a very high quarter.”). His retelling also describes the
identity of the thief (minister D), and the importance of
the document (line 9–10: “that the paper gives its holder
a certain power in a certain quarter where such power is
immensely valuable.”).
9. The correct answer is D. We can infer from his advice (line
45: “To make a thorough re-search of the premises.”) that
Dupin has an idea about the whereabouts of the letter,
which he wants to confirm through the details of the search.
Dupin listens carefully, demonstrating that he is interested

in the specifics of the investigation. The details confirm his
own ideas.
10. The correct answer is B. The word “cant” is used in a statement made by the unnamed narrator. He is commenting
about the language of the Prefect’s statement.“Jargon”(the
language or idiom of a particular group) is the only choice
related to language.
11. The correct answer is D. We know that the royal lady is
being blackmailed. Therefore, the contents must reveal
something she is hiding from the man in the room with
her, most likely her husband. Choice D is the only one that
contains information that is to be kept from the prince.
12. The correct answer is B. The passage describes the space
economy and the many products that were developed
because of space exploration and then applied to commercial markets. Lines 6–8 explain that these products
are now part of our economy. The space program was
responsible for introducing new products to the market
and new products that people want and buy increase
economic activity.
13. The correct answer is A. The passage discusses the inclusion of private investment (lines 11–12). Commercial
companies are already gearing up to provide new space
services based on the ability to travel in space and explore
places beyond Earth. Formerly, space travel was only done
by NASA.
14. The correct answer is D. People have always been interested in space but the passage talks about how the next
phase of space exploration will include ordinary citizens
who will be able to take advantage of the commercialization
of space exploration and travel.
15. The correct answer is D. This statement represents the
best summary because it states the main idea by providing
the two most important aspects of the commercialization
of space exploration.
16. The correct answer is B. The table shows that the “Be a
Martian” project involves tagging on the Internet, and the
table caption explains that participants are provided with
any needed software. According to the descriptions of the
other projects, they all require some additional skill, for
example, translating text or doing more complex searches.

18. The correct answer is C. The first line of Passage 1 sets
the tone of danger (feeling of uneasiness) that continues
after the crack in the ice. Although the men aren’t killed,
the path to the boat was nevertheless full of danger. Passage 2 begins with Bingham’s observations that show his
appreciation of the beauty of the place. He marvels at the
Indians who make their way down the steep precipice on
a regular basis (lines 22–23). As Bingham makes his way
through the jungle and sees Machu Picchu for the first
time, detailed descriptions show awe and surprise at the
“wonderful ruins” (line 29).
19. The correct answer is D. Shackleton describes the scene
and how he warned to “look carefully for cracks” (lines 3–4),
which shows he was checking the ice to make sure he and
his men were safe. When he saw the ice cracking, his first
instinct was to throw himself down at the edge (lines 8–9)
and save one of the men. He made sure the others were
safe, leading them back to the boat; he was the last in line
to board.
20. The correct answer is B. Both passages record the experiences of men who are drawn to exploring places that most
people don’t ever see. Both use detailed descriptions of
the physical terrain of remote places that are void of other
people. There is no call inviting others to join them—just
descriptions that inform.
21. The correct answer is C. The surrounding text describes
the condition of the man (who was wet down to the waist),
and we know from the rest of the text that the water was
bitter cold. The word “otherwise” shows that what follows
is an exception, so he is—not harmed—except for being
wet (and presumably cold).
22. The correct answer is C. In lines 32–35, Bingham describes
the others who may have come before him. He remarks on
the beauty of the ancient city and how incredible it is that
others missed it. Surely the conquistadors would have told
others about it; Weiner never found it; and the muleteer
didn’t understand what he was looking at.

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23. The correct answer is B. Both passages describe places
about which little was known at the time. Both men undertook the trip and wrote detailed accounts of their experiences so that others could learn about them and perhaps
follow their footsteps to gain even more knowledge.

17. The correct answer is C. Reading the descriptions, the skill
required in all of the projects is how to use the Internet to
conduct searches. Even for projects that involve language
translation, a participant would have to access the Internet
to get the text that needs to be translated.

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SUMMING IT UP
• You do not need information beyond what is provided by the passage to answer any question.
• Complete familiar and interesting passages first.
• Read the introduction; it often provides background that will help form an overview of the passage.
• Read the questions and their answer choices.
• Read the passage as quickly as possible, marking the text to note information that may be relevant to the questions. Let the
questions drive your focus and the notations you make.

• Don’t get bogged down in details. Most questions will not be focused on details unless they are used to support a major idea
or theme.

• Go back into the passage to find answers as needed.
• Reading Test questions are ordered from those related to the central ideas and overall themes to those involving structural
and language-related concepts. Approach each passage by first looking for the big ideas and then the more structural
concepts.

• For any question you’re not sure of, eliminate obviously wrong answers and take your best guess. Wrong answers are not
counted against you, so answer every question.

• Answer every question for a passage before starting the next passage.

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• Advanced Question Types
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• Analyzing Structure
• Common Question Types
• Hard Prose Passages
• Hard Science Passages
• Other Reading Questions
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• Point of View
• Quantitative Information
• Reading Actively
• Understanding Difficult Text
• Wrong Answer Types
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part iv: writing
strategies for the sat ®

Chapter 4: Writing and Language Test Strategies
Chapter 5: English Language Conventions

Chapter 4:
Writing and Language
Test Strategies
OVERVIEW
A Closer Look at the Writing and Language Test
The Three Most Common Multiple-Choice Editing
Questions
Expression of Ideas Questions: Words in Context
Expression of Ideas Questions: Adding or
Deleting Text
Expression of Ideas Questions: Reordering
Sentences
Expression of Ideas Questions: Combining
Sentences and Using Transitional Words and
Phrases Correctly
Graphic Organizer Questions
Exercise: Writing and Language Test
Answer Keys and Explanations
Summing It Up

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A CLOSER LOOK AT THE WRITING AND LANGUAGE TEST
The Writing and Language Test gives you opportunities to demonstrate your college and career readiness by revising and editing
four passages. Each passage is 400–450 words long. There are 11 multiple-choice questions about each one—a total of 44
questions in all. You have 35 minutes to complete this section; that is about 48 seconds per question. If that doesn’t sound like
enough time to you, be assured that you will be able to answer many of the questions in fewer than 48 seconds. You can save up
your extra seconds for the harder questions you’ll encounter.
One passage of the four will be career-related; for example, this chapter includes a passage on technical writing careers. Another
passage will be humanities-related; it might be about visual art, music, theater arts, or literature. For example, this chapter includes
a short critical essay on a nineteenth-century novel by British author Jane Austen. The other two passages will be about history/
social studies and science. Some of the passages will be accompanied by graphic organizers such as tables, charts, or graphs.
The passages’ writing modes will include argument, informative/explanatory text, and nonfiction narrative. For example, this
chapter’s sample passage on technical writing is an informative/explanatory text. The essay on Jane Austen that appears later in
this chapter includes both argument and nonfiction narrative.
Answering the multiple-choice questions on each passage will place you in an editor’s role. You will be revising and editing the
work of an unspecified writer. You will be asked to improve each passage’s development, organization, and use of language. Your

Peterson’s SAT® Prep Guide 2017

tasks will include making sure that each passage conforms to standard rules of English grammar, usage, and punctuation. When a
passage is accompanied by one or more graphic organizers, you may need to correct the passage’s inaccurate interpretation of data.
These editing and revising goals may sound overwhelming, but don’t worry. Every answer is right there on your test page. All you
have to do is to select one out of four possible solutions [A, B, C, or D] to choose the best use of language.

THE THREE MOST COMMON MULTIPLE-CHOICE EDITING QUESTIONS
The Writing and Language Test contains three primary categories of multiple-choice questions:

1. Expression of Ideas Questions
More than half of the questions fall into this category. This group includes questions about the following:

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•
•
•
•
•
•
•
•
•

Words in context
Adding, deleting, or changing text
Transitional language that smoothly and clearly takes the reader from one idea to another
Relevant (or irrelevant) details
Combining sentences to make text more concise and less “choppy-sounding”
Eliminating awkward language and wordiness
Reordering sentences so that paragraphs make better sense
Consistency of style and tone
Cohesion and precision of language

You will learn how to approach Expression of Ideas questions later in this chapter.

2. Standard English Conventions Questions
About 45 percent of the questions fall into this category of grammar, usage, and punctuation rules. It includes questions that
require you to demonstrate your knowledge of the following:

•
•
•
•

Consistent (or inconsistent) verb tenses
Punctuation
Sentence structure
Correct (or incorrect) word usage

The Exercise section of this chapter provides practice with answering some questions from this category. In Chapter 5, you will
learn more about grammar, usage, and punctuation questions.

3. Graphic Organizer Questions
This is the smallest percentage of the three primary categories. There may be only a few questions that deal with graphic organizers on the Writing and Language Test. This type of question asks you to make revising and editing decisions about passages
in light of information and ideas conveyed through graphic organizers such as tables, charts, and graphs. However, you will not
need to do any mathematical computation in order to answer the questions in this category. Later in this chapter, you will learn
how to approach Graphic Organizer questions.

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EXPRESSION OF IDEAS QUESTIONS: WORDS IN CONTEXT
Words-in-Context questions are perhaps the easiest type of question on the Writing and Language Test. Most of the questions
on the test, including words in context, do not have a question right before the answer choices. Instead, you should choose the
response that corrects the sentence or paragraph and makes the writing stronger. Review the example below.

Jane Austen and Fanny Price

4

In real life, one 1 was sometimes left out because of

A.

NO CHANGE

actual purposeful wrongdoing on the parts of those doing

B.

Fairy tales

the leaving out, but sometimes one is simply left out—and

C.

Stories

D.

Articles

2 realism is beside the point. 3 The second case is
much less satisfying to the left-out person, but it is also much
more usual. 4 Novels such as “Cinderella” are satisfying
because, in them, it is clearly the nastiness of the villains
and villainesses that causes the heroes and heroines to be
excluded from pleasurable activities.

First, quickly read the paragraph to get a general sense of its meaning—you don’t need to understand every word. Right now
you are answering Question 4 only, so don’t worry about any errors you might spot in the rest of the paragraph—you will deal
with those later in the “Exercise” portion of this chapter.
Question 4 asks you to decide which word best fits the context. Your four choices are:

157

A. NO CHANGE (Novels); B. Fairy tales; C. Stories; and D. Articles.
Use the context clue Cinderella to make the best choice. Right away you can eliminate choice D: “Cinderella” is not an article—
most articles are nonfiction. Choice C is a possibility: “Cinderella” is a story—set this choice aside for a moment. Is “Cinderella” a
novel? No—novels are long, and the story of Cinderella is fairly short. Eliminate choice A. Is Cinderella a fairytale? Yes, it is: it has
magical elements, and it is a traditional tale that has been passed down for many generations. Is Fairy tales a better answer than
Stories? Yes, it is, because this description is more precise. Fairy tales are a specific type of story. The correct answer is choice B.

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Let’s try a words-in-context question that goes with a different passage.

A Technical Writing Career

11

Technical 11 communicators, better known as “technical

A.

NO CHANGE

writers,” plan, write, and edit instruction manuals, print and

B.

engineers

online articles, and other documents that transform 12

C.

manuals

D.

workers

intense technical information into simpler language that end
users can understand.

Peterson’s SAT® Prep Guide 2017

Quickly read the one-sentence paragraph to figure out what it is mainly about. Again, focus on the question you’re answering
(Question 11).
Here are some clue words and phrases that will help you figure out which word belongs after the numeral 11 : writers; write,
and edit; and simpler language that . . . users can understand. To which answer choice do these clues direct you? You can eliminate
choice C, manuals: a manual is a thing (a set of instructions), not a person. Might engineers, choice B, and workers, choice D, be
writers who write and edit text?
Maybe, but the word communicators is the best choice because it is most precise—a writer’s main job is to communicate with
others by writing understandable sentences, paragraphs, and so on.
Since the word communicators already appears in the paragraph after the 11 , the correct answer is choice A, NO CHANGE.

If you are somewhat stuck, but not totally, make an educated guess. If you can
eliminate at least one answer choice that you know is wrong (eliminating two
is even better), you won’t be guessing at random. Remember that you will not
be penalized for wrong answers, so educated guessing is a fine strategy to use.

EXPRESSION OF IDEAS QUESTIONS: ADDING OR DELETING TEXT
Here is a question that asks you whether it is a good or bad idea to delete a certain sentence.
When I was in elementary school, I was a shy little girl. . . .

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The stories I made up always had the same plot: I was carried

5

The writer is considering deleting the underlined sentence. Should the writer do this?

off by a prince (David) on a flying horse to his castle where

A.

Yes, because it adds unnecessary technical details.

I became his wife and got to live in the lap of luxury. 5

B.

Yes, because it makes the grown-up writer seem
silly and childish.

C.

No, because it adds funny details and helps to
show that the narrator was a child at the time.

D.

No, because, without this sentence, the last
sentence in the paragraph would not make sense.

From there I would go into elaborate detail about the décor
of the castle, how many horses we owned, what colors
they were, and so on. It was a silly, childish fantasy, but it
comforted me.

This question truly casts you in an editor’s role. You’re asked whether the paragraph would be better or worse if this sentence
were deleted. Not only that, you’re asked why your choice is the correct one.
You can eliminate choice B. The underlined sentence doesn’t make the grown-up writer seem “silly and childish”—it makes her
subject (her younger self ) seem so. The writer explicitly states this in the last sentence of the paragraph—she is having fun making
fun of her younger self.
Also eliminate choice A: This sentence does contain details, but they are not technical. How about choice D? Would the paragraph’s
last sentence fail to make sense if the underlined sentence were deleted? Read the first two sentences and the last one without
the middle (underlined) one. The last sentence still makes sense, so you can eliminate choice D.
You are left with choice C. Even if you disagree that the underlined sentence is funny, it definitely “helps to show that the narrator
was a child at the time.” The correct answer is choice C.

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EXPRESSION OF IDEAS QUESTIONS: REORDERING SENTENCES
Next, let’s try a question that asks you to reorder sentences in a paragraph.

[1] A few years ago, U.S. government experts predicted that
employment of technical writers would grow 1.5 percent

16 For the sake of cohesion, sentence 4 should
be placed

from 2012 to 2022, a gain of about 7,400 jobs per year.

A.

where it is now.

[2] (This is faster than the average for all occupations on

B.

before sentence 1.

C.

after sentence 1.

D.

after sentence 2.

which the U.S. Department of Labor gathers statistics.) [3]
This causes a greater need for professionals with the talent
and skills to write instructional manuals that users can
easily understand and follow. 16 [4] The high-tech and
electronics industries continue to change and grow.

In this type of question, each sentence in a paragraph is numbered with a numeral in brackets. Your job is to decide whether a
specific sentence should stay where it is; move to a different spot in the paragraph; and if it should move, where it should go.
The best way to answer this question would be to read the paragraph out loud to yourself. During the SAT®, you won’t be able
to do that, but try it now if possible. You will probably notice that sentence 4 sounds odd at the end of the paragraph—it seems
“tacked on” somehow. So it needs to be moved, but where?
Try each option one at a time. You will discover that this sentence belongs right after sentence 2—before sentence 3. Sentence
3 says that “This causes a greater need . . . [for good technical writers].” What causes this need? The cause is the fact that “the
high-tech and electronics industries continue to change and grow.” If sentences 3 and 4 were to swap places, the paragraph would
be “most logical”—it would make better sense than it does now, for sure. The correct answer is choice D.

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If a particular question seems overwhelming, circle it on
your test booklet and go on to the next question. When
you’re done with all of the easier ones, come back to any
questions you circled.

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Peterson’s SAT® Prep Guide 2017

EXPRESSION OF IDEAS QUESTIONS: COMBINING SENTENCES AND
USING TRANSITIONAL WORDS AND PHRASES CORRECTLY
Here is a question that asks you to complete two tasks at once: combine two shorter sentences into one longer one and choose
the transitional word or phrase that makes the best sense in the new, longer sentence.
Most novel readers do not like it when story events seem

7

artificially “rigged” by the author in order to teach a moralistic

Which choice most effectively combines the two
sentences at the underlined portion?

lesson. In this novel, Jane Austen rigs 7 events. Whenever

A.

events, but whenever Fanny is left out,

Fanny is left out, we can be sure that one or some of the

B.

events so that whenever Fanny is left out,

C.

events, because, whenever Fanny is left out,

D.

events, and the cause is that whenever Fanny is
left out,

other characters are engaged in something sinful. This is an
unrealistic (fairytale) element in what is otherwise a highly
developed, realistic novel.

Again, the best way to solve this editing problem is to imagine reading the paragraph out loud. If you try one answer choice at a
time, you will soon find the one that makes the best sense. The writer is saying that Jane Austen heavy-handedly rigs events in
her novel in order that such and such happens. The transitional words and phrases in the other answer choices (A: but; C: because;
D: and the cause is that) do not relay that same idea; only so that communicates this thought clearly. The correct answer is choice B.

GRAPHIC ORGANIZER QUESTIONS

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Every Writing and Language Test contains one or more passages with graphic organizers such as tables, charts, and graphs. One
or more of the questions following such a passage deals with its organizers. This type of question asks you to compare information given in the passage to similar information or data that the graphic organizer presents. If the two sets of information are
inconsistent, you will need to make editing changes to the passage.
Here is an example:

A TECHNICAL WRITING CAREER
Quick Facts: Technical Writers
2012 Median Pay

$65,500 per year
$31.49 per hour

Entry-level Education

Bachelor’s degree

Work Experience in a Related Occupation

Less than 5 years

On-the-job Training

Short-term on-the-job training

Number of Jobs, 2012

49,500

Job Outlook, 2012–22

15% (faster than average)

Employment Change, 2012–22

7,400

[Source: http://www.bls.gov/ooh/Media-and-Communication/Technical-writers.htm]

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Technical communicators, better known as “technical
writers,” plan, write, and edit instruction manuals, print and

17 Which choice most accurately and effectively represents
the information in the chart?

online articles, and other documents. Requirements for this

A.

NO CHANGE

well-paid profession include 17 a bachelor’s degree and

B.

a bachelor’s degree and less than ten years of
on-the-job work experience in a related
occupation

C.

a master’s degree, less than five years of work
experience in a related occupation, and a short
period of on-the-job training

D.

a bachelor’s degree, less than five years of work
experience in a related occupation, and a short
period of on-the-job training

five to ten years of on-the-job training in a technical field.

To find the correct answer choice, you will need to pay close attention to details in the chart. In this case, carefully read the three
rows beginning with “Entry-level Education” and ending with “On-the-job Training.” Choice A is incorrect because the chart does
not specify “five to ten years of on-the-job training in a technical field.” Choice B is incorrect because the chart does not specify
“less than ten years of on-the-job work experience in a related occupation.” Choice C is incorrect because the chart lists a bachelor’s
degree, not a master’s, as a prerequisite for an entry-level technical writing job. The only answer choice that perfectly matches
the chart is choice D. The correct answer is choice D.

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EXERCISE: WRITING AND LANGUAGE TEST
18 Minutes—22 Questions
DIRECTIONS: Each of the following passages is accompanied by a set of questions. For some questions, you will consider
how the passage might be revised to improve the expression of ideas. For other questions, you will consider how the
passage might be edited to correct errors in sentence structure, usage, or punctuation. A passage or a question may be
accompanied by one or more graphics (such as a table, chart, graph, or photograph) that you will consider as you make
revising and editing decisions.
Some questions will direct you to an underlined portion of a passage. Other questions will direct you to a location in a
passage or ask you to think about the passage as a whole.
After reading each passage, choose the answer to each question that most effectively improves the quality of writing in
the passage or that makes the passage conform to the conventions of Standard Written English. Many questions include a
“NO CHANGE” option. Choose that option if you think the best choice is to leave the relevant portion of the passage as it is.

Questions 1–11 are based on the following passage.

1

Jane Austen and Fanny Price
. . . In real life, one 1 was sometimes left out because of
actual purposeful wrongdoing on the parts of those doing

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the leaving out, but sometimes one is simply left out—and
2 realism is beside the point. 3 The second case is

NO CHANGE

B.

is

C.

will be

D.

may have been

A.

NO CHANGE

B.

purpose

C.

culpability

D.

satisfaction

2

much less satisfying to the left-out person, but it is also
much more usual. Fairy tales such as “Cinderella” are
satisfying because, in them, it is clearly the nastiness of the
villains and villainesses that causes the heroes and heroines
to be excluded from pleasurable activities.
3

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A.

The writer is considering deleting the underlined sentence. Should the writer do this?
A.

Yes, because it does not provide a good transition
between the first and third sentences in the
paragraph.

B.

Yes, because it fails to support the main argument
of the passage: that Mansfield Park presents a
fairytale conception of being left out.

C.

No, because it identifies important distinctions
among three different “cases.”

D.

No, because it provides a good transition between
the first and third sentences in the paragraph.

[1] When I was in elementary school, I was a shy little girl

4

not unlike Fanny Price of Jane Austen’s 4 Mansfield Park.
[2] Sometimes when I was feeling left out, I would sit on a

The writer is considering adding the phrase “magnificent
work of art” here. Should the writer do this?
A.

Yes, because it identifies the genre of Mansfield
Park.

B.

Yes, because it sums up the writer’s true opinion of
the book.

C.

No, because it is an unnecessary and possibly
confusing addition.

D.

No, because it contradicts the writer’s previous,
more critical statement.

bench at the very edge of the playground and put spit on my
cheeks to simulate tears, in case David Gould, 5 a boy I
knew, should pass by and take pity on me. 6 [3] I also had
a fairytale conception of being left out (as Fanny and Jane
Austen have in this novel). [4] He never did, but it didn’t really
7 matter. My imagination would take over from that point.
[5] The stories I made up always had the same plot: I was
carried off by a prince (David) on a flying horse to his castle

5

where I became his wife and got to live in the lap of luxury.
[6] (From there I would go into elaborate detail about the
décor of the castle, how many horses we owned, 8 what
colors they were and so on.

6

7

Which choice provides the most relevant detail?
A.

NO CHANGE

B.

the shortest boy in my classroom

C.

the handsomest boy in the school

D.

my sister’s best friend’s boyfriend

To make this paragraph most logical, sentence 3 should
be placed
A.

where it is now.

B.

after sentence 1.

C.

after sentence 4.

D.

after sentence 6.

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Chapter 4

Which choice most effectively combines the two sentences at the underlined portion?
A.

matter, because my imagination

B.

matter; for example, my imagination

C.

matter; in other words, my imagination

D.

matter—consequently, my imagination

A.

NO CHANGE

B.

what colors they are and so on.

C.

what colors they will be, and so on.)

D.

what colors they were, and so on.)

Writing and
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8

Peterson’s SAT® Prep Guide 2017

In the course of Mansfield Park, Fanny goes through one big

9

suffer-and-be-vindicated cycle (her 9 ultimate marriage to

A.

NO CHANGE

B.

ill-advised

C.

initial

D.

volatile

A.

NO CHANGE

B.

yet,

C.

predictably,

D.

on the other hand,

A.

NO CHANGE

that I felt at the moment in my David Gould fantasy when

B.

sinful, this

he would suddenly appear and sweep me up and away from

C.

sinful: this

my wrongfully left-out state into one of well-deserved bliss.

D.

sinful (this

Edmund after many, many years of being left out) and many
little suffer-and-be-comforted cycles:
“Edmund . . . going quietly to another table
. . . brought a glass of Madeira to Fanny [who had
a headache as a result of being deprived of proper
exercise by the thoughtless disregard of others],
and obliged her to drink the greater part. She
wished to be able to decline it, but the tears which
a variety of feelings created, made it easier to
swallow than to speak.” [Mansfield Park, page 513;
from The Complete Novels of Jane Austen, Modern
Library edition]
The little gush of passionate, passive gratefulness that Fanny

10

11

feels when Edmund “obliges” her to drink the Madeira feels
to me similar to the sweet rush of vindicated personal pathos

This way of thinking is childish, self-pitying, and self-deluded;

164

10 accordingly, Jane Austen (who is usually far more
astute) lets Fanny get away with it.

Chapter 4
The author rigs things so that whenever Fanny is left out,
Writing and
Language
Test
Strategies

we can be sure that one or some of the other characters
are engaged in something 11 sinful; this is an unrealistic
(fairytale) element in what is otherwise a highly developed,
realistic novel.

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Questions 12–22 are based on the following passage and
supplementary material.

12

A Technical Writing Career
Technical communicators, better known as “technical writers,”
plan, write, and edit instruction manuals, print and online
articles, and other documents that transform 12 intense

A.

NO CHANGE

B.

ordinary

C.

complex

D.

varied

A.

NO CHANGE

B.

determines end users’ needs.

C.

determines and decides end users’ needs.

D.

determines the needs of end users and people
who will be using the product.

13

technical information into simpler language that end users
can understand.
A typical technical writer

•

13 determines end users’ needs and requirements.

• studies product samples and discusses them with
designers and developers.

• outlines, writes, and edits documents that support a
variety of technical products.

• gathers and/or creates graphics that illustrate instructions
and other technical documents.

14 Which choice provides the most relevant list of
details?
A.

NO CHANGE

B.

(books, magazines, newspapers, television news
shows)

C.

(drawing, painting, sculpture, computer-generated graphics)

D.

(keyboards, monitors, hard drives, software, cloud
computing)

• writes scripts for online instructional videos.
• decides which medium 14 (how-to manuals, “frequently asked questions” pages, online videos) will most
effectively convey the information.

• gathers feedback on products’ usability from customers,

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designers, and manufacturers.

• revises documents to fit product changes.
• works with customer service specialists to improve the
end-user experience through product design changes.

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Technical writers must be able to fully comprehend complex
information. Their work colleagues may include computer

15 Which change most effectively ends this paragraph and
is also consistent with information provided in the chart?

hardware engineers, scientists, computer support specialists,

A.

NO CHANGE

and software developers.

B.

Prior knowledge about such colleagues’ fields
enables technical writers to understand and
translate “tech speak” into clear, useful instructions
for users.

C.

It takes years of graduate study to enable a
technical writer to understand various work
colleagues’ fields—a master’s degree or Ph.D. in
technical writing is advisable.

D.

DELETE the underlined sentence and do not
replace it.

15 How else can they be expected to communicate with
people from such an array of professional backgrounds?

Quick Facts: Technical Writers
2012 Median Pay

$65,500 per year
$31.49 per hour

Entry-Level Education

Bachelor’s degree

Work Experience in a Related Occupation

Less than 5 years

Chapter 4

On-the-job Training

Short-term on-the-job training

Writing and
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Number of Jobs, 2012

49,500

Job Outlook, 2012–22

15% (Faster than average)

Employment Change, 2012–22

7,400

166

[Source: http://www.bls.gov/ooh/Media-and-Communication/Technical-writers.htm]

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16 [1] Why is the technical writing field growing so quickly?

16 For the sake of cohesion, sentence 1 should be placed

[2] A few years ago, U.S. government experts predicted
that employment of technical writers would grow 17 1.5
percent from 2012 to 2022, a gain of about 7,400 jobs per
year. [3] (This is faster than the average for all occupations on
which the U.S. Department of Labor gathers statistics.) [4] As
the high-tech and electronics industries continue to change
and grow, they 18 reflect a greater need for professionals

A.

where it is now.

B.

after sentence 2.

C.

after sentence 3.

D.

after sentence 4.

17 Which choice most accurately and effectively represents
the information in the chart (on the previous page)?

with the talent and skills to write instructional manuals that
users can easily understand and follow.

A.

NO CHANGE

B.

7.5 percent from 2012 to 2022, a gain of about
7,500 jobs per year.

C.

15 percent from 2012 to 2022, a gain of about
7,400 jobs per year.

D.

15 percent from 2012 to 2022, a gain of about
7,400 jobs total.

A.

NO CHANGE

B.

generate

C.

effect

D.

obligate

18

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Job opportunities 19 abound. This is especially true
for applicants with both technical skills and writing skills.

19 Which choice most effectively combines the two sentences
at the underlined portion?

Increasingly, consumers rely on technologically sophisticated

A.

abound, especially for applicants with

products in the home as well as in the workplace. 20 In

B.

abound, especially is this the case for applicants
with

C.

abound; especially positively impacted are
applicants with

D.

abound—and especially for those lucky applicants
who have acquired

A.

NO CHANGE

B.

And another thing: for many people,

C.

I need to make one last point—for many people—

D.

In contrast, for many people,

A.

NO CHANGE

B.

As older technical writers retire,

C.

Baby Boomers are retiring these days, so

D.

Smarter job seekers will recall that, as older
workers retire,

A.

NO CHANGE

B.

competitive

C.

apparent

D.

stagnant

addition—for many people—ordinary daily life requires us
to understand complex medical and scientific information.
All of these factors are combining to create many new
opportunities for technical writers. 21 Remember now, that
as older workers retire, their jobs will become vacant. Yet,
competition among freelance technical writers will remain

20

22 lively.

21

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22

ANSWER KEYS AND EXPLANATIONS
PASSAGE 1

1. B

4. C

6. B

8. D

10. B

2. C

5. C

7. A

9. A

11. A

3. D

1. The correct answer is B. Except for this verb, was, the
writer uses the present tense throughout the paragraph.
Therefore, to be consistent, the correct answer is choice B.
2. The correct answer is C. The writer is making a distinction
between times when a person is left out because other(s)
have misbehaved and times when a person is left out, but
no one else is at fault, or culpable. Since a noun is needed
in the sentence here, culpability, choice C, is the correct
answer.
3. The correct answer is D. The writer should not delete the
sentence because it provides a good transition between
the first and third sentences in the paragraph.
4. The correct answer is C. The writer should not add this
phrase here because it is unnecessary and may confuse
readers into thinking Mansfield Park is a work of visual art,
such as a painting.
5. The correct answer is C. It makes sense that a girl would
cast the handsomest boy in the school in the role of a prince
who saves her from her unhappy state. Choices A, B, and
D are incorrect because, while they are possible choices,
they are not as relevant as choice C.

6. The correct answer is B. Sentence 3 makes the best sense
when it follows sentence 1.
7. The correct answer is A. The linking word because makes
sense in context: the writer is explaining why it did not
matter that David Gould never noticed or pitied her when
she was a shy little girl.
8. The correct answer is D. Choice D correctly includes a
needed comma following the word were and includes a
missing close parenthesis mark.
9. The correct answer is A. The writer is referring to an event
that occurs near the end of the novel (“after many, many
years of being left out,” Fanny finally marries Edmund). The
correct choice is A, NO CHANGE: ultimate, meaning “final.”
10. The correct answer is B. The transitional word yet makes
sense in context: the writer is explaining that, even though
Jane Austen is usually astute about her characters, in this
case, the author “lets Fanny get away with” childishness,
self-pity, and self-delusion.
11. The correct answer is A. No change is needed because
this phrase is correctly punctuated with a semicolon.

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PASSAGE 2

12. C

15. B

17. D

19. A

21. B

13. B

16. C

18. B

20. A

22. A

14. A
12. The correct answer is C. The writer is referring to technical
ideas that are the opposite of “simple.” The correct choice
is C: complex.
13. The correct answer is B. Only choice B offers clear and
concise wording without redundancy. Choice A is incorrect
because end users’“needs and requirements”is redundant.
Choice C eliminates the original redundancy, but then
it creates a new one: “determines” and “decides” mean
the same thing. Choice D is incorrect because it, too, is
redundant: “end users” and “people who will be using the
product” mean the same thing.
14. The correct answer is A. No change is needed because
the writer is using the word medium to refer to a genre or
method of effectively conveying technical information.
15. The correct answer is B. This sentence clarifies information
presented earlier in the paragraph.

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16. The correct answer is C. Sentence 1 makes the most sense
when it follows sentence 3.
17. The correct answer is D. The last two lines of the chart say
that the projected job outlook for technical writers during
the ten years from 2012 to 2022 is 15 percent growth and
that, during those ten years, the technical writing field
will gain 7,400 jobs total. Choices A, B, and C are incorrect
because the statistics they give do not match those shown
in the chart.

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18. The correct answer is B. The writer is referring to certain
changes in the U.S. economy that will produce, result in, or
generate more technical writing jobs. The correct choice
is B: generate.
19. The correct answer is A. This is the simplest, clearest way
to combine the two sentences. Choices B, C, and D are
incorrect because each is either awkward or too wordy or
does not match the rest of the passage’s style and tone.
20. The correct answer is A. No change is needed because
the existing wording is the simplest, clearest way to make
this point. It matches the style and tone of the passage
while including necessary transitional language.
21. The correct answer is B. This phrase is appropriately specific
and best matches the rest of the passage’s style and tone.
22. The correct answer is A. The writer’s point is that even
though many older technical writers will be vacating their
jobs as they retire, freelance technical writers will still face
competition as they vie for available jobs; thus, the competition will be lively.

SUMMING IT UP
• In the Writing and Language Test section of the SAT®, there are four passages and 44 multiple-choice questions (eleven questions about each passage). You will have 35 minutes to complete this section.

• The multiple-choice questions in this section put you in an editor’s role. Each question consists of an “editing problem” with
four possible solutions.

• Some passages are accompanied by informational graphics, such as graphs, charts, tables, and photographs. You’ll need to
refer to these supplementary graphics to answer a question or two, but no math will be required in this section.

• There are three main categories of multiple-choice questions:
1.

Expression of Ideas questions, including questions on words in context; adding, deleting, or changing text; transitional language; relevant details; combining sentences; eliminating awkward language and wordiness; reordering
sentences so that paragraphs make better sense; consistency of style and tone; cohesion and precision of language

2.

Standard English Conventions questions covering grammar, usage, and punctuation

3.

Graphic Organizer questions that ask you to compare information that is in the passage to similar information or
data that are presented in a chart, graph, table, or photograph

• In addition to your Writing and Language Test score, the following subscores and cross-test scores will be provided:
ºº Command of Evidence: Questions that ask you to interpret and use evidence in the passages and informational
graphics, such as graphs, tables, and charts

ºº Expression of Ideas: Questions that focus on topic development, organization, and rhetorically effective use of
ºº
ºº
ºº
ºº

language
Words in Context: Questions that address the meaning of a particular word or phrase in context
Standard English Conventions: Questions that focus on sentence structure, usage, and punctuation
Analysis in History/Social Studies: Questions based on history/social studies–related passages
Analysis in Science: Questions based on science-related passages

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• Frequently Confused Words
• Grammar Question Types
• Organization Questions
• Pacing in Writing
• Parallelism
• Pronouns
• Punctuation
• Style Questions
• Verb Tenses
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Chapter 5:
English Language
Conventions
OVERVIEW
Sentence Formation
Verb Tense, Mood, and Voice
Conventions of Usage
Agreement
Frequently Confused Words
Conventions of Punctuation
Summing It Up

This chapter reviews standard English language conventions. While we’re not attempting to teach you all the rules of punctuation
and grammar, we do want you to review those rules that may be tested or that you may need to call on for your own writing.
Each of the three main domains covered in this chapter—sentence structure, conventions of usage, and conventions of punctuation—is broken down into smaller sections. These instructional sections are followed by exercises with answers and explanations.
Be sure to read all the answer explanations, even for the questions you answered correctly. Review is an important part of your
SAT® exam preparation.

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SENTENCE FORMATION
Sentence Boundaries
Fragments
Basic Rule
Every sentence must have a complete subject and verb and express a full idea. A group of words that is missing one of these
elements is called a sentence fragment or an incomplete sentence.
There are three ways to correct incomplete sentences:
1.

Add the fragment to the sentence that precedes it.
Incorrect: Zoologists and wildlife biologists study animals and other wildlife. How they interact with their
ecosystems.
Correct: Zoologists and wildlife biologists study animals and other wildlife and how they interact with their
ecosystems.
Explanation: The fragment is added to the sentence that precedes it by adding the word and.

2.

Add the fragment to the sentence that follows it.
Incorrect: By studying animal behaviors. Wildlife biologists seek to understand how animals interact with their
ecosystems.
Correct: By studying animal behaviors, wildlife biologists seek to understand how animals interact with their
ecosystems.

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Explanation: The fragment is added to the sentence that follows it by inserting a comma.

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3.

Add a subject and verb to the fragment.
Incorrect: Considerable time studying animals in their natural habitats.
Correct: Wildlife biologists may spend considerable time studying animals in their natural habitats.
Explanation: A subject (wildlife biologists) and verb (may spend) are added to the fragment.

Run-ons
A run-on sentence occurs when a writer fails to use either end-stop punctuation to divide complete thoughts or suitable conjunctions to join two ideas.
The following rules will help you avoid and fix run-on sentences:
1.

Though the result can be short, choppy sentences, the most common way to correct a run-on sentence is to simply divide
the sentence using end-stop punctuation.
Incorrect: Zoologists need a bachelor’s degree for entry-level positions a master’s degree or Ph.D. is often needed
for advancement.

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Correct: Zoologists need a bachelor’s degree for entry-level positions. A master’s degree or Ph.D. is often needed
for advancement.
2.

A more advanced technique is to create a compound sentence by joining independent clauses using a coordinating conjunction (e.g., and, but, or so).
Incorrect: Zoologists need a bachelor’s degree for entry-level positions a master’s degree is often needed for
advancement.
Correct: Zoologists need a bachelor’s degree for entry-level position, but a master’s degree is often needed for
advancement. (Remember that a comma is required when you use a coordinating conjunction to join two independent clauses.

3.

Another option is to create a complex sentence by adding a subordinating conjunction (e.g., because, although, or while),
making one of the independent clauses a dependent clause.
Incorrect: Zoologists need only a bachelor’s degree for entry-level positions a master’s degree is often needed for
advancement.
Correct: Zoologists need only a bachelor’s degree for entry-level positions although a master’s degree is often
needed for advancement. (In general, commas are not required when the dependent clause follows the
independent clause.)
Also Correct: Although a master’s degree is often needed for advancement, zoologists need only a bachelor’s
degree for entry-level positions. (Commas are required when the dependent clause precedes the independent
clause.)

4.

Use a semicolon when ideas are very closely related in meaning and full end-stop punctuation seems too strong.
Incorrect: Zoologists and wildlife biologists study how animals and other wildlife interact with their ecosystems
these scientists work in offices, laboratories, or outdoors.

175

Correct: Zoologists and wildlife biologists study how animals and other wildlife interact with their ecosystems;
these scientists work in offices, laboratories, or outdoors.

Chapter 5
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Skill Builder: Fragments and Run-ons
DIRECTIONS: Revise the following sentences to correct fragments and eliminate run-ons.

1. Zoologists and wildlife biologists perform a variety of scientific tests and experiments for example, they take blood samples
from animals to assess their levels of nutrition, check animals for disease and parasites, and tag animals in order to track
them.

2. In order to track potential threats to wildlife. Wildlife biologists often use computer programs.

3. Zoologists and wildlife biologists work to expand our knowledge of wildlife species. Work closely with public officials to
develop wildlife management and conservation plans.

Peterson’s SAT® Prep Guide 2017

4. Herpetologists study reptiles, such as snakes. And amphibians, such as frogs.

5. Some wildlife biologists develop conservation plans and make recommendations on conservation and management issues.
To policymakers and the general public.

6. Ecologists study ecosystems. And the relationships between organisms and the surrounding environments.

7. Evolutionary biologists study the origins of species. The changes in their inherited characteristics over generations.

8. Zoologists and wildlife biologists conduct experimental studies they also collect biological data for analysis.

Answers
In some cases, there are many possible correct answers. Here are some examples:
1. Zoologists and wildlife biologists perform a variety of scientific tests and experiments. For example, they take blood samples
from animals to assess their levels of nutrition, check animals for disease and parasites, and tag animals in order to track
them.
2. In order to track potential threats to wildlife, wildlife biologists often use computer programs.

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3. Zoologists and wildlife biologists work to expand our knowledge of wildlife species and work closely with public officials to
develop wildlife management and conservation plans.
4. Herpetologists study reptiles and amphibians, such as snakes and frogs.
5. Some wildlife biologists develop conservation plans and make recommendations on conservation and management issues
to policymakers and the general public.
6. Ecologists study ecosystems and the relationships between organisms and the surrounding environments.
7. Evolutionary biologists study the origins of species and the changes in their inherited characteristics over generations.
8. Zoologists and wildlife biologists conduct experimental studies and collect biological data for analysis.

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Coordination and Subordination
Basic Rule
Coordinating and subordinating conjunctions are used to join together clauses and form compound and complex sentences.
Some common coordinating and subordinating conjunctions are:

Coordinating conjunctions

Subordinating conjunctions

and, but, for, nor, or, so, yet

after, although, as, as if, because, before, even if,
even though, if, if only, rather than, since, that, though,
unless, until, when, where, whereas, wherever,
whether, which, while

Basic Rule of Coordinating Conjunctions
Coordinating conjunctions are used to join independent clauses to make compound sentences. In these sentences, each piece
of information carries the same weight.
Independent clauses: There was a Treaty of Paris signed in 1763. There was also one signed in 1783.
Joined together: There were Treaties of Paris signed in 1763 and 1783.
When two clauses are joined, if the second remains an independent clause, a comma must be used before the coordinating
conjunction.

177

Independent clauses: There was a Treaty of Paris signed in 1763. There was also one signed in 1783.
Chapter 5
Joined together: There was a Treaty of Paris signed in 1763, but there was another Treaty of Paris signed in 1783.

Basic Rule of Subordinating Conjunctions

English
Language
Conventions

Subordinating conjunctions are used to join independent and dependent clauses to make complex sentences. In these sentences,
the dependent clause establishes a place, a time, a reason, a condition, a concession, or a comparison for the independent clause.
Independent clauses: A tax on imported goods from another country is called a tariff. A tax on imported goods
from another country to protect a home industry is called a protective tariff.
Joined together: A tax on imported goods from another country is called a tariff, while a tax on imported goods
from another country to protect a home industry is called a protective tariff.

Peterson’s SAT® Prep Guide 2017

Here, the subordinate clause is at the end. You can also place a subordinate clause at the beginning of a sentence, as long as you
use a comma.
Independent clauses: A tax on imported goods from another country is called a tariff. A tax on imported goods
from another country to protect a home industry is called a protective tariff.
Joined together: While a tax on imported goods from another country is called a tariff, a tax on imported goods
from another country to protect a home industry is called a protective tariff.

Skill Builder: Subordination and Coordination
DIRECTIONS: Join the following sentences using subordinating or coordinating conjunctions.

1. A democracy is a form of government that is run for the people. It is also run by the people.

2. A primary source is an original record of an event. A secondary source is something that was written later.

3. The Industrial Revolution ushered in a time of unparalleled human progress. People often forget the damage that this
progress did, and continues to do, to the environment.

178

4. Elizabeth Cady Stanton became famous as an advocate of women’s rights. During the Civil War, she was also an ardent
abolitionist.

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Answers
In some cases, there are many possible correct answers. Here are some examples:
1. A democracy is a form of government that is run for the people, and it is also run by the people.
2. Whereas a primary source is an original record of an event, a secondary source is something that was written later.
3. While the Industrial Revolution ushered in a time of unparalleled human progress, people often forget the damage that this
progress did, and continues to do, to the environment.
4. Elizabeth Cady Stanton became famous as an advocate of women’s rights, and, during the Civil War, she was also an ardent
abolitionist.

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Parallel Structure
Parallel structure is the repetition of a grammatical form within a sentence. Parallel structure is a hallmark of effective writing
and is often used to emphasize ideas and present compared items in an equal light. Coordinating conjunctions are often used
in parallel constructions.
Non-parallel structure: As a child, George Washington Carver enjoyed reading, learned about plants, and he
made art.
Parallel structure: As a child, George Washington Carver enjoyed reading, learning about plants, and making art.

Modifier Placement
A modifier is a word, phrase, or clause that adds detail to a sentence. In order to avoid confusion, modifiers should be placed as
close as possible to the things they modify.
Examples of different modifiers are underlined in the sentences below.
Within the field of marine biology, employment is highly competitive. (The phrase “within the field of marine biology” modifies
the subject of the sentence, which is “employment.” The word “highly” modifies our understanding of the competitive nature of
finding employment. )
The abundant supply of marine scientists far exceeds the demands, and the number of federal and state government jobs is
limited. (“Abundant” modifies “supply.” “Marine” modifies “scientists.” “Limited” modifies our understanding of “jobs.”)
When the subject of a modifier is unclear or is not included in the sentence, it is considered a dangling modifier.

179

Incorrect: Not realizing that the job title of marine biologist rarely exists, “marine biology” is a term recognized by
most people. (What is the first phrase modifying?)

Chapter 5

Possible revision: Not realizing that the job title of marine biologist rarely exists, most people recognize the term
“marine biology.”

English
Language
Conventions

Misplaced modifiers occur when a modifier is poorly placed and it doesn’t express the writer’s intent accurately.
Incorrect: The term “marine biologist” is used to almost describe all of the disciplines and jobs that deal with the
study of marine life, not just those that deal with the physical properties of the sea.
Possible revision: The term “marine biologist” is used to describe almost all of the disciplines and jobs that deal
with the study of marine life, not just those that deal with the physical properties of the sea.

Peterson’s SAT® Prep Guide 2017

Skill Builder: Modifier Placement
DIRECTIONS: Revise the following sentences to eliminate problems with modifier placement.

1. Critical for getting a competitive edge in the job market, fishery science requires a strong background in advanced mathematics and computer skills.

2. A fishery scientist studies population dynamics of fish and marine mammals after taking course work in the animal and
aquatic sciences.

3. Another increasingly important field within marine biology, more universities are starting to offer programs in fisheries or
wildlife management.

4. As well as their interactions, biological oceanographers study both the biological and physical aspects of the sea.

5. A student may take course work weighted heavily in physics, mathematics, and computer modeling in the field of physical
oceanography.

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Answers
In some cases, there are many possible correct answers. Here are some examples:

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Conventions

1. A strong background in advanced mathematics and computer skills is critical for getting a competitive edge in the fishery
science job market.
2. After taking course work in the animal and aquatic sciences, a fishery scientist studies fish and marine mammal population
dynamics.
3. More universities are starting to offer programs in fisheries or wildlife management, another increasingly important field
within marine biology.
4. Biological oceanographers study both the biological and physical aspects of the sea, as well as their interactions.
5. A student in the field of physical oceanography may take course work weighted heavily in physics, mathematics, and computer modeling.

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VERB TENSE, MOOD, AND VOICE
Basic Rule
Use the same verb tense whenever possible within a sentence or paragraph. Do not shift from one tense to another unless there
is a valid reason.
Incorrect: The Magna Carta was signed in 1215 by King John of England and has been the first document of its
kind to limit the power of the British monarchy.
Correct: The Magna Carta was signed in 1215 by King John of England and was the first document of its kind to
limit the power of the British monarchy.

When to Use the Perfect Tenses
Basic Rule
Use present perfect for an action begun in the past and extended to the present.
Example: Scientists at NASA have seen an alarming increase in the accumulation of greenhouse gases.
Explanation: In this case, scientists at NASA saw would be incorrect. What they have seen (present perfect) began
in the past and extends to the present.

181

Basic Rule
Use past perfect for an action begun and completed in the past before some other past action.

Chapter 5

Example: Despite their preparations, Lewis and Clark had never encountered the kinds of challenges that awaited
them before their expedition.

English
Language
Conventions

Explanation: In this case, never encountered would be incorrect. The action had never encountered (past perfect) is
used because it is referring to events prior to their expedition.

Basic Rule
Use future perfect for an action begun at any time and completed in the future.
Example: When the American astronauts arrive, the Russian cosmonauts will have been on the International Space
Station for six months.
Explanation: In this case, although both actions occur in the future, the Russian cosmonauts will have been on the
space station before the American astronauts arrive. When there are two future actions, the action completed first
is expressed in the future perfect tense.

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Skill Builder: Verb Tense
DIRECTIONS: Circle the word with the correct verb tense for each sentence.

1. (was, has been) Founded in Jamestown, Virginia, the House of Burgesses ___ the first representative body founded in the
new world.
2. (have been, were) There ___ many great American explorers, but some scholars argue that none is as historically significant
as Lewis and Clark.
3. (had never been, never was) Before 1804, Meriwether Lewis ___ on an expedition of any significance, let alone led one.
4. (will have added, has added) By the time this article is published, the United States ___ 250,000 new jobs.
5. (was, has been) Civil Disobedience, or the refusal to obey a government law or laws, ___ one of Martin Luther King, Jr.’s key
tactics during the Civil Rights Movement.

Answers
1. The correct answer is was. The past tense is used because the action occurred in the past.
2. The correct answer is have been. The present perfect tense is used because the sentence refers to action that began in the
past and extended to the present.
3. The correct answer is had never been. The past perfect tense is used because the sentence contains a past tense action
that occurred before another action.

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4. The correct answer is will have added. The future perfect tense is used because the sentence refers to action begun at
any time and completed in the future.
5. The correct answer is was. The past tense is used because the action occurred in the past.

Mood
Basic Rule
Mood, as it relates to verb forms, refers to the kind of message the writer intends to communicate.
The indicative mood is the most common mood and is used to state facts or opinions.
Example: Zora Neale Hurston’s novel Their Eyes Were Watching God was forgotten for many years but is now
considered a literary classic.
The imperative mood is used when a writer wants to give a directive or make a request. Though not stated, the subject of an
imperative sentence is you.
Example: Stop pretending that it doesn’t matter.
Example: George Washington peered across the Potomac as the frigid wind lashed his face. “Hurry!” he exclaimed.
(Peered is in the indicative. Hurry is in the imperative.)

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The subjunctive mood expresses a condition contrary to fact, a wish, a supposition, or an indirect command. Although it is going
out of use in English, the subjunctive can still be seen in the following forms:

• To express a wish not likely to be fulfilled or impossible to be realized
Example: I wish it were possible for us to approve his transfer at this time. (It is not possible.)

• In a subordinate clause after a verb that expresses a command, a request, or a suggestion
Example: It was recommended by the White House that the Office of Homeland Security be responsible for preparing the statements.

• To express a condition known or supposed to be contrary to fact
Example: If Ann were chosen to be our company’s president, women would earn more than their male
counterparts.

• After as if or as though. In formal writing and speech, as if and as though are followed by the subjunctive, since they
introduce as supposition something not factual. In informal writing and speaking, the indicative is sometimes used.
Example: Before defecting to the British Army, Benedict Arnold talked as if he were a true American patriot.
(He was not.)

Avoid shifts in mood. Once you have decided on the mood that properly expresses
your message, use that mood throughout the sentence or the paragraph. A shift in
mood is confusing to the listener or reader; it indicates that the speaker or writer
himself has changed his way of looking at the conditions.
Incorrect: It is requested by the White House that a report of Congressional
proceedings be prepared and copies should be distributed to all citizens. (Be is
subjunctive; should be, indicative.)
Correct: It is requested by the White House that a report of the Congressional
proceedings be prepared and that copies be distributed to all citizens.

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Voice
Basic Rule
Voice tells us whether the subject of a sentence is the actor or is acted upon. In formal writing, active voice is preferred because
it is more immediate and places the reader closer to the action.
Active voice example: According to legend, George Washington chopped down the whole cherry tree.
Passive voice example: According to legend, the cherry tree was chopped down by George Washington.

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Skill Builder: Mood and Voice
DIRECTIONS: In the following sentences, choose the correct mood or voice.

1. (was, were) The team of Russian engineers ___ unable to prevent the nuclear reactor in Chernobyl from melting down.
2. (was, were) If climate change ___ not such a threat to life on this planet, the scientific community would not be making such
a big deal about carbon emissions.
3. (is, were) If inflation ___ to continue to rise, the effects on the economy would be disastrous.
4. (be, should be) The President asked that the Speaker of the House ___ present when the special announcement was made.
5. (passive or active voice) The stony coral polyps were placed in a cup made of calcium carbonate.
6. (passive or active voice) For over 40 years, Henry Clay played a central role on the national political stage.

Answers and Explanations
1. The correct answer is was. This sentence uses the indicative mood and requires the simple past tense.
2. The correct answer is were. The subjunctive is correct because the sentence is making a supposition.
3. The correct answer is were. The subjunctive is correct because the sentence is making a supposition.
4. The correct answer is be. The subjunctive is correct because the subordinate clause after the verb is making a request.
5. The correct answer is passive voice. This sentence is passive because the subject of the sentence is acted upon.
6. The correct answer is active voice. This sentence is active because the subject of the sentence is the actor.

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CONVENTIONS OF USAGE
Pronouns
Pronouns substitute for nouns.
Examples: George Washington was born on February 22, 1732, in Pope’s Creek, Virginia; he was the first American
president.
Did you know that Besty Ross and George Washington both went to the same church? It was called Christ Church,
and it was located in Philadelphia.

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The following pronoun chart may prove helpful:

Number

Singular

Plural

Person

Subjective
Case

Objective
Case

Possessive
Case

1st person

I

me

mine

2nd person

you

you

yours

3rd person

he, she, it, who

him, her, it, whom

his, hers, whose

1st person

we

us

ours

2nd person

you

you

yours

3rd person

they, who

them, whom

theirs, whose

Basic Rule
A pronoun uses the subjective case when it is the subject of the sentence or when it renames the subject as a subject complement.
Incorrect: That night, George Washington, Robert Morris, and him asked Betsy Ross to sew the first flag.
Correct: That night, George Washington, Robert Morris, and he asked Betsy Ross to sew the first flag. (He is part of
the subject of the sentence.)
Incorrect: George Ross is him.

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Correct: George Ross is he. (He renames the subject.)

Basic Rule

Chapter 5

If a pronoun is the object of a verb or preposition, it is placed in the objective case.

English
Language
Conventions

Incorrect: After the plot was discovered, they accused Benedict Arnold and he of treason.
Correct: After the plot was discovered, they accused Benedict Arnold and him of treason. (Him is the object of the
verb accused.)
Incorrect: Despite the fact that we turned in our marine biology paper late, “A” grades were given to Franklin and I.
Correct: Despite the fact that we turned in our marine biology paper late, “A” grades were given to Franklin and
me. (Me is the object of the verb given.)

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Pronoun Clarity
Avoid ambiguity and confusion by placing a pronoun as close as possible to its antecedent (the word it refers to) and by making
sure that the antecedent is clear.
Incorrect: At the height of his career, Frank Lloyd Wright told an architectural scholar that he thought his work was
improving. (Is Wright talking about his own work or the work of the scholar?)
Correct: At the height of his career, Frank Lloyd Wright told an architectural scholar that he thought his own work
was improving.
Incorrect: Frank Lloyd Wright and his wife Olgivanna founded a school for aspiring artists in Spring Green,
Wisconsin, where they could “learn by doing.” (Does they refer to the Wrights or the artists?)
Correct: Frank Lloyd Wright and his wife Olgivanna founded a school in Spring Green, Wisconsin, where aspiring
artists could “learn by doing.”

Possessive Determiners
When a pronoun expresses ownership, it is placed in the possessive case.

Possessive determiners (its, your, their), contractions (it’s, you’re, they’re), and adverbs
(there) are often confused. Remember that personal pronouns that express ownership
never require an apostrophe.

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Incorrect: Frank Lloyd Wright believed that an architectural structure should be
in harmony with it’s environment.

Chapter 5

Correct: Frank Lloyd Wright believed that an architectural structure should be
in harmony with its environment.

English
Language
Conventions

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Skill Builder: Pronouns
DIRECTIONS: In the space provided, identify and explain the pronoun error contained in each statement.

1. The Battle of Yorktown was an important turning point in the American Revolution and the British defeat signaled
the end of it.

2. The American Bill of Rights was based on the English Bill of Rights; it protected the rights of the citizens.

3. While the Articles of Confederation is less famous than other historical documents like the Declaration of Independence and
the Constitution, it’s historical significance cannot be overstated.

4. Federalists and Anti-Federalists felt differently about the division of power between national and state governments. They
preferred more power be given to the states.

5. Thomas Paine was an American patriot who’s pamphlets Common Sense and The Crisis helped stir the American independence movement.

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Answers and Explanations

Chapter 5

1. It is unclear whether the “it” at the end of the sentence refers to the American Revolution or the Battle of Yorktown.
2. It is unclear whether the “it” that follows the semicolon refers to the American or English Bill or Rights or both.
3. “It’s” is a contraction; the correct word should be the possessive pronoun “its.”

English
Language
Conventions

4. It is unclear whether “they” refers to the Federalists or the Anti-Federalists.
5. “Who’s” is a contraction; the correct word should be the possessive pronoun “whose.”

Peterson’s SAT® Prep Guide 2017

AGREEMENT
Pronoun-Antecedent Agreement
A pronoun agrees with its antecedent in both person and number.
Example: The archaeologists examined the fossilized bone with great care to make sure they didn’t damage it.
Explanation: The pronoun they refers to archaeologists, its antecedent. The pronoun it refers to bone, its
antecedent.

Remember to use a singular pronoun when you refer to indefinite pronouns such as
everyone, everybody, each, every, anyone, anybody, nobody, none, no one, one, either,
and neither.
Examples:
Although Union High School’s male lacrosse players operate as a team, each
knows it’s his (not their) responsibility to arrive on time and in uniform.
Despite the fact that many of the women came from wealthy families,
everyone who attended the Seneca Falls conference on women’s rights risked
her (not their) life and reputation.
When the programmers were questioned, neither could be certain if it was
his or her (not their) mistake that caused the computer network to crash.

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Subject-Verb Agreement

Chapter 5
English
Language
Conventions

Basic Rule
A verb agrees in number with its subject. A singular subject takes a singular verb. A plural subject takes a plural verb.
Examples: Coral reefs are an important part of the marine ecosystem.
V
S
My teacher believes that coral reefs are an important part of the marine ecosystem.
S
V
V
S

For the following sentence, choose the correct verb: (is, am, are)
Booker T. Washington, Frederick Douglass, and W.E.B. DuBois _____ all important historical figures.
Explanation: Remember that the verb must agree with the subject. Since the subject is plural—subjects joined
by and are plural—a plural verb is needed. The correct response therefore should be:
Booker T. Washington, Frederick Douglass, and W.E.B. DuBois are all important historical figures.

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Sometimes the subject comes after the verb, but the rule still applies.
Choose the correct verb: (is, are)
While the lecture has lasted 2 hours already, there ____ still 3 more speakers.
Explanation: The correct choice is are since the subject speakers is plural and requires a plural verb.

There is one major exception to this rule. When the sentence is introduced by the
word “there” and the verb is followed by a compound (double) subject, the first part
of the subject dictates whether the verb should be singular or plural.
Example: There is one American astronaut in the shuttle and four Russian
astronauts in the space station.

When compound subjects are joined by either-or or neither-nor, the verb agrees with the subject closest to the verb.
Examples:
Neither the violinist nor the other musicians have had much experience performing for an audience.
Neither you nor I am willing to make the sacrifices required of a professional musician.
Explanation:
In the first example, musicians (plural) is closest to the verb; in the second example, I (singular) is closest
to the verb.
Sometimes a word or a group of words may come between the subject and the verb. The verb still must agree with the simple
subject, and the simple subject is never part of a prepositional phrase.

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Example:
Stephen King, the author of hundreds of best-selling novels, novellas, and short stories, is also a guitarist
and singer in a band.
The simple subject is Stephen King, a singular noun. The verb must be is.
Choose the correct verb: (was, were)
The causes of the deterioration of coral reefs ____ not known until recently.
Explanation:
The simple subject is causes; “of the deterioration of coral reefs” is a prepositional phrase. Since the
subject is plural, the plural verb were is required. So, the correct answer is were.
The third person singular of most verbs ends in “s.” First person: I, we speak;
second person: you speak; third person: he, she, it speaks. Examples:
He runs. She jogs. It jumps. The man sees. Mary laughs. The child walks.

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Collective Nouns
Collective nouns present special problems. A collective noun names a group of people or things. Although usually singular in
form, it is treated as either singular or plural according to the sense of the sentence:

• A collective is treated as singular when members of the group act, or are considered, as a unit:
Example: The citizens’ assembly is drafting a petition that would seek to protect local aquifers from chemical run-off
and hazardous waste.

• A collective is treated as plural when the members act, or are considered, as multiple individuals:
Example: After one of the longest and most fabled droughts in baseball history, the Boston Red Sox have finally
overcome the “Curse of the Bambino” to win another World Series.
Common collective nouns include:
assembly, association, audience, board, cabinet, class, commission, committee, company, corporation, council,
counsel, couple, crowd, department, family, firm, group, jury, majority, minority, number, pair, press, public, staff,
United States
The following short words—though seldom listed as collective nouns—are governed by the same rules. They are singular or
plural according to the intended meaning of the sentence.
all, any, more, most, some, who, which

Skill Builder: Agreement

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Conventions

DIRECTIONS: Follow the principles of agreement and choose the correct word.
1. (is, are) The President and Vice President of the United States of America ___ expected to attend tomorrow’s historic ceremony
honoring Rosa Parks.
2. (have, has) Either the chair of the department or one of the professors ___ the necessary paperwork.
3. (was, were) In the time of the first settlers, there ___ no antibiotics to prevent outbreaks of disease.
4. (its, their) Because of ___ biodiversity, coral reefs are often called “the rainforests of the sea.”
5. (her, their) After a brief introduction, each of the doctors presented ___ findings at the medical conference.
6. (is, are) Many young people are surprised to find out that the music of Verdi’s operas ____ as vibrant and fun as anything on
the radio.
7. (his, their) As the conductor took the podium, the musicians finished tuning ___ instruments.
8. (know, knows) Neither Sherlock Holmes nor the detectives of Scotland Yard ___ who the perpetrator is.
9. (is, are) According to preliminary market reports, either Xiaomi or Huawei ___ the biggest smartphone provider in China.
10. (is, are) However, neither the Republicans nor the Democrats ___ satisfied with the language of the new nuclear weapons
treaty.

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Answers and Explanations
1. The correct answer is are. The plural subject President and Vice President requires the plural are.
2. The correct answer is has. The verb must agree with the subject closest to it, in this case the singular one.
3. The correct answer is were. The plural subject antibiotics requires the plural were.
4. The correct answer is their. The plural subject coral reefs requires the plural their.
5. The correct answer is her. The singular subject each requires the singular her.
6. The correct answer is is. The singular subject music requires the singular is.
7. The correct answer is their. The plural antecedent musicians requires the plural their.
8. The correct answer is know. In a neither-nor construction, the verb is governed by the closest subject, detectives.
9. The correct answer is is. The singular determiner either requires the singular verb is.
10. The correct answer is are. In a neither-nor construction, the verb is governed by the closest subject, Democrats.

Skill Builder: Agreement
DIRECTIONS: Revise the underlined words to eliminate agreement errors. If the underlined word is grammatically
correct, write C above it. If a change is necessary, indicate the change and give a grammatical reason for it. Do not make
unnecessary changes.

Joseph, one of my best friends, are considering becoming a medical doctor. He and I believe that medicine, compared to
1
2
other professions, are an exciting and fulfilling field. In order to help him decide, he asked each of his friends to give his or her
3
opinion about why medicine would or would not be a fulfilling career choice. After that, he also asked his parents for their

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Language
Conventions

opinions. It seems that his friends and his mother is in agreement, but his father do not agree.
5
4
Jacob’s father feels strongly that the medical profession, unlike other professions, requires an excessive amount of study and is
6
too emotionally taxing. On the other hand, his friends and his mother agree that while the schooling is rigorous, the practice
7
itself would be very rewarding.

Jacob is still deciding what he wants to be, but he and I has learned that there is always many possible answers to a question.
8
9
10

Peterson’s SAT® Prep Guide 2017

Answers and Explanations
1. The correct answer is is. The subject, Joseph, is singular.
2. The correct answer is C. The subject, He and I, is plural.
3. The correct answer is is. The subject, medicine, is singular.
4. The correct answer is are. The subject, his friends and his mother, is plural.
5. The correct answer is does. The subject, father, is singular.
6. The correct answer is C. The subject, medical profession, is singular.
7. The correct answer is C. The subject, friends and mother, is plural.
8. The correct answer is C. The subject, Jacob, is singular.
9. The correct answer is have. The subject, he and I, is plural.
10. The correct answer is are. The subject, answers, is plural.

FREQUENTLY CONFUSED WORDS
The following pages review groups of words that are similar in sound and/or meaning and are generally found to be confusing
to students and adults alike. Misunderstanding what they mean or how they are used results in various usage problems. The
word groups have been broken down into manageable sections to help you learn them more easily. Do not try to master all the
information at once. Study one section at a time.

192

At the end of each section there is a practice exercise. See how well you do on the exercise by checking your answers against
the answers and explanations given. If you do well, go on to the next section. If you find that you have made a number of errors,
review the section. It is important that you master each section before moving on to the next one.

Chapter 5
English
Language
Conventions

Frequently Confused Words: Group 1
a is used before words that start with a consonant sound
an is used before words that start with a vowel sound
Please give the baby a toy.
He is an only child. We put up a united front. (United begins with a consonant sound—y.)
We spent an hour together. (Hour begins with a vowel sound, since the h is silent.)
and is used to join words or ideas
We enjoy shopping and sightseeing.
She is a very serious student, and her grades are the best in the class.

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accept means to receive or to agree to something
except means to exclude or excluding
I’ll accept the gift from you.
Everyone except my uncle went home.
My uncle was excepted from the group of losers.
advice means counsel (noun), opinion
advise means to offer advice (verb)
Let me give you some free advice.
I’d advise you to see your doctor.
affect means to influence (verb)
effect means to cause or bring about (verb) or a result (noun)
The pollution can affect your health.
The landmark decision will effect a change in the law.
The effect of the storm could not be measured.
all ready means everybody or everything ready
already means previously
They were all ready to write when the test began.
They had already written the letter.
all together means everybody or everything together
altogether means completely
The boys and girls stood all together in line.

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His action was altogether strange for a person of his type.
desert (DEZZ-ert) means an arid area
desert (di-ZERT) means to abandon, or a reward or punishment (usually plural)
dessert (di-ZERT) means the final course of a meal
I have seen several movies set in the Sahara desert.
The soldier was warned not to desert his company.
We’re certain that execution is a just desert for his crime.
He received his just deserts.
We had strawberry shortcake for dessert.

Peterson’s SAT® Prep Guide 2017

in is used to indicate inclusion, location, or motion within limits
into is used for motion toward one place from another
The spoons are in the drawer.
We were walking in the room.
I put the spoons into the drawer.
She walked into the room.
it’s is the contraction of it is or it has
its is a possessive pronoun meaning belonging to it
It’s a very difficult assignment.
We tried to analyze its meaning.
lay means to put
lie means to recline
To lay:
(present)		

I lay

(past)			

I laid the gift on the table.

(present perfect)

I have laid

To lie:

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(present)		

I lie

(past)			

I lay on my blanket at the beach.

(present perfect)

I have lain

lets is third person singular present of let
let’s is a contraction for let us
He lets me park my car in his garage.
Let’s go home early today.
loose means not fastened or restrained, or not tight-fitting
lose means to mislay, to be unable to keep, to be defeated
The dog got loose from the leash.
Try not to lose your umbrella.
passed is the past tense of to pass
past means just preceding or an earlier time
The week passed very quickly.
The past week was a very exciting one.

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principal means chief or main (adjective), or a leader, or a sum of money (noun)
principle means a fundamental truth or belief
His principal support comes from the real estate industry.
The principal of the school called a meeting of the faculty.
He earned 10 percent interest on the principal he invested last year.
As a matter of principle, he refused to register for the draft.
quiet means silent, still
quit means to give up, to discontinue
quite means very or exactly, to the greatest extent
My brother is very shy and quiet.
I quit the team last week.
His analysis is quite correct.
raise means to lift, to erect
raze means to tear down
rise means to get up, to move from a lower to a higher position, to increase in value
The neighbors helped him raise a new barn.
The demolition crew razed the old building.
The price of silver will rise again this month.
set means to place something down (mainly)
sit means to seat oneself (mainly)
To set:
(present)		

He sets

(past)			

He set the lamp on the table.

(present perfect)

He has set

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To sit:
(present)		

He sits

(past)			

He sat on the chair.

(present perfect)

He has sat

stationary means standing still
stationery means writing material
In ancient times, people thought that the earth was stationary.
We bought our school supplies at the stationery store.

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suppose means to assume or guess
supposed is the past tense and also past participle of suppose
supposed also means ought to or should (when followed by to)
I suppose you will be home early.
I supposed you would be home early.
I had supposed you would be there.
I am supposed to be in school tomorrow.
than is used to express comparison
then is used to express time or a result or consequence
Jim ate more than we could put on the large plate.
I knocked on the door, and then I entered.
If you go, then I will go, too.
their means belonging to them
there means in that place
they’re is the contraction for they are
We took their books home with us.

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Your books are over there on the desk.
They’re coming over for dinner.
though means although or as if
thought is the past tense of to think, or an idea (noun)
through means in one side and out another, by way of, finished
Though he is my friend, I can’t recommend him for this job.
I thought you were serious!
We enjoyed running through the snow.
to means in the direction of (preposition); it is also used before a verb to indicate the infinitive
too means very, also
two is the numeral 2
We shall go to school.
It is too hot today.
We shall go, too.
I ate two sandwiches for lunch.

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use means to employ, put into service
used is the past tense and the past participle of use
I want to use your chair.
I used your chair.
used meaning in the habit of or accustomed to, is followed by to
used is an adjective meaning not new
I am used to your comments.
I bought a used car.
weather refers to atmospheric conditions
whether introduces a choice; it should not be preceded by of or as to
I don’t like the weather in San Francisco.
He inquired whether we were going to the dance.
were is a past tense of be
we’re is a contraction of we are
where refers to place or location
They were there yesterday.
We’re in charge of the decorations.
Where are we meeting your brother?
who’s is the contraction for who is (or who has)
whose means of whom, implying ownership

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Who’s the next batter?
Whose notebook is on the desk?
your is a possessive, showing ownership
you’re is a contraction for you are
Please give him your notebook.
You’re very sweet.

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Skill Builder: Usage
DIRECTIONS: Circle the correct word to complete the sentence while adhering to formal American-English conventions.

1. The patriot Samuel Adams was one of the (principal, principle) organizers of the Boston Tea Party.
2. Merchants in Boston refused to (accept, except) the taxes imposed upon them by the Tea Act of 1773.
3. Though the significance of the Tea Act of 1773 cannot be overrestimated, (weather, whether) or not the Tea Act of 1773 led
to the American Revolution is hard to say.
4. In late November of 1773, the ship the Dartmouth sailed (in, into) Boston Harbor.
5. Governor Hutchinson was determined to collect the taxes and (adviced, advised) the tea consignees not to back down.
6. More (than, then) 40 tons of tea were thrown into the water during the Boston Tea Party.
7. Tea smuggling was (all ready, already) a significant problem, especially in New York and Philadelphia.
8. The overall (affect, effect) of the Boston Tea Party was to bolster the revolutionary fervor that was sweeping New England.
9. The British Crown reacted swiftly and harshly in order to dispel the idea that they were (loosing, losing) control of the colonies.
10. (All together, Altogether) 342 chests of tea, weighing over 92,000 pounds, were dumped into the water.

Answers
1. principal
2. accept

198

3. whether

Chapter 5

5. advised

English
Language
Conventions

4. into

6. than
7. already
8. effect
9. losing
10. altogether

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Frequently Confused Words: Group 2
abbreviate means to shorten by omitting
abridge means to shorten by condensing
New York is abbreviated to NY.
In order to save time in the reading, the report was abridged.
ad is used informally, but in formal usage advertisement is correct; similarly: exam (examination), auto (automobile), phone
(telephone), gym (gymnasium)
advantage means a superior position
benefit means a favor conferred or earned (as a profit)
He had an advantage in experience over his opponent.
The rules were changed for his benefit.
aggravate means to make worse
annoy means to bother or to irritate
Your nasty comments aggravated a bad situation.
Your nasty comments annoyed him. (Not: Your nasty comments aggravated him.)
ain’t is an unacceptable contraction for am not, are not, or is not, although ain’t is sometimes heard in very informal speech
alibi is an explanation on the basis of being in another place
excuse is an explanation on any basis
The accused man’s alibi was that he was in another town when the robbery occurred.
Whenever he is late, he makes up a new excuse.

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all ways means in every possible way
always means at all times
He was in all ways acceptable to the voters.
He was always ready to help.
almost means nearly, not quite
most refers to the greatest amount or number or to the largest part, a majority
We are almost finished writing the book.
Most of the credit should be given to his uncle.
alongside of means side by side with
alongside means parallel to the side

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He stood alongside of her at the corner.
Park the car alongside the curb.
allot means to give or apportion
I will allot 3 hours for painting the table.
alot is a misspelling of a lot
He earned a lot of money. (Better: He earned a great deal of money.)
alright is now often employed in common usage to mean all right (In formal usage, all right is still preferred by most authorities.)
all right means satisfactory, very well, uninjured, or without doubt
I’m alright, thank you.
It was his responsibility, all right.
alternate, as a noun, means a substitute or second choice
alternate, as a verb, means to perform by turns
alternative means a choice between two things, only one of which may be accepted
She served as an alternate delegate to the convention.
The cook alternated green beans and cauliflower on the menu.
Is there an alternative to the proposition? (In less formal usage, alternative is not always limited to a choice between two.)

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Language
Conventions

alumna means a female graduate (plural: alumnae; ae rhymes with key)
alumnus means a male graduate (plural: alumni; ni rhymes with high)
She is an alumna of Mrs. Brown’s School for Young Women.
He is an alumnus of City College.
among is used to discuss more than two items
between is used to discuss two items only
The work was divided among the four brothers.
She divided the pie between Joe and Marie.
amount is used to refer to a quantity not individually countable
number is used to refer to items that can be counted individually
A tremendous amount of work had piled up on my desk.
We ate a great number of cookies at the party.
annual means yearly
biannual means twice a year (also semiannual)

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biennial means once in two years or every two years
Are you going to the annual holiday party?
I received biannual car insurance statements in April and in October.
He gets a new car biennially.
anxious means worried
eager means keenly desirous
We were anxious about our first airplane flight.
I am eager to see you again.
anyways is an incorrect form for anyway
anywheres is an incorrect form for anywhere
I didn’t want to go anyway.
I couldn’t locate her anywhere.
aren’t I is used informally, but in formal usage am I not is correct
Am I not entitled to an explanation?
as is not always as clear as because, for, or since (also see like)
She wants to cry because she is very sad.

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as, used as a conjunction, is followed by a verb
like, used as a preposition, is not followed by a verb

Chapter 5

Do as I do, not as I say.

English
Language
Conventions

Try not to behave like a child.
as … as is used in an affirmative statement
so … as is used in a negative statement
She is as talented as any other actress in the show.
He is not so reliable as his older brother.
as good as is used for comparisons, not to mean practically
This bicycle is as good as the other one.
They practically promised us a place in the hall. (Not: They as good as promised us a place in the hall.)
astonish means to strike with sudden wonder
surprise means to catch unaware
The extreme violence of the hurricane astonished everybody.
A heat wave in April would surprise us.

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at should be avoided when it does not contribute to the meaning of an idea
Where do you live at? may be heard in informal usage, but Where do you live? is the correct form.
The group will arrive about noon. (not at about noon)
awfully is sometimes heard in informal usage. In formal usage, very is correct.
This pie is very good. (not awfully good)
a while is used after a preposition (noun)
awhile is used in other cases (adverb)
I coached the team for a while.
I coached the team awhile.
backward and backwards both may be used as adverbs
We tried to skate backward. (Or: We tried to skate backwards.)
bad is used after verbs that refer to the senses, such as look, feel (adjective)
badly means greatly, in a bad manner (adverb)
He felt bad that he could not attend the meeting.
The young man needs a part-time job very badly.

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been is the past participle of to be, used after helping verbs have, has, or had
being is the -ing form of to be, usually used after helping verbs is, am, are, was, and were
I have been living here for six years.
He was being a troublemaker, so we told him to stay away from us.
being as and being that should not be used in standard English. Because and since are preferable.
Since it was dark, we turned on the lights.
Because he is my friend, I give him a gift.

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Skill Builder: Usage
DIRECTIONS: Circle the correct word to complete the sentence while adhering to formal American-English conventions.
1. The (abbreviated, abridged) jobs report omitted jobs in the arts in order to make the report more manageable.
2. Detectives often ask suspects if they have an (alibi, excuse) for where they were on the night of a given crime.
3. Animal trainers must (allot, a lot) a significant amount of their work time to building a relationship with their animals.
4. The famous dog trainer Felix Ho is an (alumna, alumnus) of La Trobe University in Melbourne, Australia.
5. Animal care and service workers often divide their time (among, between) training, feeding, grooming, and exercising their
animals.
6. Learning how to be an effective animal trainer requires a tremendous (amount, number) of work.
7. Animals form very real bonds with their trainers; likewise, trainers are often (anxious, eager) to see their animals when they
have been away.
8. While horse trainers rarely have formal schooling in their field, they can’t get their training just (anywhere, anywheres).
9. Becoming a good animal trainer requires experience that can be (awfully, very) difficult to attain.
10. Many animal trainers feel (bad, badly) when they have to leave their animals for long periods of time.
11. (Because, Being that) zookeepers and marine mammal trainers require formal education, they tend to make more money
than other animal care professionals.

Answers
1. abbreviated

203

2. alibi

Chapter 5

3. allot

English
Language
Conventions

4. alumnus
5. among
6. amount
7. eager
8. anywhere
9. very
10. bad
11. Because

Peterson’s SAT® Prep Guide 2017

Frequently Confused Words: Group 3
beside means at the side of
besides means in addition to
In our tennis game, he played beside me at the net.
We entertained Jim, Sue, and Louise, besides the members of the chorus.
better means recovering
well means completely recovered
better is used with the verb had to show desirability
He is better now than he was a week ago.
In a few more weeks, he will be well.
He had better (not he better) follow instructions or pay the penalty.
between you and I is the incorrect form, since the object of the preposition between should be the objective case me, not the
subjective case I
Between you and me, he has not been very helpful this week.
both means two considered together

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each means one of two or more
Both of the applicants qualified for the position.
Each applicant was given a good reference.
bring means to carry toward the speaker
take means to carry away from the speaker
Bring the coat to me.
Take money for carfare when you leave.
bunch is used informally to describe a group of people, but in formal usage group is preferred
When he returned to his office, he learned that a group of students was waiting for him.
burst is used in present and past tenses to mean to explode (or to break)
bust and busted are incorrect forms of burst
I do hope the balloon will not burst.
He cried when the balloon burst. (not busted)
but that is sometimes heard in informal usage, but in formal usage that is correct
He never doubted that she would visit him.

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can means able
may implies permission or possibility
I can eat both desserts.
May I eat both desserts?
It may snow tonight.
cannot seem is sometimes used informally, but in formal usage seems unable is correct
My elderly uncle seems unable to remember his own phone number.
complected should not be used for complexioned
At the beach, the fair-complexioned boy stayed under an umbrella.
consistently means in harmony
constantly means regularly, steadily
If you give me advice, you should act consistently with that advice.
I constantly warned him about leaving the door unlocked.
continual means happening again and again at short intervals
continuous means without interruption
The teacher gave the class continual warnings.

205

Noah experienced continuous rain for forty days.
could of is an incorrect form of could have, which can be contracted to could’ve in speech or informal writing
I wish that I could’ve gone. (Better: I wish that I could have gone.)

Chapter 5
English
Language
Conventions

couple refers to two; several or a few refers to more than two
Alex and Frieda are the most graceful couple on the dance floor.
A few of my cousins—Mary, Margie, Alice, and Barbara—will be at the reunion tonight.
data is the Latin plural of datum, meaning information (data is preferred with plural verbs and pronouns, but is now acceptable
in the singular)
These data were very significant to the study. (Or: This data was very significant to the study.)
did is the past tense of do
done is the past participle of do
I did whatever was required to complete the job.
I have done what you requested.

Peterson’s SAT® Prep Guide 2017

different than is often used informally, but in formal usage different from is correct
Jack is different from his brother.
disinterested means impartial
uninterested means not interested
The judge must be a disinterested party in a trial.
I’m an uninterested bystander, so I find the proceedings boring.
doesn’t is a contraction of does not (third person singular)
don’t is a contraction of do not and is not a substitute for doesn’t
She doesn’t go to school.
They don’t go to school.
doubt whether is often heard in informal usage, but doubt that is the correct form
I doubt that I will be home this evening.
due to is sometimes used informally at the beginning of a sentence, but in formal usage because of, on account of, or some similar
expression is preferred
Because of (not due to) the rain, the game was postponed. (But: The postponement was due to the rain.)
each other refers to two persons

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Language
Conventions

one another refers to more than two persons
Jane and Jessica have known each other for many years.
Several of the girls have known one another for many years.
either … or is used to refer to choices
neither … nor is the negative form
Either Lou or Jim will drive you home.
Neither Alice nor Carol will be home tonight.
else than is sometimes heard in informal usage, but in formal usage other than is correct
Shakespeare was rarely regarded by students as anything other than the writer of plays.
enthuse or enthused should be avoided; use enthusiastic
We were enthusiastic when given the chance to travel abroad.
equally as good is an incorrect form; equally good or just as good is correct
This bicycle is just as good as that one.

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etc. is the abbreviation for the Latin term et cetera, meaning and so forth, and other things. In general, it is better to be specific
and not use etc.
I think that oranges, peaches, cherries, etc., are healthful. (Etc. is not preceded by and)
everyone, written as one word, is a pronoun
every one, written as two words, is used to refer to each individual
Everyone present voted for the proposal.
Every one of the voters accepted the proposal.
every bit is incorrect usage for just as
You are just as (not every bit as) clever as she is.
ever so often means frequently or repeatedly
every so often means occasionally or now and again
He sees his brother ever so often, practically every day.
Since tickets are so expensive, we only attend the theater every so often.
expect is sometimes used incorrectly to mean assume or presume
I assume (not expect) that he won the race.

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Language
Conventions

Peterson’s SAT® Prep Guide 2017

Skill Builder: Usage
DIRECTIONS: Circle the correct word to complete the sentence while adhering to formal American-English conventions.
1. (Beside, Besides) needing a bachelor’s degree in marine biology, animal science, biology, or a related field, marine mammal
trainers often need to have SCUBA certification.
2. Between you and (me, I), training lions does not sound like fun!
3. After a rigorous application process, (both, each) of the trainers were hired.
4. A dog trainer might begin with a simple task, such as training a dog to (bring, take) a rubber ball back to him.
5. Young students often ask zookeepers, “(Can, May) I help you feed the animals?”
6. While job opportunities in kennels, grooming shops, and pet stores are increasing, many advanced animal care professionals,
such as zookeepers and marine mammal trainers, (cannot seem, seem unable) to find work.
7. In jobs surveys, the percentage of nonfarm animal caretakers (consistently, constantly) outnumbers the number of actual
animal trainers.
8. Here are a (couple, few) examples of what an animal care specialist might do: give food and water to animals; clean equipment and the living spaces of animals; monitor animals and record details of their diet, physical condition, and behavior;
and examine animals for signs of illness or injury.
9. Even though they sound the same, the job of a pet groomer is very (different from, different than) the job of a horse groom.
10. (Because of, Due to) employment growth and high job turnover, job opportunities in the animal care and services field will
continue to grow.

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Language
Conventions

11. The starting salary for animal care and service workers is (equally as good, just as good) as the starting salary for nonfarm
animal caretakers.

Answers
1. Besides
2. me
3. both
4. bring
5. May
6. seem unable
7. constantly
8. few
9. different from
10. Because of
11. just as good

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Frequently Confused Words: Group 4
fewer is used to refer to items that can be counted
less is used to refer to something viewed as a mass, not as a series of individual items
I made fewer repairs on the new car than on the old one.
After the scandal, the company enjoyed less prestige than it had the previous year.
finalized is used to mean concluded or completed, usually in informal usage; in formal usage, completed is preferred
Labor and management completed arrangements for a settlement.
flaunt means to make a display of
flout means to show contempt, scorn
He flaunted his new wealth in an ostentatious manner.
She flouted the policeman’s authority.
former means the first of two
latter means the second of two
The former half of the story was in prose.
The latter half of the story was in poetry.
good is an adjective; good is often used informally as an adverb, but the correct word is well
She is a good singer.
She sings well.
graduated is followed by the preposition from when it indicates completion of a course of study
graduated also means divided into categories or marked intervals

209
Chapter 5
English
Language
Conventions

He graduated from high school last year. (Or: He was graduated from high school last year.)
A graduated test tube is one that has markings on it to indicate divisions.
guess is sometimes used informally to mean think or suppose, but it is incorrect in formal use
I think (not guess) I’ll go home now.
habit means an individual tendency to repeat a thing
custom means group habit
He had a habit of breaking glasses before each recital.
The custom of the country was to betroth girls at an early age.

Peterson’s SAT® Prep Guide 2017

had ought is an incorrect form for ought or should
hadn’t ought is an incorrect form for should not or ought not
The men ought (not had ought) to go to the game now.
He ought not (not hadn’t ought) to have spoken.
He should not (not hadn’t ought) have spoken.
hanged is used in reference to a person
hung is used in reference to a thing
The prisoner was hanged in the town square.
The drapes were hung unevenly.
have got is incorrect usage; got should be omitted
I have an umbrella.
healthful is used to express whatever gives health
healthy is used to express whatever has health
He follows a healthful diet.
He is a healthy person.
hisself is a misspelling of himself

210
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Language
Conventions

Let him do it himself.
humans is used informally to refer to human beings, but in formal usage human beings is correct
He says that love is a basic need of all human beings. (But used as an adjective: He says that love is a basic human need.)
if introduces a condition
whether introduces a choice
I shall go to Greece if I win the prize.
He asked me whether I intended to go to Greece.
if it was implies that something might have been true in the past
if it were implies doubt or indicates something that is contrary to fact
If your book was there last night, it is there now.
If it were summer now, we would all go swimming.
imply means to suggest or hint at (the speaker implies)
infer means to deduce or conclude (the listener infers)
Are you implying that I have disobeyed orders?
From your carefree tone, what else are we to infer?

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in back of means behind
in the back of (or at the back of) means in the rear of
The shovel is in back of (behind) the barn.
John is sitting in the back of the theater.
in regards to is an incorrect form for in regard to
He called me in regard to your letter.
instance where is sometimes used informally, but the correct term is instance in which
Can you tell me of one instance in which such a terrible thing occurred?
irregardless in an incorrect form for regardless
I’ll be your friend regardless of what people say, even if the people are accurate.
is when and is where are sometimes used informally, but in formal usage occurs when and is a place where are correct
The best scene occurs when the audience least expects it.
My favorite vacation spot is a place where there are no telephones.
kind of and sort of are informal expressions that should be rephrased in formal writing—for instance, somewhat or rather are
preferable
I am rather sorry he retired.
He was somewhat late for the meeting.
kid is used informally to mean child (noun) or to make fun of (verb) but is incorrect in formal usage
My cousin is a very sweet child.
They always laugh when you make fun of me.
learn means to acquire knowledge

211
Chapter 5
English
Language
Conventions

teach means to give knowledge
We can learn many things just by observing carefully.
He is one actor who likes to teach his craft to others.
least means the smallest in degree or lowest rank
less means the smaller or lower of two
This is the least desirable of all the apartments we have seen.
This apartment is less spacious than the one we saw yesterday.

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leave means to go away from (a verb is NOT used with leave)
let means to permit (a verb IS used with let)
Leave this house at once.
Let me remain in peace in my own house.
lend is a verb meaning to give to
loan is a noun denoting what is given
borrow means to take from
The bank was willing to lend him $500.
He was granted a loan of $500.
I’d like to borrow your electric drill for an hour.
liable means responsible according to the law
likely suggests probable behavior
If he falls down the stairs, we may be liable for damages.
A cat, if annoyed, is likely to scratch.
libel is a written and published statement injurious to a person’s character

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Conventions

slander is a spoken statement of the same sort
The unsubstantiated negative comments about me in your book constitute libel.
When you say these vicious things about me, you are committing slander.
like is a preposition used to introduce a phrase
as if is used to introduce a clause (a subject and a verb)
as is a conjunction used to introduce a clause
like if is an incorrect form for like, as, or as if
It seems like a sunny day.
It seems as if it is going to be a sunny day.
He acted as he was expected to act.
many refers to a number
much refers to a quantity or amount
How many inches of rain fell last night?
Much rain fell last night.
may of is an incorrect form for may have
might of is an incorrect form for might have

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NOTE: Contractions of these terms are unacceptable in formal usage.
He may have been there, but I didn’t see him.
I might have gone to the party if I hadn’t been ill.

Skill Builder: Usage
DIRECTIONS: Circle the correct word to complete the sentence while adhering to formal American-English conventions.
1. As of 2012, the median annual wage for nonfarm animal caretakers is (less, fewer) than $20,000.
2. During the interrogation, the lawyer (flaunted, flouted) his authority as he questioned the accused thief.
3. In addition to being patient with animals, animal caretakers must also work (good, well) with people!
4. She is one of 2,200 people to (graduate, graduate from) the National Association of Professional Pet Sitters’ certification program.
5. While in captivity, many animals develop nervous (customs, habits), such as pacing and over-grooming.
6. While it is usually not required, animal caretakers in shelters (ought, had ought) to attend training programs through the
Humane Society of the United States and the American Humane Association.
7. Keepers in zoos plan diets, monitor eating patterns, and clean enclosures in order to maintain (healthful, healthy) animals.
8. Often, the difference between the needs of animals and (human beings, humans) is not as great as you think.
9. (Irregardless, Regardless) of one’s education, the number one trait an animal trainer must have is a love for animals.
10. While it does happen, it is (rather, sort of ) rare for a groom to be hired who does not have significant prior experience with
horses.

213

11. Animal trainers take good care of their animals because when animals are neglected, they are more (liable, likely) to become
aggressive and dangerous.

Chapter 5

Answers

English
Language
Conventions

1. less
2. flaunted
3. well
4. graduate from
5. habits
6. ought
7. healthy
8. human beings
9. Regardless
10. rather
11. likely

Peterson’s SAT® Prep Guide 2017

Frequently Confused Words: Group 5
maybe means perhaps, possibly (adverb)
may be shows possibility (verb)
Maybe he will meet us later.
He may be here later.
mighty means powerful or great; it should not be used in formal writing to mean very
He was very (not mighty) sleepy.
media is the Latin plural of medium; it refers to a means of mass communication or artistic expression and is used with a plural verb
Most media that report the news realize their responsibility to the public.
That artist’s favorite medium is watercolor.
must of is an incorrect form for must have
I must have been sleeping when you called. (A contraction of this term is unacceptable in formal usage.)
myself is used as an intensifier if the subject of the verb is I
myself instead of I or me is not correct
Since I know myself better, let me try it my way.
My daughter and I (not myself) will play.

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Language
Conventions

They gave my son and me (not myself) some food.
nice is used informally to mean pleasing, good, fine, but a more exact, less overused word is preferred
This is sunny (or good or fine) weather (not nice weather).
He is a good (or kind) person.
nowheres is incorrect usage for nowhere
The dog was nowhere to be found.
off of is sometimes used informally, but off is correct in formal usage
Joe was taken off the team.
okay (O.K.) is used informally but is to be avoided in formal writing
Informal: His work is okay.
Formal: His work is acceptable (or good).
on account of is an incorrect form of because
We could not meet you because we did not receive your message in time.
oral means spoken
verbal means expressed in words, either spoken or written

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Instead of writing a note, she gave him an oral message.
Shorthand must usually be transcribed into verbal form.
outdoor is an adjective
outdoors is an adverb
We spent the summer at an outdoor music camp.
We played string quartets outdoors.
owing to is used informally, but in formal usage because is preferred
Because of a change of management, his company canceled the takeover attempt.
people comprise a united or collective group of individuals
persons are individuals who are separate and unrelated
The people of our city will vote next week.
Only ten persons remained in the theater after the first act.
per is a Latin term used mainly in business: per diem (by the day), per hour (by the hour). In formal writing, according to or by the
is preferred
As per your instructions... (Better: According to your instructions...)
plan on is used informally, but in formal usage plan to is correct
Do you plan to go (not plan on going) to the lecture?
plenty means abundance (noun)
plenty is incorrect as an adverb or adjective
There is plenty of room in that compact car.
That compact car is very large (not plenty large).

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Conventions

prefer that than is the incorrect form for prefer that to
I prefer that to anything else you might suggest.
put in is incorrect for to spend, make, or devote
Every good student should spend (not put in) several hours a day doing homework.
Be sure to make (not put in) an appearance at the meeting.
quit is sometimes used informally to mean stop, but in formal usage stop is preferred
Please stop your complaining.
quite is used to mean very in informal usage, but in formal usage very is preferred
Your comment was very (not quite) intelligent.

Peterson’s SAT® Prep Guide 2017

quite a few is used to mean many in informal usage, but in formal usage many is preferred
My car has many (not quite a few) dents.
read where is heard in informal usage, but in formal usage read that is correct
I read that the troops were being reviewed today.
real is sometimes used informally instead of really or very, but in formal usage really is correct
He’s a very (not real) good ballplayer.
He plays really (not real) well with the band.
reason is because is used informally in speech, but in formal usage the reason is that is correct
The reason she calls is that (not because) she is lonely. (Or: She calls because she is lonely.)
refer back/report back: since re means back or again, the word back is redundant and should be omitted
Please refer to your notes.
Please report to the supervisor.
repeat again is redundant; again should be omitted
Please repeat the instructions.
respectfully means with respect and decency
respectively means as relating to each, in the order given

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Conventions

The students listened respectfully to the principal.
Jane and Lena are the daughters respectively of Mrs. Smith and Mrs. Jones.
run is used informally to mean conduct, manage, but in formal usage conduct or a similar word is preferred
He wants to conduct (not run) the operation on a profitable basis.
said is sometimes used in business or law to mean the or this; in formal usage, the or this is correct
said is also used incorrectly to mean told someone
When the (not said) coat was returned, it was badly torn.
The professor told us (not said) to study for the examination.
same as is an incorrect form for in the same way as or just as
The owner’s son was treated in the same way as any other worker.
says is present tense of say
said is past tense of say
He says what he means.
He said what he meant. (Goes or went should not be used instead of says or said.)

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Skill Builder: Usage
DIRECTIONS: Circle the correct word to complete the sentence while adhering to formal American-English conventions.
1. If you are interested in the social and organizational side of businesses, the field of organizational psychology (may be,
maybe) the field for you.
2. As the economy continues to grow and more businesses are created, organizational psychology is becoming a (mighty, very)
important field.
3. (Because of, On account of ) their collaborative work with social workers and healthcare professionals, demand for psychologists is expected to increase in tandem with the healthcare industry overall.
4. (Nowhere, Nowheres) is the demand for psychologists growing more than in the field of organization psychology.
5. Because of the importance of talking with patients, clinical psychologists are expected to have good (oral, verbal) communication skills.
6. Psychologists typically work in offices, so if you prefer working (outdoor, outdoors), this might not be the job for you.
7. Over the next decade, the job market for psychologists will continue to grow in large part (because of, owing to)
increased demand for psychological services in schools, hospitals, mental health centers, and social services agencies.
8. Most (people, persons) don’t realize that there is a difference between a Ph.D. in psychology (a research degree) and a Psy.D.
(a clinical degree).
9. While a master’s degree is sufficient for some positions, most aspiring psychologists (plan on getting, plan to get) a Ph.D. or
a Psy.D.
10. In addition to formal schooling, most clinical psychologists (put in, spend) hundreds of unpaid hours as interns in their fields
of specialization.
11. The reason most aspiring psychologists get Ph.D.’s and Psy.D.’s is (because, that) there are more jobs available for psychologists
with terminal degrees.

Answers

217
Chapter 5
English
Language
Conventions

1. may be
2. very
3. Because of
4. Nowhere
5. oral
6. outdoors
7. because of
8. people
9. plan to get
10. spend
11. that

Peterson’s SAT® Prep Guide 2017

Frequently Confused Words: Group 6
saw is the past tense of see
seen is the past participle of see
We saw a play yesterday.
I have never seen a Broadway show.
seem is used in informal speech and writing in the expressions I couldn’t seem to and I don’t seem to but in formal usage:
We can’t find the address. (Not: We can’t seem to find the address.)
seldom ever is used informally, but in formal usage ever is redundant and should be omitted, or if should be inserted
I seldom swim in January.
I seldom if ever swim in January.
shall is used with I and we in formal usage; informally, I will (would) may be used
will is used with you, he, she, it, they
When an emphatic statement is intended, the rule is reversed
I shall be there today.
We shall pay the rent tomorrow.
I certainly will be there.

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Language
Conventions

They shall not pass.
shape is incorrect when used to mean state or condition
The refugees were in serious condition (not shape) when they arrived here.
should of is an incorrect form for should have, which can be contracted to should’ve in speech or informal writing
You should’ve returned that sweater. (Better: You should have returned that sweater.)
sink down is sometimes heard in informal usage, but down is redundant and should be omitted
You can sink into the mud if you are not careful.
some time means a segment of time
sometime means at an indefinite time in the future
sometimes means occasionally
I’ll need some time to make a decision.
Let’s meet sometime next week.
Sometimes I have an urge to watch a late movie on television.

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stayed means remained
stood means took or remained in an upright position or erect
He stayed in bed for three days.
The scouts stood at attention while the flag was lowered.
still more yet is redundant; yet should be omitted
There is still more to be said.
sure is used informally to mean surely or certainly, but in formal usage surely or certainly is preferred
She certainly (not sure) is pretty!
We will surely be in trouble unless we get home soon.
testimony means information given orally
evidence means information given orally or in writing; an object that is presented as proof
He gave testimony to the grand jury.
He presented written evidence to the judge.
than any is used informally in a comparison, but in formal usage than any other is preferred
He is smarter than any other boy in the class.
the both is used informally, but in formal usage the should be omitted

219

I intend to treat both of you to lunch.
their, in informal usage, often appears in the construction, “Anyone can lose their card,” but because anyone takes a singular
personal pronoun, his or her is the correct form

Chapter 5
English
Language
Conventions

theirselves is an incorrect form for themselves
They are able to care for themselves while their parents are at work.
them is the objective case of they; it is not used instead of those (the plural of that) before a noun
Give me those (not them) books!
try and is sometimes used informally instead of try to, but in formal usage try to is correct
My acting teacher is going to try to attend the opening of my play.
unbeknownst to is unacceptable for without the knowledge of
The young couple decided to get married without the knowledge of (not unbeknownst to) their parents.
upwards of is an incorrect form for more than
There are more than (not upwards of) 60,000 people at the football game.

Peterson’s SAT® Prep Guide 2017

valuable means of great worth
valued means held in high regard
invaluable means priceless
This is a valuable manuscript.
You are a valued friend.
A good name is an invaluable possession.
wait on is sometimes used informally, but in formal usage wait for is correct
We waited for (not on) him for over an hour.
which is sometimes used incorrectly to refer to people; it refers to things
who is used to refer to people
that is used to refer to people or things
He decided to wear his orange striped tie, which had been a gift from his daughter.
I am looking for the girl who made the call.
He finally returned the books that he had borrowed.
while is unacceptable for and, but, whereas, or though
The library is situated on the south side, whereas (not while) the laboratory is on the north side.

220

Though (not while) I disagree with you, I shall not interfere with your right to express your opinion.
who is, who am is used with these constructions:

Chapter 5
English
Language
Conventions

It is I who am the most experienced.
It is he who is...
It is he or I who am...
It is I or he who is...
It is he and I who are...
who, whom To determine whether to use who or whom (without grammar rules), use he, him:
(Who, Whom) do you think should represent our company?
Step 1:

Change the who—whom part of the sentence to its natural order:

		

Do you think (who, whom) should represent our company?

Step 2:

Substitute he for who, and him for whom:

		

Do you think (he, him) should represent our company?

Step 3:

Since he would be used in this case, the correct form is:

		

Who do you think should represent our company?

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whoever, whomever (see who, whom above)
Give the chair to whoever wants it (subject of verb wants).
Speak to whomever you see (object of preposition to).
win is used when you win a game
beat is used when you beat another player; beat is incorrect usage for swindle
We won the contest.
We beat (not won) the other team.
The hustler swindled the gambler out of twenty dollars.
without is incorrect usage for unless
You will not receive the tickets unless (not without) you pay for them in advance.
worst kind and worst way are incorrect usages for terms such as very badly or extremely
The school is greatly in need of more teachers (not needs teachers in the worst way).
would of is an incorrect form for would have, which can be contracted to would’ve in informal usage
He would’ve treated you to the movies. (Better: He would have treated you to the movies.)
would have is not used instead of had in an if clause
If I had (not would have) gone, I would have helped him.

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Peterson’s SAT® Prep Guide 2017

Skill Builder: Usage
DIRECTIONS: Circle the correct word to complete the sentence while adhering to formal American-English conventions.
1. In Sir Arthur Conan Doyle’s story, “A Scandal in Bohemia,” Sherlock Holmes (can’t believe, can’t seem to believe) that he’s been
outsmarted by the beautiful and alluring Irene Adler.
2. Forest conservation workers are (seldom, seldom ever) required to have education beyond a basic bachelor’s degree.
3. According to his fictional biography, Sherlock Holmes’s friend, John H. Watson, returns to England in serious (condition,
shape) after being wounded in the Second Anglo-Afghan War.
4. (Some time, Sometime) in the near future, the U.S. Forest Service may be forced to undertake its own fire suppression duties,
which will result in higher levels of employment.
5. At the end of all Sherlock Holmes stories, Holmes presents (evidence, testimony) that shows how a crime was committed
and whom it was committed by.
6. According to some scholars, Sherlock Holmes is more recognizable (than any, than any other) fictional detective in the world.
7. The complexity of the Tube, Britain’s underground railway system, makes it possible for anyone to lose (his or her, their) way.
8. In the event that they get lost, forest service workers need to be able to take care of (theirselves, themselves) in the wild.
9. If formal schooling is unavailable, aspiring forest and conservation services workers should (try and, try to) gain experience
working in forestry-related fields.
10. As of 2012, there were (more than, upwards of ) 10,000 jobs in the forest and conservation services field.

222
Chapter 5
English
Language
Conventions

Answers
1. can’t believe
2. seldom
3. condition
4. Some time
5. evidence
6. than any other
7. his or her
8. themselves
9. try to
10. more than

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Conventional Expressions
A conventional expression is a phrase or clause that has become a characteristic way of expressing a certain idea. Ironically,
despite these expressions being conventional, they are often misused.
Here is a list of commonly misused conventional expressions and their correct usages.

Incorrect Usage

Correct Usage

It’s a doggy-dog world

It’s a dog-eat-dog world

For all intensive purposes

For all intents and purposes

I’m suppose to go running

...supposed to…

statue of limitations

statute of limitations

I could care less

I couldn’t care less

Fall by the waste side

Fall by the wayside

Irregardless

Regardless

Escape goat

Scapegoat

I guess we’ll make due

...make do...

Peak my interest

Pique my interest

The criminal got away scott free

...scot free...

I’m waiting with baited breath

...bated breath...

Without further adieu

Without further ado

The boy had free reign

...free rein...

Hunger pains

Hunger pangs

I should of called

...should have...

Don’t step foot on this carpet

...set foot...

Nipped in the butt

Nipped in the bud

The waiter was at his beckon call

...beck and call...

The lawyer made a mute point

...moot point...

Case and point

Case in point

The cops were starting to hone in

...home in...

One in the same

One and the same

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Language
Conventions

Peterson’s SAT® Prep Guide 2017

Logical Comparison
Basic Rule
In order for a comparison to make sense, it must be logical and complete.
In an incomplete comparison, what is being compared is unclear.
Incomplete: According to some scholars of U.S. history, the Magna Carta is more important because it was the first
document to limit the powers of the King of England.
Complete: According to some scholars of U.S. history, the Magna Carta is more important than the Declaration of
Independence because it was the first document to limit the powers of the King of England.
Incomplete: At 864,000 miles across, the Sun is more than 100 times the size.
Complete: At 864,000 miles across, the Sun is more than 100 times the size of Earth.
For a comparison to be logical, you must be comparing the same things.
Illogical: Despite popular opinion, many scholars believe that the political achievements of Thomas Jefferson were
much more significant to American history than George Washington. (Thomas Jefferson’s achievements are being
compared to the person George Washington.)
Logical: Despite popular opinion, many scholars believe that the political achievements of Thomas Jefferson were
much more significant to American history than the achievements of George Washington.

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Illogical: The distance from the Sun to the planet Neptune is 30 times greater than the Sun and Earth. (This distance
from the Sun to Neptune is being compared to the Sun and Earth, not the distance between them.)

Chapter 5

Logical: The distance from the Sun to the planet Neptune is 30 times greater than the distance from the Sun to
Earth.

English
Language
Conventions

Skill Builder: Logical Comparisons
DIRECTIONS: Revise the following comparisons to make sure they are logical.

1. According to scientists, the brain mass of a dolphin is actually slightly greater than a human.

2. Although they are on the same hemisphere, the average rainfall in South America differs greatly from North America.

3. In the 2015 World Happiness Report, the people of Switzerland ranked happier than America.

4. The tallest mountain in the United States is still 9,000 feet shorter than Nepal.

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Answers
1. According to scientists, the brain mass of a dolphin is actually slightly greater than the brain mass of a human.
2. Although they are on the same hemisphere, the average rainfall in South America differs greatly from the average rainfall
in North America.
3. In the 2015 World Happiness Report, the people of Switzerland ranked happier than the people of America.
4. The tallest mountain in the United States is still 9,000 feet shorter than the tallest mountain in Nepal.

CONVENTIONS OF PUNCTUATION
End-of-Sentence Punctuation
There are three types of punctuation used to end a sentence: the period, the question mark, and the exclamation mark.
1.

A period is used at the end of a sentence that makes a statement.
Examples:
Toni Morrison’s first novel, The Bluest Eye, was published in 1970.
In 1620, the Pilgrims in Plymouth signed the Mayflower Compact.

2.

A question mark is used after a direct question. A period is used after an indirect question.
Examples:
Direct Question—Were The Federalist Papers written by James Madison, John Jay, or Alexander Hamilton?
Indirect Question—Profession Mahin wanted to know if you knew who wrote The Federalist Papers.

3.

An exclamation mark is used after an expression that shows strong emotion or issues a command. It may follow a word, a
phrase, or a sentence.
Examples:

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Koko the gorilla knows more than 1,000 sign-language signs and can communicate with humans.
Amazing!
One of the most famous quotes in American history is Patrick Henry’s, “Give me liberty or give me
death!”

Peterson’s SAT® Prep Guide 2017

Within-Sentence Punctuation
The Colon
Basic Rule
The colon is used to precede a list of three or more items or a long quotation.
Examples:
Christopher Columbus led three ships to the New World: La Nina, La Pinta, and La Santa Maria.
In the United States, there are three branches of government: the Executive, the Legislative, and the
Judicial.

Avoid using the colon after a verb. It can interrupt the natural flow of language.
Incorrect: The Louisiana Purchase included territory that would become:
Montana, South Dakota, Nebraska, Kansas, Oklahoma, Arkansas, Louisiana, and
Missouri.
Correct: The Louisiana Purchase included territory that would become many of
today’s states: Montana, South Dakota, Nebraska, Kansas, Oklahoma, Arkansas,
Louisiana, and Missouri.

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The Semicolon
Basic Rule
A semicolon may be used to separate two complete ideas (independent clauses) in a sentence when the two ideas have a close
relationship and are not connected with a coordinating conjunction.
Example: “Inalienable rights” are basic human rights that many believe cannot and should not be given up or taken
away; life, liberty, and the pursuit of happiness are some of those rights.
The semicolon is often used between independent clauses connected by conjunctive adverbs such as consequently, therefore,
also, furthermore, for example, however, nevertheless, still, yet, moreover, and otherwise.
Example: In 1867, critics thought William H. Seward foolish for buying the largely unexplored territory of Alaska
for the astronomical price of $7 million; however, history has proven that it was an inspired purchase.
Do not use the semicolon between an independent clause and a phrase or
subordinate clause.
Incorrect: While eating ice cream for dessert; Clarence and Undine
discussed their next business venture.
Correct: While eating ice cream for dessert, Clarence and Undine discussed
their next business venture.

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Skill Builder: Punctuation
DIRECTIONS: Decide whether the colons and semicolons are correctly placed in the following sentences or whether another
mark of punctuation would be better. Write the correct punctuation in the space provided.

1. He is an excellent student and a fine person; as a result, he has many friends.

2. Because he is such an industrious student; he has many friends.

3. We tried our best to purchase the books; but we were unsuccessful.

4. The students were required to pass the following exit tests: English, science, math, and social studies.

5. The rebuilt vacuum cleaner was in excellent condition; saving us a good deal of expense since we didn’t have to purchase
a new one.

6. Marie has a very soft voice; however, it is clear and distinct.

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7. Don’t open the door; the floor is still wet.

Chapter 5

8. Don’t open the door; because the floor is still wet.

English
Language
Conventions

9. To the campers from the city, every noise in the night sounded like a bear: a huge, ferocious, meat-eating bear.

10. We worked for three days painting the house; nevertheless, we still needed more time to complete the job.

11. The telephone rang several times, as a result; his sleep was interrupted.

12. Peter was chosen recently to be vice president of the business; and will take over his duties in a few days.

Peterson’s SAT® Prep Guide 2017

Answers
1. Correct.
2. Substitute a comma for the semicolon.
3. Substitute a comma for the semicolon.
4. Correct.
5. Substitute a comma for the semicolon.
6. Correct.
7. Correct.
8. Delete the semicolon.
9. Substitute a comma for the colon.
10. Correct.
11. The telephone rang several times; as a result, his sleep was interrupted. (Note the two punctuation changes. The semicolon
is placed in front of the conjunctive adverb and the comma after it.)
12. Delete the semicolon; no punctuation is necessary in its place.

The Em-Dash
Basic Rule

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Em-dashes are used to set off parenthetical material that you want to emphasize.
Example: Benjamin Franklin’s many intellectual pursuits—from printmaking to politics—exemplify his eclectic
personality.
Em-dashes can also be used when you are renaming a nearby noun. Typically, you would use a comma to set this clause off, but
since it includes commas already, you use an em-dash.
Example: Benjamin Franklin—a printer, writer, inventor, and statesman—was the son of a soap maker.
An em-dash also indicates a list, a restatement, an amplification, or a dramatic shift in tone or thought.
Example: Eager to write for his brother’s newspaper, young Benjamin began submitting letters to the editor under
the pseudonym, Silence Dogood—they were a hit!

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Skill Builder: Punctuation
DIRECTIONS: Revise the following sentences to eliminate end-of-sentence and within-sentence punctuation errors.
1. Architects design houses, office buildings, and other structures!
2. On any given day, architects might perform the following tasks; prepare structural specifications; meet with clients to determine objectives and structural requirements; direct workers to prepare drawings and documents.
3. Architects are responsible for designing the places where we live: work: play: learn: shop: and eat?
4. Architects—design both indoor and outdoor—spaces on public and private projects.
5. Architects often provide various predesign services: from environmental impact studies to cost analyses: depending on a
project’s needs.
6. For actual blueprints, traditional paper-and-pencil drafting has been replaced by computer-aided design and drafting (CADD):
however, hand-drawing skills are still important during the conceptual stages of a project.
7. Did you know that BIM stands for business information modeling.
8. Architects often collaborate with workers in related fields; civil engineers, urban planners, interior designers, and landscape
architects.
9. In addition to years of schooling, being a good architect requires a mix of artistic talent and mathematical ability: it’s not
easy!
10. The path to becoming an architect requires a college education: a five-year Bachelor of Architecture degree program is
typical.

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Answers

Chapter 5

1. Architects design houses, office buildings, and other structures.
2. On any given day, architects might perform the following tasks: prepare structural specifications, meet with clients to determine objectives and structural requirements, and direct workers to prepare drawings and documents.

English
Language
Conventions

3. Architects are responsible for designing the places where we live, work, play, learn, shop, and eat.
4. Architects design both indoor and outdoor spaces on public and private projects.
5. Architects often provide various predesign services—from environmental impact studies to cost analyses—depending on
a project’s needs.
6. For actual blueprints, traditional paper-and-pencil drafting has been replaced by computer-aided design and drafting (CADD);
however, hand-drawing skills are still important during the conceptual stages of a project.
7. Did you know that BIM stands for business information modeling?
8. Architects often collaborate with workers in related fields: civil engineers, urban planners, interior designers, and landscape
architects.
9. In addition to years of schooling, being a good architect requires a mix of artistic talent and mathematical ability—it’s not
easy!
10. The path to becoming an architect requires a college education; a five-year Bachelor of Architecture degree program is
typical.

Peterson’s SAT® Prep Guide 2017

The Apostrophe
The apostrophe is usually either misused or omitted because of the writer’s failure to proofread his paper or because he is not
certain about its use. The apostrophe is used in the following situations:

• To indicate the possessive case of nouns: If the noun does not end in s—whether singular or plural—add an ’s; if the noun ends
in s simply add the ’. Some writers like to add ’s to all nouns, even those that already end in s.
Examples:
The impact of Allen Ginsberg’s poem “Howl” on the cultural landscape of the United States cannot be overstated.
A car’s headlights are typically wired in parallel so that if one burns out the other will keep functioning.
The women’s club sponsored many charity events.
Charles Mingus’ (or Mingus’s) skill as a jazz musician is widely recognized.

Do not use apostrophes with possessive pronouns such pronouns as yours, hers, ours, theirs,
and whose, which indicate possession already.

• To indicate a contraction—the omission of one or more letters: Place the apostrophe exactly where the missing letters occur.

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Examples:
can’t = cannot
it’s = it is
we’re = we are

• To indicate plurals of letters, abbreviations, and numbers: Usually, the apostrophe is used to form the plurals of lowercase letters
(a’s, b’s, c’s, and so on) and numbers (3’s, 6’s). With capital letters, abbreviations without periods (PhD, RN), and even with
numbers when no confusion results, you have a choice. In either case, the writer should be consistent in his or her style.
Examples:
Tiffani signed her texts with x’s and o’s.
The class learned their multiplication tables for 2’s and 4’s.
BUT:
Jocelyn recited her ABCs for her parents.
The room was filled with those who had earned their PhDs.

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Skill Builder: Apostrophe Use
DIRECTIONS: Revise the following sentences to correct any apostrophe errors.

1. Poet William Wordsworths most famous work is The Prelude, which was published in 1850.
2. Its how Wordsworth uses the language of the “common man” that strikes most readers.
3. While most of his poems are considered classics, The Prelude stands as one of the crowning achievements of British Romanticism.
4. Wordsworths poem The Prelude was published by his wife Mary three months after his death.
5. Sonnet 18s theme revolves around the idea of expressing ones self through language.
6. What proportion of humans exposure to plastic ingredients and environmental pollutants occurs through seafood?
7. The use of student’s personal information for anything other than educational purposes is a violation of privacy.

Answers
1. Poet William Wordsworth’s most famous work is The Prelude, which was published in 1850.
2. It’s how Wordsworth uses the language of the “common man” that strikes most readers.
3. While most of his poems are considered classics, The Prelude stands as one of the crowning achievements of British Romanticism.
4. Wordsworth’s poem The Prelude was published by his wife Mary three months after his death.
5. Sonnet 18’s theme revolves around the idea of expressing one’s self through language.
6. Do you know what proportion of humans’ exposure to plastic ingredients and environmental pollutants occurs through
seafood?
7. The use of students’ personal information for anything other than educational purposes is a violation of privacy.

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Peterson’s SAT® Prep Guide 2017

Skill Builder: Apostrophe Use
DIRECTIONS: Circle the word with the correct spelling in the following sentences.

1. According to (statistics, statistic’s), 55 percent of the nursing workforce holds a (bachelors, bachelor’s) degree or higher.
2. Over the past decade, the average age of (nurses, nurse’s) has increased by almost two years for (RNs, RN’s) and 1.75 years
for (LPNs, LPN’s).
3. According to the Board of Registered (Nurse’s, Nurses’) list of (regulations, regulation’s), registered (nurses, nurse’s) must have
a high school diploma, appropriate pre-licensure schooling, and required certification.
4. Despite teaching similar (skills, skill’s), one nursing (program, program’s) (requirements, requirement’s) can be very different
from another (programs, program’s) (requirements, requirement’s).
5. Many future (RNs, RN’s) attend the following types of pre-licensure degree (programs, program’s): Associate Degree in Nursing
(ADN); Bachelor of Science in Nursing (BSN); Entry-Level Master’s Program in Nursing (ELM).
6. Based on recent (studies, studies’), an (RNs, RN’s) salary is about $50,000 a year.

Answers
1. statistics; bachelor’s
2. nurses; RNs; LPNs
3. Nurses’; regulations; nurses

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4. skills; program’s; requirements; program’s; requirements
5. RNs; programs
6. studies; RN’s

Items in a Series
Basic Rule
Use a comma between items in a series when three or more items are present. Items can be expressed as words, phrases, or clauses.
Example: The following wildlife biologists study animals based on where they live: limnologists, marine biologists,
and terrestrial biologists.
When the items themselves contain commas, use a semicolon to separate them.
Example: Some kinds of biologists study specific species of animals. For example, cetologists study marine mammals,
such as whales and dolphins; entomologists study insects, such as beetles and butterflies; and ichthyologists study
wild fish, such as sharks and lungfish.

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Skill Builder: Items in a Series
DIRECTIONS: Revise the following sentences to make sure the items in a series are punctuated correctly.
1. A typical architectural program includes courses on such topics as architectural history, building design, computer-aided
design, and math.
2. Architects must possess certain qualities: analytical skills, in order to understand the content of designs, communication skills,
in order to communicate with clients, creativity, in order to develop attractive and functional structures, and organizational
skills, in order to keep track of big projects.
3. In order to be hired as an architect, you typically need to: complete a professional degree in architecture: gain relevant
experience through a paid internship: and pass the Architect Registration Exam.
4. Because of growing concerns about the environment, today’s architects need to understand sustainable design, which
emphasizes the efficient use of resources, such as energy and water conservation, waste and pollution reduction, and environmentally friendly specifications and materials.
5. In addition to structural plans architects often provide drawings of the air-conditioning heating and ventilating systems
electrical systems communications systems plumbing and possibly site and landscape plans.

Answers
1. No corrections are needed. The sentence reads fine as: A typical architectural program includes courses on such topics as
architectural history, building design, computer-aided design, and math.
2. Architects must possess certain qualities: analytical skills, in order to understand the content of designs; communication skills,
in order to communicate with clients; creativity, in order to develop attractive and functional structures; and organizational
skills, in order to keep track of big projects.
3. In order to be hired as an architect, you typically need to complete a professional degree in architecture, gain relevant experience through a paid internship, and pass the Architect Registration Exam.
4. Because of growing concerns about the environment, today’s architects need to understand sustainable design, which
emphasizes the efficient use of such resources as energy and water; waste and pollution reduction; and environmentally
friendly specifications and materials.

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5. In addition to structural plans, architects often provide drawings of the air-conditioning, heating, and ventilating systems;
electrical systems; communications systems; plumbing; and possibly site and landscape plans.

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Nonrestrictive and Parenthetical Elements
Basic Rule
Nonrestrictive and parenthetical elements provide extra information that is not essential to the meaning or grammatical correctness of a sentence. A nonrestrictive or parenthetical element can be removed from a sentence without making the sentence
grammatically incorrect and without interfering with the rest of the sentence’s meaning.
Parenthetical elements are identified by commas, parentheses, or em-dashes. While each of these punctuation marks serves a
similar purpose, the difference between them is one of emphasis.
Commas indicate a slight interruption.
Example: Toni Morrison, who won the Nobel Prize in Literature in 1993, is a Professor Emeritus at Princeton University.
Parentheses are seen as “quieter” than commas and are reserved for asides that are less important and more tangential than
those indicated by commas. They also allow the inclusion of material that doesn’t have a specific grammatical connection to the
rest of the sentence.
Example: Toni Morrison’s novel Beloved (1987) explores the themes of love and the supernatural.
Example: While at Princeton, Toni Morrison (the writer) established a special creative workshop for writers and
performances called the Princeton Atelier.
If parentheses are used for “quiet” asides, dashes are used when you want to call attention to something. Dashes interrupt the
flow of your sentence, thereby calling attention to the information they contain.

234

Example: Toni Morrison—winner of both the Pulitzer and Nobel Prizes in Literature—is considered one of the
greatest writers of her generation.

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Conventions

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Skill Builder: Nonrestrictive and Parenthetical Elements
DIRECTIONS: Select a comma, parentheses, or em-dash to set off the underlined portion of each sentence.

1. Samuel Clemens 1835–1910 was born in Florida‚ Missouri, to John and Jane Clemens.
2. Mark Twain whose given name was Samuel Clemens was the sixth of seven children.
3. In 1861, Sam’s dreams of becoming a steamboat pilot ended abruptly the Civil War started.
4. Sam’s commitment to the Confederate cause was short-lived he quit the army after just two weeks.
5. Twain’s “big break” came with the publication of his short story‚ “Jim Smiley and His Jumping Frog” 1865, which was picked
up by papers across the country.
6. After the success of his story, “Jim Smiley and His Jumping Frog,” Clemens was hired by the Sacramento Union to visit and
report on the Sandwich Islands now Hawaii.
7. Clemens’ writings for the Sacramento Union were so popular that upon his return he was asked to undertake a lecture tour
across the United States.
8. Like all good writers‚ Mark Twain Samuel Clemens spent his life observing and writing about life as he saw it, with all of its
joys and horrors.

Answers
1. Samuel Clemens (1835–1910) was born in Florida‚ Missouri, to John and Jane Clemens.
2. Mark Twain, whose given name was Samuel Clemens, was the sixth of seven children. (Em-dashes might also be acceptable
here if the writer wished to emphasize the information.)
3. In 1861, Sam’s dreams of becoming a steamboat pilot ended abruptly—the Civil War started.
4. Sam’s commitment to the Confederate cause was short-lived—he quit the army after just two weeks.
5. Twain’s “big break” came with the publication of his short story‚ “Jim Smiley and His Jumping Frog” (1865), which was picked
up by papers across the country.

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6. After the success of his story, “Jim Smiley and His Jumping Frog,” Clemens was hired by the Sacramento Union to visit and
report on the Sandwich Islands (now Hawaii).
7. Clemens’ writings for the Sacramento Union were so popular that, upon his return, he was asked to undertake a lecture tour
across the United States.
8. Like all good writers‚ Mark Twain (Samuel Clemens) spent his life observing and writing about life as he saw it, with all of its
joys and horrors.

Peterson’s SAT® Prep Guide 2017

The Comma
Basic Rule
In previous sections, we covered the way that commas are used to separate the following:

• Independent clauses that are connected by a coordinating conjunction
• Items in a series
• Nonrestrictive elements
In addition to these uses, commas have several other purposes:

• To set off introductory clauses and phrases.
Example: The year after winning her Nobel Prize, Toni Morrison published the novel Jazz.

• To set off nouns that are being addressed directly.
Example: Toni Morrison, the Nobel Prize committee commends you on your achievements and thanks you for your
contribution to world literature.

• To separate the different parts of dates, addresses, and geographical names.
Example: Toni Morrison was born on February 18, 1931, in Lorain, Ohio, to Ramah and George Wofford.

• To introduce a titles and quotations.

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Example: Toni Morrison began her lecture, “The Future of Time: Literature and Diminished Expectations,” with the
line, “Time, it seems, has no future.”

• To separate contrasted elements or to indicate a distinct pause or shift.
Examples: To her handlers, Koko the gorilla seemed thoughtful, almost human.
You’re one of the senator’s close friends, aren’t you?

• To separate coordinate adjectives that precede the noun they describe. Coordinate adjectives are of equal importance and
related meaning.
Examples: Toni Morrison is rumored to be a fun, entertaining speaker.

Note how the word and can be substituted for the comma. If you cannot
substitute and without changing the meaning, the adjectives are not
coordinate, and no comma is needed.
Example: Toni Morrison is a well-respected American writer.

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Unnecessary Punctuation
Unnecessary punctuation can break a sentence into confusing and illogical fragments.
Here are some common mistakes to look out for.

• Don’t use a comma to connect independent clauses. This is called a comma splice.
Incorrect: Toni Morrison grew up in an integrated neighborhood, she did not become fully aware of racial divisions
until she was in her teens.
Possible revision: Toni Morrison grew up in an integrated neighborhood and did not become fully aware of racial
divisions until she was in her teens.

• Don’t use a comma between compound elements that are not independent clauses.
Incorrect: In 1998, Oprah Winfrey, and Danny Glover starred in a film adaptation of Morrison’s novel Beloved.
Possible revision: In 1998, Oprah Winfrey and Danny Glover starred in a film adaptation of Morrison’s novel Beloved.

• Do not use an apostrophe when making a noun plural.
Incorrect: In 2006, the New York Times Book Review named Beloved the best American novel published in the last
25 year’s.
Possible revision: In 2006, the New York Times Book Review named Beloved the best American novel published in
the last 25 years.

237
While any punctuation mark can be misused, overused commas tend to
be a common problem.

Chapter 5
English
Language
Conventions

Peterson’s SAT® Prep Guide 2017

Skill Builder: Commas and Unnecessary Punctuation
DIRECTIONS: Revise the following sentences to correct errors, with special attention to unnecessary, misused, and missing
punctuation marks. In some cases, there are multiple ways to fix these errors. Consider the answers a partial list of possible
revisions.

1. The job of an art director is a creative one he or she is responsible for the visual style and images in magazines newspapers
product packaging, and movie and television productions.

2. An art director’s job includes creating the overall design of a project; and directing others who develop artwork and layouts.

3. People interested in becoming art director’s often work as graphic designer’s, illustrator’s, copy editor’s, or photographer’s,
or in some other types of art and design occupations.

4. Some art directors work for advertising and public relations firms and others work in print media and entertainment.

5. In order to become an art director you typically need at least a bachelor’s degree in an art or design subject, and previous
work experience.

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6. Art direction is a management position, that oversees the work of other designers and artists.

Chapter 5
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Conventions

7. An art director might choose the overall style or tone, desired for a project, and communicate this vision to the artists he or
she manages.

8. In the movie industry an art director might collaborate with a director; in order to determine the look and style of a movie.

9. As of 2012: art directors held about 74,800 jobs.

10. Even though the majority of art directors are self-employed they often work under pressure to meet strict deadlines.

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Answers
1. The job of an art director is a creative one; he or she is responsible for the visual style and images in magazines, newspapers,
product packaging, and movie and television productions.
2. An art director’s job includes creating the overall design of a project and directing others who develop artwork and layouts.
3. People interested in becoming art directors often work as graphic designers, illustrators, copy editors, or photographers or
in some other types of art and design occupations.
4. Some art directors work for advertising and public relations firms, and others work in print media and entertainment.
5. In order to become an art director, you typically need at least a bachelor’s degree in an art or design subject and previous
work experience.
6. Art direction is a management position that oversees the work of other designers and artists.
7. An art director might choose the overall style or tone desired for a project and communicate this vision to the artists he or
she manages.
8. In the movie industry, an art director might collaborate with a director in order to determine the look and style of a movie.
9. As of 2012, art directors held about 74,800 jobs.
10. Even though the majority of art directors are self-employed, they often work under pressure to meet strict deadlines.

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SUMMING IT UP
• The SAT® Writing and Language Test is designed to test your mastery of Standard English conventions. On the test, you will
read multiple passages that may cover careers, science, history, or the humanities. The questions will require you to read the
passages and select the answers that improve the writing in the passage. The correct answer will be the one that best
follows Standard English conventions.

• The Standard English conventions reviewed in this lesson are keys to good writing. When you utilize proper sentence
structure, grammar, and punctuation, your writing is stronger, clearer, and more focused. That is why the use of standard
English conventions is important for both college writing and any writing you will do in your future career.

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Access more practice questions, lessons, helpful tips, and expert strategies for the following English language conventions
topics in Peterson’s SAT® Online Course:

• Comparisons
• Modifiers
• Noun Agreement
• Parallelism
• Pronoun Case
• Pronouns

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• Punctuation
• Sentence Improvements
• Subject-Verb Agreement
• Verb Tenses
• What Makes a Sentence?
To purchase and access the course, go to www.petersons.com/sat.

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part v:
sat ® writing strategies
Chapter 6: The SAT® Essay

Chapter 6:

The SAT® Essay
OVERVIEW
A Closer Look at the Essay Question
Pacing Your Writing
Prewriting
Writing the Introduction
Developing Your Ideas
Writing the Conclusion
The Scoring Rubric for the SAT® Essay
Exercise: Practicing Your Essay Skills
Additional Essay Writing Practice
Summing It Up

The Essay section of the SAT® exam is 50 minutes long. In this time, you need to read the essay prompt and plan and write your
essay. It doesn’t need to be—and isn’t supposed to be—a final, polished version. The high school and college English teachers
who will score your essay are trained to view the essays as first drafts. They will be assessing your essay and hundreds of others
against a rubric that guides them to look at the essays holistically. They are reading for an overall general impression of your
reading, writing, and analyzing skills. Later in this chapter, you will analyze a rubric that is similar to the one the scorers will use.

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A CLOSER LOOK AT THE ESSAY QUESTION
You will be given one essay prompt and asked to write an analytical essay in response. You won’t have a choice of questions to
answer. This is good because it saves you time, as you don’t have to decide which one to choose. The essay section is made up of
a prompt that directs you to read and analyze a high-quality source text. In this text, the author makes an argument or examines
a current debate, idea, or trend.
Once you have closely read and analyzed the source text, you can begin your planning. You don’t need any specific subject-area
knowledge to write your essay. The purpose of the essay is to demonstrate for the scorers that you can then analyze it in writing
using your own critical reasoning and evidence drawn from the source text.

PACING YOUR WRITING
You want to use everything you’ve been taught in English class about the writing process—but sped up to fit within 50 minutes.
Pacing yourself is important so that you are able to get your ideas down on paper in a complete, coherent, and unified essay. As
you practice writing essays in this chapter and in the practice tests in this book, work out a pacing schedule for yourself. Begin by
trying out the following timetable and see how it works for you. If necessary, adjust it as you practice, but be sure to give yourself
enough time to finish a complete draft.

244

•
•
•
•
•
•

Prewriting: 10 to 15 minutes
Writing the introduction: 4 to 5 minutes
Writing the body of the essay: 15 to 20 minutes
Writing the conclusion: 3 to 4 minutes
Revising: 3 to 5 minutes
Proofing: 3 to 5 minutes

NOTE: Remember that the readers do not take off points for specific errors in grammar,
usage, and mechanics, but they will take note of a pattern of errors. These can contribute
to a lower score. Check the rubric (later in this chapter).

Chapter 6
The SAT®
Essay

PREWRITING
Before you begin to write your analysis, read the prompt and the source text. Pay attention to the text author’s key claims.
Underline the author’s key claims. Then circle or highlight evidence the author uses to support those claims. Finally, take notes
on important stylistic features or persuasive techniques the author uses in the source text.
You want to spend about 10 to 15 minutes on prewriting, as this will give you time to carefully read the prompt and then read
and analyze the source text. Your goals in this planning stage are to:

• Identify and underline the author’s key claims
• Find and then circle or highlight specific evidence the author uses to support his or her key claims
• Take notes on ways the author has used logic, reasoning, rhetoric (persuasive language techniques), and evidence to
convince readers that his or her key claims are valid

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Familiarizing Yourself with the Prompt and the Passage
The Essay prompt will not change much at all, no matter when and where you are taking the SAT®. This is great news because it
means you can become very familiar and comfortable with the prompt before you take the test. The source text (passage) that
accompanies the prompt will be new to you; however, it will share important qualities with other passages you will read and
analyze as you prepare to write the SAT® Essay.
All passages:

•
•
•
•
•
•
•
•
•
•

Come from high-quality, previously published sources
Contain arguments written for a broad audience
Examine ideas, opinions, views, debates, or trends
Discuss topics in the arts; the sciences; or civic, cultural, or political life
Are interesting, relevant, and accessible to college-bound students at your grade level
Tend NOT to consist of simple pro/con debates on issues
Strive to convey nuanced views on complex subjects
Use evidence, logical reasoning, and/or stylistic and persuasive elements
Are similarly complex: all are challenging—but not too difficult—for readers at your grade level
Do NOT require test-takers to possess prior knowledge of specific topics

The prompt that introduces the passage will be identical or very similar to this:

As you read the passage below, consider how [the author] uses the following:

• Evidence, such as facts or examples, to support claims

245

• Reasoning to develop ideas and to connect claims and evidence
• Stylistic or persuasive elements, such as word choice or appeals to emotion, to add power to the ideas
expressed

Chapter 6
The SAT®
Essay

After the passage, the prompt will continue as follows:

Write an essay in which you explain how [the author] builds an argument to persuade [his/her] audience
that [the author’s claim is true or valid]. In your essay, analyze how [the author] uses one or more of the
features listed above [see bullet points above] (or features of your own choice) to strengthen the logic
and persuasiveness of [his/her] argument. Be sure that your analysis focuses on the most relevant aspects
of the passage. Your essay should not explain whether you agree with [the author’s] claims, but rather
explain how [he/she] builds an argument to persuade [his/her] audience.

It is vital that you read, reread, and thoroughly understand everything the prompt asks you to do. Pay particular attention to the
bullet points that tell you exactly what to look for in the passage. Also note that the prompt gives you the option of mentioning
other “features of your own choice.” Finally, let’s take a close look at the last sentence in the prompt (we have capitalized and
underlined the word not):

Your essay should NOT explain whether you agree with [the author’s] claims, but rather explain how [he/
she] builds an argument to persuade [his/her] audience.

Peterson’s SAT® Prep Guide 2017

In other words, your task is not to present your own arguments—the passage author has already done that. Your task is to analyze
the author’s argument: “explain how the author builds an argument to persuade [his/her] audience.”
The people who develop the SAT® believe that if you complete this essay-writing task, your work will exhibit three types of skills:
1.

Close reading skills

2.

Analyzing skills

3.

Writing skills

Organizing Your Essay
Decide how many paragraphs you need to write to develop your analysis of the passage. Remember that length is not a valid
substitute for strength. Your answer booklet provides a certain number of pages for your essay. You can’t write more than the
lines provided, but you can write less. It is more important to do a good job of analyzing the passage than it is to fill up all the
lines. However, an essay of five sentences won’t earn you a high score.
A safe number of paragraphs is five. Use the first paragraph for the introduction and the last one for the conclusion. That gives
you three paragraphs to develop your ideas. That doesn’t mean that you can’t write a fourth or even fifth paragraph in the body
of your essay to develop your ideas. It’s more important to have a tightly written and well-developed shorter essay than a longer,
rambling, repetitious one that you didn’t have time to revise or proofread. There is a limit to what you can write in 50 minutes.
Use the opportunity that you have for practice in this book to work on your pacing and see how much you can plan and write
well in 50 minutes.

WRITING THE INTRODUCTION

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Essay

®

Now it’s time to write. You’ve analyzed the prompt and the passage in your prewriting step and identified the author’s key claims,
use of evidence and reasoning, and stylistic or persuasive language features. Now begin writing your introduction.
In your introduction, it is important to introduce the source text to your reader. State the author’s name and the title of the passage.
Then give a brief summary of the author’s key claim. Two to four sentences should accomplish these tasks.
As you practice writing essays in this book, and as you write the real one on test day, keep the following five ideas in mind:
1.

In writing your introduction, keep the key words and phrases of the prompt in mind.

2.

Avoid being cute, funny, ironic, satiric, overly emotional, or too dramatic. Set the tone or attitude in your first sentence.
You want to be sincere, clear, and straightforward.

3.

Don’t bother repeating the key claims from the source text word for word. A paraphrase in your own words is far
better than just copying the words of the source text.

4.

In your first paragraph, in addition to introducing the passage’s author, title, topic, and the author’s key claim about
the topic, make it clear to your readers that you are about to analyze the passage and explain how the author accomplishes his or her persuasive purpose. This can be accomplished in a clear topic sentence.

5.

Each sentence should advance your topic and be interesting to your reader.

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Skill Builder: Topic Sentence
DIRECTIONS: Which of the following is the best topic sentence?
1.

I agree with the author that the use of motorized boats and watercraft should be limited in freshwater streams and lakes.

2.

The author believes that the use of motorized boats and other watercraft should be limited in freshwater ecosystems in
order to decrease pollution.

3.

In this essay, I will examine the author’s argument in favor of limiting the use of motorized boats and watercraft in freshwater
streams and lakes.

4.

The author introduces her argument with startling statistics on the negative effects of motorized watercraft on freshwater
ecosystems; she then builds on those statistics with careful reasoning to reach the logical conclusion that we should limit
the use of motorized watercraft in freshwater ecosystems.

Answer and Explanation
Of the four sentences presented, choice 4 is the best answer since it states the topic clearly, limits the scope of the essay, and
presents the key points of the analysis. Sentence 4 also implies that the essay writer will expand on these key points in the body
of the essay.
ALERT: Remember that you are writing an analysis of the passage author’s work. You are NOT writing a persuasive
essay on the author’s topic.

Recognizing Effective Introductions
An effective introduction often refers to the subject of the essay, explains the value of the topic, or attracts the attention of the
reader by giving a pertinent illustration. Ineffective beginnings often contain unrelated material, ramble, and lack clarity.

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Skill Builder: Effective Introductions

The SAT®
Essay

DIRECTIONS: Examine the following excerpts from five introductory paragraphs and decide whether each is effective
or ineffective. Be able to defend your decision.
1.

The pollution of freshwater streams and lakes has become a pretty big problem in the United States. Part of the problem is
due to the unrestricted use of motorized boats and other stuff. I mostly agree with the author that something needs to be
done about it, but I don’t know that you should tell people where they can drive their boats.

2.

Freshwater lakes and streams are popular recreational destinations for many Americans. These bodies of water are also
the principal ecosystems of many different animals and plants. In her essay, “Bringing Fresh Back to Freshwater Lakes and
Streams,” activist River Pura makes the claim that the unrestricted use of motorized watercraft is to blame for polluting these
ecosystems.

3.

River Pura makes a very persuasive argument to persuade her audience that motorized boats should be restricted in freshwater streams and lakes. She persuades her audience with some facts and emotional appeals.

4.

The author makes an argument to persuade her audience about what she is thought about. It’s pollution.

5.

Pollution in freshwater ecosystems has been growing over the past decade. Author River Pura, in her essay titled “Bringing
Fresh Back to Freshwater Streams and Lakes,” claims that motorized watercraft are the primary source of that pollution.
While Pura presents her readers with startling statistical information, her argument is anticlimactic and builds to no logical
conclusion.

Peterson’s SAT® Prep Guide 2017

Answers and Explanations
1.

Ineffective. This paragraph is focused on the topic of the source text, not on the author’s writing. Although the prompt specifically says not to present opinions on the topic, the writer states that he or she “mostly agrees” with the passage author.
Also, the writer uses vague, informal expressions: “pretty big” and “mostly agree.”

2.

Effective. This paragraph effectively introduces the author, the passage title, and the topic. The paragraph also identifies the
author’s key claim and begins to analyze the author’s work.

3.

Ineffective. While this paragraph identifies the author’s key claim, nonspecific phrases like “some facts” and “emotional appeal”
weaken the paragraph. Also, the writer uses three different forms of the word persuade, which sounds repetitious.

4.

Ineffective. The paragraph does not introduce the author, the passage title, or the author’s key claim. It does not explain that
the writer is going to analyze the passage. The writing is poorly constructed and misuses verb tense.

5.

Effective. This paragraph clearly introduces the source text, highlights the author’s key claim, and explains how the passage
author fails to build an effective argument.

DEVELOPING YOUR IDEAS
The heart of your essay is the development, or middle paragraphs. In these paragraphs, you must use explanations, details, and
examples from the source text to support the main ideas in your essay. All the sentences in the development paragraphs must
explain and support your analysis of the source text and must not digress.
In the limited time you have on the Essay section of the SAT®, you can take only 15 to 20 minutes to write the body of your essay. In
this time, you need to support your analysis of the author’s work with careful reasoning, and back up your analysis with evidence
from the source text. Your writing must be coherent, logical, unified, and organized.
Avoid the following three pitfalls in the development of your essay:

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Chapter 6
The SAT®
Essay

1.

Using sentences that are irrelevant and contain extraneous material

2.

Using sentences that do not follow a logical sequence of thought but seem to jump from one idea to another

3.

Using sentences that do not relate to the topic sentence or do not flow from the preceding sentence

Using Transitions
The successful writer uses transitional words and phrases to connect thoughts and provide a logical sequence of ideas. Become
familiar with the following list of transitions and use them in your practice essays. They will help make your writing smoother:

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therefore

for example

nevertheless

first of all

in any case

but

then

consequently

still

moreover

for instance

yet

second

on the other hand

also

indeed

of course

in addition

however

finally

furthermore

Skill Builder: Using Transitions
DIRECTIONS: In the following three samples, the transition is missing. Supply a transitional word or phrase that will allow
the second sentence to follow smoothly or logically from the first.

1.

Freshwater ecosystems hold only 0.01% of Earth’s water supply. Over half of the people on Earth live near freshwater
ecosystems.

2.

Human activity comprises the primary threat to freshwater ecosystems. Damming lakes, extracting water, and filling shallow
wetlands all lead to the destruction of these ecosystems.

3.

Constructing dams and levees can lead to a significant loss of habitat for land animals and plant species. The restricted
water flow changes natural water temperatures and impacts marine life.

Answers and Explanations
1.

The sentences require a transition that indicates contrast, such as yet, but, however, still, although and either . . . or.
Although freshwater ecosystems hold only 0.01% of Earth’s water supply, over half of the people on Earth live near these
ecosystems.

2.

These sentences require a transition that indicates an example is to follow.
Human activity comprises the primary threat to freshwater ecosystems; for example, damming lakes, extracting water, and
filling shallow wetlands all lead to the destruction of these ecosystems.

3.

These sentences require a transition that indicates additional information is to follow.
The construction of dams and levees can lead to a significant loss of habitat for land animals and plant species. Furthermore,
the restricted water flow changes natural water temperatures and impacts marine life.

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Writing Effectively

Chapter 6

There are three important elements that will be considered in scoring an SAT® essay:
1.

Reading

2.

Analysis

3.

Writing

The SAT®
Essay

Essays are scored according to how well they meet these three basic criteria. To improve an essay you are writing, ask yourself
these questions:

Reading
•
•
•
•

Does my essay demonstrate a thorough understanding of the source text?
Does my essay identify the author’s key claims?
Does my essay explain how the author uses evidence to support his or her key claims?
Does my essay effectively use evidence from the source text?

Peterson’s SAT® Prep Guide 2017

Analysis
• Does my essay offer an in-depth evaluation of the author’s use of evidence in building and supporting an argument?
• Does my essay offer an in-depth evaluation of the author’s use of stylistic or persuasive language features to build and
support his or her argument?

• Does my essay use supporting evidence from the passage that is relevant and focused on my task (analyzing the
passage)?

Writing
• Does my essay include a precise central claim that is supported with body paragraphs?
• Does my essay include an effective introduction and a strong conclusion?
• Does my essay incorporate a variety of sentence structures? Is each of my sentences clearly written? Does each sentence
flow well?

• Is my essay virtually error-free in spelling, grammar, usage, and mechanics?

WRITING THE CONCLUSION
Lewis Carroll, the author of Alice in Wonderland, once gave some very good advice for writers. He said, “When you come to the
end, stop!”
When you come to the end of your ideas, stop writing the development—and begin writing your conclusion. You can’t just end your
essay with your last development paragraph. You need to draw your comments together in a strong, clear concluding paragraph.

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The SAT®
Essay

A good concluding paragraph for your essay should assure your scorers that you have successfully read, understood, and analyzed
the source text. You should be able to do this in three to six sentences written in 3 to 4 minutes. The following are three possible
ways to end your essay:
1.

Through a restatement of your most important or most central idea

2.

Through a summary of the material covered in the essay

3.

Through a clear statement about the effectiveness of the passage author’s work

Keep in mind that a good conclusion is an integral part of your essay. It may be a review or a restatement, or it may leave your
readers with an intriguing question to think about (one that is closely related to your essay, of course). In any case, your conclusion
must be strong, clear, and effective.

What Not to Say in Your Conclusion
Just as there are good techniques, there are also some very ineffective methods that essay writers may be tempted to use in
drawing a composition to a close. Try to avoid falling into the following three traps:
1.

Apologizing for your inability to more thoroughly analyze the passage in the allotted time

2.

Complaining that the source text did not interest you or that you don’t think it was fair to be asked to write an analysis
of the source text without giving your own opinions on the topic

3.

Introducing material that you will not develop, rambling on about nonpertinent matters, using material that is trite
or unrelated, or making a sarcastic joke that indicates your disdain for the topic you just spent 50 minutes writing
about

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Recognizing Effective Conclusions
Remember that an effective concluding paragraph may restate or summarize main idea(s) in your essay, draw a logical conclusion,
or offer a strong opinion about the effectiveness of the author’s work. An ineffective final paragraph introduces new material in a
scanty fashion, apologizes for the ineffectiveness of your essay, or is illogical or unclear. Use the following skill-builder exercises
to test yourself.

Skill Builder: Effective Conclusions
DIRECTIONS: Why are the following sentences ineffective in a concluding paragraph?
1.

I wish I had more time to write a better, more in-depth analysis, but I find that in the allotted time this is all that I could do.

2.

Although I have not mentioned this before, my family enjoys boating on freshwater lakes. We often pick up litter from the
water in the hopes of decreasing pollution.

3.

This passage was incredibly difficult to understand, and I felt like the author droned on and on about nothing.

DIRECTIONS: Examine the following five excerpts from concluding paragraphs and decide whether each is effective or
ineffective.
4.

That’s all I have to say about the topic. I know I’m not an expert, but at least this is an actual analysis of the work. I also used
a lot of supporting details from the source text. So, I think you should give me at least a 3.

5.

While River Pura is a passionate spokesperson for freshwater ecosystems, her argument for limiting the use of motorized
watercraft lacks cohesion. Furthermore, a careful examination of the issue shows that motorized watercraft are not the
main cause of pollution in freshwater ecosystems. Had Pura taken the time to structure her impassioned pleas around valid
statistical evidence, her argument would be stronger.

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I forgot to mention earlier that the author uses a variety of descriptive words and phrases to appeal to the emotions of
readers. She also uses figurative language and makes some illusions to other articles the reader may have read.

Chapter 6

6.
7.

Protecting the biodiversity of freshwater ecosystems is a topic of importance for all of those who depend upon these
ecosystems for survival. River Pura makes a solid case for the restriction of motorized watercraft in freshwater areas. Pura
begins her argument with startling facts and statistics designed to capture the reader’s attention. She then moves into a
well-reasoned discourse that effectively negates any counter-claims opponents might make.

8.

In conclusion, the author makes a pretty good case for protecting freshwater ecosystems. However, most people I know
aren’t going to stop boating because an environmentalist says they should.

The SAT®
Essay

Answers and Explanations
1.

Ineffective. Don’t apologize for doing a poor job. (Maybe your essay is better than you think. And if it is bad, why would you
want to call attention to its weaknesses?) Also, don’t blame the fact that you did a poor job on the time limit.

2.

Ineffective. Don’t include asides like this anywhere in your essay—but especially not in your conclusion. You’re supposed to
be concluding your analysis, not cramming in extra details that are only loosely related to the topic.

3.

Ineffective. Do not complain about the task in the conclusion. Your essay should maintain a formal tone throughout.

4.

Ineffective. Do not speak directly to or “butter up” the scorers in your concluding paragraph. Maintain a formal tone.

5.

Effective. This paragraph sums up the key points of the analysis and states the essay writer’s opinion of the passage author’s
persuasive writing.

Peterson’s SAT® Prep Guide 2017

6.

Ineffective. Do not introduce new ideas into a concluding paragraph. If you have more information to add that would improve
or add depth to your analysis, consider adding another body paragraph. Also, the essay writer has misused the word illusion:
he or she should have used allusion.

7.

Effective. This paragraph effectively summarizes the key points of the analysis.

8.

Ineffective. This paragraph gives only a vague summary of the analysis and offers an opinion that is irrelevant to the essaywriting task.

THE SCORING RUBRIC FOR THE SAT® ESSAY
The SAT® Essay will be scored based on a 4-point rubric. Points will be awarded in three areas: reading, analysis, and writing. Each
essay will be scored by two graders who will give a score of 1 to 4 in each of the three areas. Scores in each area will be reported
separately from the other two. For example, a test-taker might earn a score of 3/4/3. This means that the test-taker scored 3 out
of 4 points in both reading and writing and 4 out of 4 points in analysis.
All the scorers read the essays against the same rubric developed by the College Board, which administers the SAT®. This rubric
guides the scorers in considering overall impression, development, organization, diction, sentence structure, grammar, usage,
and mechanics. The rubric also directs scorers in evaluating essay writers’ comprehension of the source text, use of relevant evidence from the passage, and their analysis of the passage author’s argument. The scoring guidelines are similar to the following:

Essay Scoring 4 (Advanced)
• Reading: shows a comprehensive understanding of the source text, including the author’s key claims, use of details and
evidence, and the relationship between the two

• Analysis: offers an “insightful” and in-depth evaluation of the author’s use of evidence and stylistic or persuasive features

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Essay

in building an argument; uses relevant supporting details that address the task

• Writing: includes all of the features of a strong essay, including a precise central claim, body paragraphs, and a strong
conclusion; incorporates a variety of sentence structures; is virtually free of all convention errors

Essay Scoring 3 (Proficient)
• Reading: shows an appropriate understanding of the source text, including the author’s key claims and use of details
in developing an argument

• Analysis: offers an effective evaluation of the author’s use of evidence and stylistic or persuasive features in building an
argument; uses appropriate supporting details and evidence that are relevant and focused on the task

• Writing: includes all of the features of an effective essay, including a precise central claim, body paragraphs, and a strong
conclusion; incorporates a variety of sentence structures and is relatively free of common grammatical errors

Essay Scoring 2 (Partial)
• Reading: shows some understanding of the source text, including the author’s key claims; uses limited textual evidence;
incorporates unimportant details

• Analysis: offers a limited evaluation of the author’s use of evidence and stylistic or persuasive features in building an
argument; supporting details are lacking and/or irrelevant to task

• Writing: does not provide a precise central claim, nor an effective introduction, body paragraphs, and conclusion;
incorporates little variety of sentence structures and contains numerous errors in grammar and conventions

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Essay Scoring 1 (Inadequate)
• Reading: demonstrates little or no understanding of the source text or the author’s use of key claims
• Analysis: offers no clear evaluation of the author’s use of evidence and stylistic or persuasive features in building an
argument; supporting details and evidence are nonexistent or irrelevant to task

• Writing: lacks any form of cohesion or structure; incorporates little variety in sentence structure and includes significant
errors in convention that make it difficult to read
Read the rubric several times. As you practice writing essays for the SAT®, keep this rubric in mind. As you write each essay, try to
focus on one or two qualities of good writing that the rubric measures. After you have finished writing your essay, come back to
the rubric and see how your essay measures up.
Use the following table to help you. Give yourself anywhere from 1 to 4 points for each quality of good writing.

PRACTICE TEST SCORING TABLE
Reading:
Analysis:
Writing:
Final Score:

__________
__________
__________
__________ / __________ / __________

EXERCISE: PRACTICING YOUR ESSAY SKILLS
Use the following prompt to practice writing an effective essay. Carefully read the prompt and write a response. Make use of the
effective writing techniques discussed in this chapter. Use the scoring rubric to evaluate your work. Then read and evaluate the
three sample responses.

As you read the passage below, consider how the writer uses the following:

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Chapter 6

• Evidence, such as facts or examples, to support claims

The SAT®
Essay

• Reasoning to develop ideas and to connect claims and evidence
• Stylistic or persuasive elements, such as word choice, emotional appeal, intellectual appeal, or
ethical appeal, to add power to the ideas expressed

Henry Clay (1777–1852) served several terms in Congress and was Secretary of State in 1825. In the following speech, given in 1818, he
argues that the United States should support South America in gaining independence from Spain.

The Emancipation of South America
1

Spain has undoubtedly given us abundant and just cause for war. But it is not every cause of war that should lead to
war. . . . If we are to have war with Spain, I have, however, no hesitation in saying that no mode of bringing it about
could be less fortunate than that of seizing, at this time, upon her adjoining province. There was a time, under certain
circumstances, when we might have occupied East Florida with safety; had we then taken it, our posture in the
negotiation with Spain would have been totally different from what it is.

2

But we have permitted that time, not with my consent, to pass by unimproved. If we were now to seize upon Florida after
a great change in those circumstances, and after declaring our intention to acquiesce in the procrastination desired by
Spain, in what light should we be viewed by foreign powers—particularly Great Britain? We have already been accused

Peterson’s SAT® Prep Guide 2017

of inordinate ambition, and of seeking to aggrandize ourselves by an extension, on all sides, of our limits. Should we not,
by such an act of violence, give color to the accusation? No, Mr. Chairman; if we are to be involved in a war with Spain, let
us have the credit of disinterestedness. Let us put her yet more in the wrong. Let us command the respect which is never
withheld from those who act a noble and generous part. I hope to communicate to the committee the conviction which I
so strongly feel, that the adoption of the amendment which I intend to propose would not hazard, in the slightest degree,
the peace of the country… .

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3

In contemplating the great struggle in which Spanish America is now engaged, our attention is fixed first by the
immensity and character of the country which Spain seeks again to subjugate. Stretching on the Pacific Ocean from about
the fortieth degree of north latitude to about the fifty-fifth degree of south latitude, and extending from the mouth of the
Rio del Norte (exclusive of East Florida), around the Gulf of Mexico and along the South Atlantic to near Cape Horn, it is
nearly five thousand miles in length, and in some places nearly three thousand in breadth… .

4

Throughout all the extent of that great portion of the world which I have attempted thus hastily to describe, the spirit
of revolt against the dominion of Spain has manifested itself. The Revolution has been attended with various degrees of
success in the several parts of Spanish America. In some it has been already crowned, as I shall endeavor to show, with
complete success, and in all I am persuaded that independence has struck such deep root, that the power of Spain can
never eradicate it. What are the causes of this great movement?

5

Three hundred years ago, upon the ruins of the thrones of Montezuma and the Incas of Peru, Spain erected the most
stupendous system of colonial despotism that the world has ever seen—the most vigorous, the most exclusive. The great
principle and object of this system have been to render one of the largest portions of the world exclusively subservient, in
all its faculties, to the interests of an inconsiderable spot in Europe….

6

Thus upon the ground of strict right, upon the footing of a mere legal question, governed by forensic rules, the Colonies,
being absolved by the acts of the parent country from the duty of subjection to it, had an indisputable right to set up for
themselves. But I take a broader and a bolder position. I maintain that an oppressed people are authorized, whenever
they can, to rise and break their fetters. This was the great principle of the English Revolution. It was the great principle of
our own… .

7

In the establishment of the independence of Spanish America, the United States have the deepest interest. I have no
hesitation in asserting my firm belief that there is no question in the foreign policy of this country, which has ever arisen,
or which I can conceive as ever occurring, in the decision of which we have had or can have so much at stake. This interest
concerns our politics, our commerce, our navigation… .

8

I would invoke the spirits of our departed fathers. Was it for yourselves only that you nobly fought? No, no! It was the
chains that were forging for your posterity that made you fly to arms, and, scattering the elements of these chains to the
winds, you transmitted to us the rich inheritance of liberty.

Chapter 6
The SAT®
Essay

Write an essay in which you explain how Henry Clay builds an argument to persuade his audience that
the United States should support South America in its efforts to secure freedom from Spain. In your essay,
analyze how Clay uses one or more of the features previously listed (or features of your own choice) to
strengthen the logic and persuasiveness of his argument. Be sure that your analysis focuses on the most
relevant aspects of the passage.
Your essay should not explain whether you agree with Clay’s claims, but rather explain how he builds
an argument to persuade his audience.

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Use the following scoring guide to help you evaluate Sample Essay 1. Then, read our analysis of the essay, as well as suggestions
for improvement.

Score Point

4
(Advanced)

3
(Proficient)

2
(Partial)

1
(Inadequate)

Reading

Analysis

Writing

The essay shows
a comprehensive
understanding of the
source text, including the
author’s key claims, use
of details and evidence,
and the relationship
between the two.

The essay offers an
“insightful” and in-depth
evaluation of the author’s
use of evidence and stylistic
or persuasive features in
building an argument.
Supporting details and
evidence are relevant and
focus on those details that
address the task.

The essay includes all of the
features of a strong essay,
including a precise central
claim, body paragraphs,
and a strong conclusion.
There is a variety of
sentence structures used in
the essay, and it is virtually
free of all convention errors.

The essay shows an
appropriate understanding
of the source text,
including the author’s key
claims and use of details in
developing an argument.

The essay offers an
“effective” evaluation of the
author’s use of evidence
and stylistic or persuasive
features in building an
argument. Supporting
details and evidence are
appropriate and focus
on those details that
address the task.

The essay includes all
of the features of an
effective essay, including
a precise central claim,
body paragraphs, and a
strong conclusion. There
is a variety of sentence
structures used in the essay,
and it is free of significant
convention errors.

The essay shows some
understanding of the
source text, including
the author’s key claims,
but uses limited textual
evidence and/or
unimportant details.

The essay offers limited
evaluation of the author’s
use of evidence and
stylistic or persuasive
features in building an
argument. Supporting
details and evidence are
lacking and/or are not
relevant to the task.

The essay does not provide
a precise central claim,
nor does it provide an
effective introduction,
body paragraphs, and
conclusion. There is
little variety of sentence
structure used in the essay,
and there are numerous
errors in grammar
and conventions.

The essay demonstrates
little or no understanding
of the source text or the
author’s use of key claims.

The essay offers no clear
evaluation of the author’s
use of evidence and
stylistic or persuasive
features in building an
argument. Supporting
details and evidence are
nonexistent or irrelevant
to the task.

The essay lacks any form
of cohesion or structure.
There is little variety
of sentence structures,
and significant errors
in convention make it
difficult to read.

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Sample Essay 1
In his speech to Congress, Henry Clay emphasizes the importance of supporting South America in their revolt against Spain. Clay
begins his argument by talking about the significance of the state of Florida, which isn’t actually a state yet. Clay urges Congress
not to invade Florida (which apparently has been captured by Spain) because it would make the U.S. look bad. Clay’s goal is for
Spain to be “in the wrong” and for the U.S. to “command the respect which is never withheld from those who act a noble and
generous part.”
Clay goes on to remind his listeners that the majority of South America is already involved in a Revolution against Spain. This
is a key point of persuasion. Finally, Clay makes the comparison between the Revolution in South America and the Revolution
against Great Britain. He even goes so far as to “invoke the spirits of our departed fathers” and uses other images like “fly to arms”
and “the rich inheritance of liberty.” While Clay’s argument is okay, it’s long-winded and focuses too much on emotional appeal.

Analysis of Sample Essay 1
This response scored a 2/2/2.

• Reading—2: The writer demonstrates some comprehension of the source text. In the first paragraph, the writer conveys
the basic central claim—the importance of supporting South America in their revolt against Spain. The writer also shows
a partial understanding of Clay’s position on Florida—Clay’s goal is for Spain to be “in the wrong”—but does not effectively
tie it to the central claim. In the following paragraph, the writer correctly identifies Clay’s comparison of the South
American Revolution to the U.S. Revolution. However, there is little demonstration of the relationship between the
central claim and the supporting details. Overall, the writer shows a partial understanding of the source text.

• Analysis—2: The response offers a limited analysis of the source text, showing only partial understanding of the task.
The writer mentions the significance of the state of Florida but then does not elaborate on this significance or explain
how Florida’s significance contributes to Clay’s argument. In the second paragraph, the writer makes note of a key point
of persuasion; however, there is no further discussion of why South America’s revolution against Spain is a key point of
persuasion. Furthermore, the writer does not develop the effect of Clay’s comparison of the South American Revolution
to the U.S. Revolution. While the writer includes the use of emotional appeal in the analysis, there is no explanation of
it. Overall, this response is only a partially successful analysis.

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• Writing—2: This response reflects limited cohesion and some skill in the use of language. There is no precise central
claim, nor is there an effective introduction and conclusion. Phrases like isn’t actually a state yet and which apparently
has been captured by Spain use an informal, almost flippant tone. Calling the source text long-winded is subjective and
inappropriate for a formal analysis. Overall, this response represents a partially developed essay.

Suggestions for Improvement
1.

The response needs a clearly developed introduction that establishes the topic and presents a precise central claim.
While the first paragraph makes a start, a precise central claim should briefly summarize the key points that the writer
will make in his or her analysis.

2.

Each body paragraph should begin with a topic sentence that summarizes the key point of the paragraph. For instance,
the writer could begin the second paragraph with the sentence: Clay begins his argument with reason rather than
rhetoric, logically appealing to those who are hesitant to engage in open warfare.

3.

The use of evidence from the source text should directly support key aspects of the essay writer’s analysis. Look at this
sentence from paragraph 2 of Sample Essay 1: He even goes so far as to “invoke the spirits of our departed fathers”
and uses other images like “fly to arms” and “the rich inheritance of liberty.” The quotations here do not support the
writer’s analysis of the source text. A better use of quotations might be:

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Clay uses emotionally powerful language as a persuasive tool in making the correlation between the U.S. quest for
independence from Great Britain and South America’s quest for independence from Spain. He wants to inspire the
same passionate support for South America’s struggle that most of his listeners feel about their own country’s revolution, so he uses emotional phrases like “invoke the spirits of our departed fathers,” “fly to arms,” and “rich inheritance of
liberty.”
4.

The response should include a strong conclusion that restates the thesis, recaps the most important parts of the
analysis, or leaves the reader with a final thought. The conclusion should provide a sense of closure on the topic.

5.

All SAT® essays should maintain a formal and objective tone. The writer should limit the use of contractions and refrain
from using derogatory adjectives like long-winded to describe the source text.

Sample Essay 2
In Henry Clay’s 1818 address to Congress, he is building an argument to persuade his listeners to support South America in their
revolution against Spain. He builds his argument in three different ways. Clay demonstrates an understanding of his audience,
underscores the action that they need to take, and closes with an emotional appeal for liberty by referencing the Revolutionary War.
Clay begins building his argument by demonstrating an understanding of his audience. He demonstrates an understanding of his
audience when he assures them at the outset of his speech that he is not calling for open warfare against Spain. This is significant
because many members of Congress still remembered the bloodshed from the Revolutionary War. Clay reassures his audience
and seeks to get them on his side by stating “it is not every cause of war that should lead to war.” Once Congresses minds are put
at ease, Clay can move onto talking about his central claim.
After Clay reassures his listeners, he then attempts to underscore the amount of involvement they would have to take in the
conflict. He is seeking to minimize the nature of the conflict in order to win support for it. Clay says that, “The Revolution has
been attended with various degrees of success in the several parts of Spanish America. In some it has been already crowned, as
I shall endeavor to show, with complete success. . . .”
Finally, Clay seeks to draw a parallel for his listeners between the revolution they had recently won and the revolution being fought
in South America. This is his emotional appeal. Clay is telling his listeners, “Remember your fight for independence? Remember
how greatly you desired freedom? This is the same thing.” By getting his listeners to remember how passionately they desired
freedom from Great Britain, he is hoping to sway their emotions in favor of supporting South America against Spain.
Henry Clay uses logic, minimizing, and emotional appeal to build an argument. He shows that he understands his audience and
seeks to give them what they need so that they will agree with him.

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Analysis of Sample Essay 2
This essay scored a 3/3/3.

• Reading—3: This response demonstrates effective understanding of the source text with appropriate use of evidence
through the analysis. In the second paragraph, the author discusses Clay’s understanding of his audience and what the
audience most fears. Although the source text does not refer explicitly to this fear, the writer picks up on it from a careful
reading of the passage. In the next paragraph, the writer cites and discusses a claim Clay makes that supporting South
America requires a minimal response. Finally, the last body paragraph paraphrases Clay’s emotional call to remember
the principles that guided the American Revolution. The writer shows an effective understanding of both the central
idea and important details.

• Analysis—3: The writer shows an effective understanding of the task by identifying three ways Clay builds his argument
(Clay demonstrates an understanding of his audience, underscores the action that they need to take, and closes with an
emotional appeal . . .) and then elaborating on each point in the body paragraphs. Each body paragraph carefully evaluates how pieces of evidence from the source text, the author’s use of reasoning, or stylistic or persuasive features are

Peterson’s SAT® Prep Guide 2017

used to develop an argument. For example, in the final body paragraph the writer claims that Clay is getting his listeners
to remember how passionately they desired freedom from Great Britain and explains that this is to sway their emotions in
favor of supporting South America against Spain. The response could have made stronger use of evidence from the text,
offering a direct quote rather than a paraphrase. However, this response shows an effective analysis of the source text
using relevant support.

• Writing—3: This essay includes most of the features of an effective essay, including a precise central claim and body
paragraphs. The introduction and conclusion lack development, although the introduction presents the central claim
and the conclusion restates it. There is appropriate variety of sentence structure, and the few errors of convention and
grammar do not detract from the overall reading of the response. Overall, this analysis is proficient.

Suggestions for Improvements
1.

The author should work to develop a stronger introduction and conclusion. For example, the introduction could
explain the significance of the topic, and the conclusion might provide a final thought or statement.

2.

Paragraph 3 could further discuss Clay’s reasons for minimizing the support needed in South America. While the
writer effectively points out Clay’s use of minimization, the importance of this technique to Clay’s argument is not
fully clear.

3.

In paragraph 4, the writer paraphrases Clay’s emotional appeal. When discussing an author’s use of emotional language
to persuade listeners to share his point of view or take a certain action, direct quotes from the source text are best.

Sample Essay 3

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Essay

In 1818, South America was rising in revolt against Spain. Congressman Henry Clay believed that it was in America’s best interest
to support their neighbors to the south. In his speech to the 1818 Congress, Clay builds an argument tailor-made for his audience,
outlining action steps Congress should take while at the same time appealing to their passionate belief in democracy. Through
a combination of careful rhetoric, logic, and emotional appeal, Clay hopes to convince Congress to adopt an amendment that
would put the U.S. at war with Spain.
Clay begins his argument with careful rhetoric designed specifically for his audience. Clay understands that some may be thinking
of the bloodshed of the American Revolution and have no desire to engage in another war. “It is not every cause of war that
should lead to war …” Clay maintains. However, Clay goes on to suggest to his audience that, had they acted prior to this moment
in time, war might have been avoided. In essence, Clay is telling his audience that he does not condone war; however, Congress
has brought about the necessity to engage in war by previous inaction. Clay tells his audience, “There was a time . . . ”when . . .
our posture in the negotiation with Spain would have been totally different . . . ” Yet, Clay maintains, that time has passed. Now
is the time for more heavy-handed action.
Clay furthers the appeal of his argument by insisting that the action steps he proposes would not lead the country into another
violent conflict. Rather, Clay maintains “ . . . ” the adoption of the amendment which I intend to propose would not hazard, in the
slightest degree, the peace of the country. . . . ” He uses logic to reason that, because of the size of South America, and because of
the success that some South American people have already had in their fight against Spain, the revolution in some areas is already
“a complete success.” Therefore, Clay is demonstrating to his audience that supporting his amendment poses no risk to them.
Finally, Clay uses an emotional appeal to ultimately convince Congress to pass the amendment. Clay draws a parallel between
the ideals of freedom and democracy that underscored the fight for American independence and the ideals of freedom and
democracy that are bolstering the fight in South America. His emotional appeal is summed up in the last paragraph of the speech:
“I would invoke the spirits of our departed fathers. Was it for yourselves only that you nobly fought? No, no! It was the chains that
were forging for your posterity that made you fly to arms . . . ””
Speaking to Congress of the imminent threat of Spain, Henry Clay seeks to persuade his listeners to pass an amendment that
would essentially put the fledgling nation at war with another European nation. With careful rhetoric, logic, and emotional appeals,
Clay seeks to convince his listeners that freedom is something for which one should always be willing to fight.

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Analysis of Sample Essay 3
This essay scored a 4/4/4.

• Reading—4: This response demonstrates thorough understanding of the source text with skillful use of paraphrases
and direct quotations. The writer briefly summarizes the main idea of Clay’s argument (Clay believed that it was in America’s
best interest to support their neighbors to the south) and presents many details from the text, including Clay’s reflection
that Congress had an opportunity to negotiate with Spain but missed it, to demonstrate why Clay’s argument is significant. There are few long direct quotations from the source text. Instead, the author accurately and precisely paraphrases
the key points of the speech.

• Analysis—4: The writer demonstrates an insightful understanding of the task by identifying three ways Clay builds his
argument (Through a combination of careful rhetoric, logic, and emotional appeal . . .) and then elaborating on each point
in the body paragraphs. Each body paragraph carefully evaluates how pieces of evidence from the source text, the
author’s use of reasoning, or stylistic or persuasive features are used to develop an argument. For example, in the final
body paragraph the writer claims that Clay “draws a parallel between the ideals of freedom and democracy that underscored the fight for American independence and the ideals of freedom and democracy that are bolstering the fight in
South America.” The response demonstrates a thorough understanding of both the source text and its effect on the
audience.

• Writing—4: This essay is cohesive and shows an effective command of language, including a precise central claim and
body paragraphs. The introduction and conclusion are well developed. There is ample variety of sentence structures
and no errors of convention and grammar that detract from the overall reading of the response. Overall, this analysis
shows advanced writing proficiency.

ADDITIONAL ESSAY WRITING PRACTICE
For more practice, carefully read the following prompt and source text. Then write an analysis of the passage. Score your analytical
response using the rubric provided.

As you read the passage below, consider how the writer uses the following:

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Chapter 6

• Evidence, such as facts or examples, to support claims

The SAT®
Essay

• Reasoning to develop ideas and to connect claims and evidence
• Stylistic or persuasive elements, such as word choice, emotional appeal, intellectual appeal, and ethical
appeal, to add power to the ideas expressed

Padre Island National Seashore separates the Gulf of Mexico from the Laguna Madre, one of a few hypersaline lagoons in the world.
The park protects 70 miles of coastline, dunes, prairies, and wind tidal flats teeming with life. It is a safe nesting ground for the Kemp’s
Ridley sea turtle and a haven for 380 bird species. It also has a rich history, including the Spanish shipwrecks of 1554.

The Importance of the 1554 Shipwrecks
1

In April, 1554, three Spanish naos (a type of cargo and passenger ship similar to Columbus’s Santa Maria) went aground
on Padre Island following a storm that had blown them across the Gulf of Mexico from the coast of Cuba. At the time this
was the greatest disaster to ever befall the Spanish fleet in the New World. Tons of treasure bound for Spain was lost in
addition to the lives of approximately three hundred passengers and crew who died from hunger, thirst, and attacks by
natives as they attempted to walk back to the port of Vera Cruz.

Peterson’s SAT® Prep Guide 2017

2

But the story of the 1554 shipwreck does not end there, nor does it end with the conclusion of the salvage operations that
took place later that year. As with any important historical event, its effects resonate through the centuries and can still be
felt today—if one looks for them.

3

First of all, the wrecks were the first documented occurrence of Europeans on the island and one of the first occurrences
of Europeans in what was to become Texas. The salvage operation was the first documented instance of Europeans
intentionally coming to the island and staying for an extended period.

4

Second, the three ships that wrecked (the Santa Maria de Yciar, the Espiritu Santo, and the San Esteban) are the oldest
shipwrecks ever found in North America (excluding the Caribbean and Latin America).

5

Third, when the remains of the ships were discovered in 1967, a private company called Platoro, Ltd. began excavating
them. This set off a long legal battle over ownership of the remains, as Texas had no laws governing antiquities at the
time. In the long run, the state won its case and the remains were turned over to the National Park Service, which has
transferred curation of the artifacts to the Corpus Christi Museum of Science and History, where they may now be viewed.

6

Historian and Marine Archeologist Dr. Donald Keith, President of the Ships of Discovery at the Corpus Christi Museum of
Science and History, notes that:

7

“The 1554 shipwrecks are important for a lot of reasons. The ‘mining’ of them by Platoro caused the state of Texas to realize
that shipwrecks and archaeological sites in general are important, and the property of the people and the state. They are
cultural resources that have to be cared for. Some of the earliest experiments in the conservation of artifacts from the
sea were done on the objects and hull remains that were recovered from the sites that Platoro and the State worked. . . .
The Platoro conflict did lead to the establishment of the Texas Antiquities Committee, which led to the Texas Historical
Commission, which led to the discovery and excavation of La Belle [the ship of the French explorer La Salle, found on the
Texas coast within the past few years] among other accomplishments.”

8

This third and last effect on our present society is undoubtedly the most important, because it resulted in new Texas laws
to protect archeological resources. These laws follow the federal Antiquities Act in spirit, which gives federal agencies
custody of relics found within their jurisdictions so that they may be properly protected and studied. Thus, instead of
ending up in private collections where they become curiosities for a fortunate few, the knowledge derived from the
artifacts goes to the public in the form of publications and exhibits in museums and on websites.

9

Bits and pieces of the 1554 wrecks and many other historical events still wash up on the island or can be found emerging
from the sands. If you discover something, please remember that the right thing to do is leave it where it is and report it
to us, so that we may conduct a proper archeological dig and learn more about the rich history of the island and share our
findings (and yours) with the world.

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Essay

Write an essay in which you explain how the writer builds an argument to persuade his or her audience
of the importance of the 1554 shipwrecks. In your essay, analyze how the author uses one or more of the
features listed previously (or features of your own choice) to strengthen the logic and persuasiveness
of his/her argument. Be sure that your analysis focuses on the most relevant aspects of the passage.
Your essay should not explain whether you agree with the writer’s claims, but rather explain how the
writer builds an argument to persuade his or her audience.

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Score Point

4
(Advanced)

3
(Proficient)

2
(Partial)

1
(Inadequate)

Reading

Analysis

Writing

The essay shows
a comprehensive
understanding of the
source text, including the
author’s key claims, use of
details and evidence, and
the relationship between
the two.

The essay offers an
“insightful” and in-depth
evaluation of the author’s
use of evidence and stylistic
or persuasive features in
building an argument.
Supporting details and
evidence are relevant and
focus on those details that
address the task.

The essay includes all
of the features of a
strong essay, including
a precise central claim,
body paragraphs, and a
strong conclusion. There
is a variety of sentence
structures used in the
essay, and it is virtually
free of all convention
errors.

The essay shows an
appropriate understanding
of the source text, including
the author’s key claims and
use of details in developing
an argument.

The essay offers an
“effective” evaluation of the
author’s use of evidence
and stylistic or persuasive
features in building an
argument. Supporting
details and evidence are
appropriate and focus
on those details that
address the task.

The essay includes all
of the features of an
effective essay, including
a precise central claim,
body paragraphs, and a
strong conclusion. There
is a variety of sentence
structures used in the essay,
and it is free of significant
convention errors.

The essay shows some
understanding of the
source text, including
the author’s key claims,
but uses limited textual
evidence and/or
unimportant details.

The essay offers limited
evaluation of the author’s
use of evidence and stylistic
or persuasive features in
building an argument.
Supporting details and
evidence are lacking and/or
are not relevant to the task.

The essay does not provide
a precise central claim,
nor does it provide an
effective introduction,
body paragraphs, and
conclusion. There is
little variety of sentence
structure used in the essay,
and there are numerous
errors in grammar
and conventions.

The essay demonstrates
little or no understanding
of the source text or the
author’s use of key claims.

The essay offers no clear
evaluation of the author’s
use of evidence and stylistic
or persuasive features in
building an argument.
Supporting details and
evidence are nonexistent or
irrelevant to the task.

The essay lacks any form of
cohesion or structure. There
is little variety of sentence
structures, and significant
errors in convention make
it difficult to read.

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SUMMING IT UP
• Remember that your essay should not explain whether you agree with the author’s claims, but, rather, it should explain how
the author builds an argument to persuade his or her audience.

• SAT® essays are scored based on how well they meet the following three criteria:
1.

Reading—Does the essay demonstrate a thorough understanding of the source text, identify the author’s key claims
and explain how he or she uses evidence to support them, and use evidence from the source text effectively?

2.

Analysis—Does the essay offer in-depth evaluations of the author’s use of evidence and stylistic or persuasive language features to build and support his or her argument? Does the essay include relevant supporting evidence from
the passage to aid in its analysis?

3.

Writing—Does the essay include an effective introduction, a precise central claim supported by body paragraphs,
and a strong conclusion? Does the essay include a variety of clearly written sentences that flow well together? Is the
essay mostly error-free in grammar, usage, and mechanics?

• Practice pacing yourself so that you are able to get your ideas down on paper in a complete, coherent, and unified essay.
• Prewriting should take 10 to 15 minutes. Use this time to:
ºº Identify and underline the author’s key claims
ºº Find and then circle or highlight specific evidence the author uses to support his or her key claims
ºº Take notes on ways the author has used logic, reasoning, and rhetoric (persuasive language techniques), and evidence
to convince readers that his or her key claims are valid

• Writing the introduction should take 4 to 5 minutes. Keep these five ideas in mind:

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Essay

1.

Keep the key words and phrases of the prompt in mind.

2.

Be sincere, clear, and straightforward. Avoid being cute, funny, ironic, satiric, overly emotional, or too dramatic.

3.

Paraphrase key claims from the source text.

4.

Write a clear topic sentence that introduces the passage’s author, title, topic and the author’s key claim about the topic,
and makes it clear to your readers that you are about to analyze the passage and explain how the author accomplishes
his or her persuasive purpose.

5.

Write sentences that advance the topic and interest the reader.

• Writing the body of the essay should take 15 to 20 minutes. Support your analysis of the author’s work with careful
reasoning, and back up your analysis with evidence from the source text. Your writing must be coherent, logical, unified,
and organized.

• Writing the conclusion should take 3 to 4 minutes. A good concluding paragraph should assure your scorers that you have
successfully read, understood, and analyzed the source text. There are three possible ways to end your essay:
1.

Through a restatement of your most important or most central idea

2.

Through a summary of the material covered in the essay

3.

Through a clear statement about the effectiveness of the passage author’s work

• Revising (3 to 5 minutes) and proofing (3 to 5 minutes) are important steps in the writing process. Be sure to leave yourself
enough time to polish your essay, even though it is considered a first draft by the test graders.

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ONLINE
PREP

Want to Know More?

Access more practice questions, valuable lessons, helpful tips, and expert strategies for the following essay writing topics in
Peterson’s SAT® Online Course:

• Avoiding Common Errors
• Effective Style in Essays
• Essay Scoring
• Essay Writing Method
• Organize and Develop Ideas
• Putting It All Together
To purchase and access the course, go to www.petersons.com/sat.

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Essay

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part vi: math strategies
for the sat ®

Chapter 7:
Chapter 8:
Chapter 9:
Chapter 10: 	
Chapter 11: 	
Chapter 12: 	
Chapter 13: 	

Multiple-Choice Math
Grid-In Strategies
Numbers and Operations
Basic Algebra
Geometry
Functions and Intermediate Algebra
Data Analysis

Chapter 7:
Multiple-Choice Math
OVERVIEW
Why Multiple-Choice Math Is Easier
Question Format
Solving Multiple-Choice Math Questions
Know When to Use Your Calculator
Learn the Most Important Multiple-Choice
Math Tips
Exercises: Multiple-Choice Math
Summing It Up

267

WHY MULTIPLE-CHOICE MATH IS EASIER
How can one kind of math possibly be easier than another? SAT® multiple-choice math is easier than the math tests you take
in class because the answers are right there in front of you. As you know from taking other standardized tests, multiple-choice
questions always give you the answer. You just have to figure out which answer is the correct one. So even if you aren’t sure and
have to guess, you can use estimating to narrow your choices and improve your odds.

Chapter 7
MultipleChoice Math

The questions in each multiple-choice math section are arranged from easiest to most difficult. The questions don’t stick to one
content area. They jump around from algebra to geometry to advanced math to data analysis to statistics and back to algebra
in no particular pattern.

Peterson’s SAT® Prep Guide 2017

QUESTION FORMAT
On the SAT® Math test, each set of multiple-choice math questions starts with directions and a reference section that look like this:
DIRECTIONS: For Questions 1–30, solve each problem, select the best answer from the choices provided, and fill in the
corresponding circle on your answer sheet. For Questions 31–38, solve the problem and enter your answer in the grid on
the answer sheet. The directions before Question 31 will provide information on how to enter your answers in the grid.

ADDITIONAL INFORMATION:

• The use of a calculator is permitted (In the Math Test—No Calculator section, this will say: “The use of a calculator is not
permitted.)

• All variables and expressions used represent real numbers unless otherwise indicated.
• Figures provided in this test are drawn to scale unless otherwise indicated.
• All figures lie in a plane unless otherwise indicated.
• Unless otherwise specified, the domain of a given function f is the set of all real numbers x for which is f(x) is a real number.

Circle:

Rectangle:

r
C = 2�r
A = �r2

l
A = lw

Chapter 7

V = lwh

MultipleChoice Math

r

c

a
b

x 2

x

b

1
A = bh
2

Cylinder:

w
l

h

w

Rectangular
Solid:

268

Triangle:

x

x

Special Right Triangles

a2 + b2 = c2
Sphere:

2x

x 3

Cone:

Rectangular-Based
Pyramid:

h
r

V = �r2h

h

h
r

4
V = �r3
3

The number of degrees of arc in a circle is 360.
The number of radians in the arc of a circle is 2 .
The sum of the measures in degrees of the angles of a triangle is 180.

1
V = �r2h
3

l

w
1
3

V = lwh

The information in the reference section should all be familiar to you from your schoolwork. Know that it’s there in case you need
it. But remember: the formulas themselves aren’t the answers to any problems. You have to know when to use them and how
to apply them.
Some multiple-choice questions ask to solve a given equation or system of equations, while others are presented in the form of
word problems. Some include graphs, charts, or tables that you will be asked to interpret. All of the questions have four answer
choices. These choices are arranged in order when the answers are numbers, usually from smallest to largest, but occasionally
from largest to smallest.

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SOLVING MULTIPLE-CHOICE MATH QUESTIONS
These five steps will help you solve multiple-choice math questions:
1. Read the question carefully and determine what’s being asked.
2. Decide which math principles apply and use them to solve the problem.
3. Look for your answer among the choices. If it’s there, mark it and go on.
4. If the answer you found is not there, recheck the question and your calculations.
5. If you still can’t solve the problem, eliminate obviously wrong answers and take your best guess.
Now let’s try out these steps on a couple of SAT®-type multiple-choice math questions.
Example:

Q

15°

x°
P

S

R

PQ = PS

269

In the figure above, x =
A.

15°

B.

30°

C.

60°

D.

75°

Chapter 7
MultipleChoice Math

Solution:
1. The problem asks you to find the measure of one angle of right triangle PQR.
2. Two math principles apply: (1) the sum of the measures in degrees of the angles of a triangle is 180, and (2)
45-45-90 right triangles have certain special properties. Since PQ = PS, PQS is a 45-45-90 right triangle.
Therefore, angle PQS = 45° and angle PQR = 45 + 15 = 60°. Therefore, angle x = 180 − 90 − 60 = 30°.
3. Look to see if 30° is among the answer choices. You’ll see that it’s choice B. No further steps are needed.
The correct answer is choice B.

Peterson’s SAT® Prep Guide 2017

Example:

If x and y are negative numbers, which of the following is negative?
A.

xy

B.

(xy)2

C.

(x − y)2

D.

x+y

Solution:
1. The problem asks you to pick an answer choice that is a negative number.
2. The principles that apply are those governing operations with signed numbers. Since x and y are negative, choice
A must be positive. As for choices B and C, as long as x and y are not equal to each other, both expressions must
be positive. (If they’re equal, the expression equals zero, and any number other than zero squared gives a positive
result.) Choice D, however, is negative since it represents the sum of two negative numbers.
3. Looking among the answer choices, you can see that the correct answer is choice D. If you have trouble working
with letters, try substituting easy numbers for x and y in each choice.

KNOW WHEN TO USE YOUR CALCULATOR

270
Chapter 7
MultipleChoice Math

Calculators are allowed in the SAT® Math Test—Calculator section, but you won’t need a calculator to solve any SAT® math questions. Calculators can be helpful in solving most of the problems, whether you use the calculator for simplifying expressions or
graphing equations. But remember that your calculator is not some sort of magic brain. If you don’t understand the questions in
the first place, the calculator won’t give you a solution.
Most calculators that you would use in class are allowed. It is best to use whichever calculator you are already comfortable using
instead of trying to learn how to use a new one.
The most important thing to remember is to set up your work on paper first, and then plug the information into the calculator.
For example, if you have a question that deals with an equation, set up the equation on your scratch paper. Then make your
number substitutions on the calculator. This way, you always have something to refer to without having to think, “Oh, nuts, how
did I set that up?” as the seconds tick by.
When you use your calculator, check the display each time you enter numbers to make sure you entered them correctly. Make
sure to hit the Clear key after each finished operation; otherwise, it could get ugly.

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LEARN THE MOST IMPORTANT MULTIPLE-CHOICE MATH TIPS
You’ve probably heard some of these tips before, but some will be new to you. Whatever the case, read them, learn them, and
remember them. They will help you.

ALERT: Don’t automatically reach for your calculator. If it can’t help you solve the problem, you’ll just waste time fiddling
with it. Save the calculator for what it does best, especially simplifying numeric expressions.

The Question Number Tells You How Hard the Question Is
Just as in most of the other SAT® sections, the questions go from easy to hard as you work toward the end. The first third of the
questions is easy, the middle third is average but harder, and the final third gets more and more difficult. Take a look at these
three examples. Don’t solve them yet (you’ll be doing that in a couple of minutes); just get an idea of how the level of difficulty
changes from Question 1 to Question 12 to Question 25.
1.

12.

If

a+5
= m, and m = 9, what is the value of a?
6

A.

24

B.

49

C.

59

D.

84

Line a intersects the x-axis at (3, 0) and the y-axis at (0, –2). Line b passes through the origin and is parallel to line a. Which
of the following is an equation of line b?
A.

3
y= x
2

B.

2
y= x
3

C.

3
y=− x
2

D.

2
y=− x
3

271
Chapter 7
MultipleChoice Math

Peterson’s SAT® Prep Guide 2017

25.

Yasmine owns a coffee shop and orders both coffee and tea from a wholesale supplier. The supplier will send no more than
600 kg in a shipment. Coffee beans come in packages that weigh 18.5 kg, and tea leaves come in packages that weigh 10 kg.
Yasmine wants to buy at least twice as many packages of coffee as packages of tea. If c stands for the number of packages
of coffee, and t stands for the number of packages of tea, which of the following systems of inequalities best represents
Yasmine’s order? Both c and t are nonnegative integers.
A.

18.5c + 10t ≤ 600
c ≥ 2t

B.

18.5c + 10t ≤ 600
2c ≥ t

C.

37c + 10t ≤ 600
c ≥ 2t

D.

37c + 10t ≤ 600
2c ≥ t

Look for shortcuts. SAT® math problems test your math reasoning, not your ability to make
endless calculations. If you find yourself calculating too much, you’ve probably missed a
shortcut that would have made your work easier.

Can you see the difference? You can probably do Question 1 very quickly. For Question 12, you might have to think for a bit.
Question 25 may cause you to wince a little and then get started on some heavy-duty thinking.

272
Chapter 7
MultipleChoice Math

Easy Questions Have Easy Answers—Difficult Questions Don’t
The easy questions are straightforward and don’t have any hidden tricks. The obvious answer is almost always the correct answer.
So for Question 1, the answer is indeed choice B.
When you hit the difficult stuff, you have to think harder. The information is not straightforward, and the answers aren’t obvious.
You can bet that your first-choice, easy answer will be wrong. If you don’t believe it, let’s take another look at Question 25.
Example:
25.

Yasmine owns a coffee shop and orders both coffee and tea from a wholesale supplier. The supplier will send no more than
600 kg in a shipment. Coffee beans come in packages that weigh 18.5 kg, and tea leaves come in packages that weigh 10 kg.
Yasmine wants to buy at least twice as many packages of coffee as packages of tea. If c stands for the number of packages
of coffee, and t stands for the number of packages of tea, which of the following systems of inequalities best represents
Yasmine’s order? Both c and t are nonnegative integers.
A.

18.5c + 10t ≤ 600
c ≥ 2t

B.

18.5c + 10t ≤ 600
2c ≥ t

C.

37c + 10t ≤ 600
c ≥ 2t

D.

37c + 10t ≤ 600
2c ≥ t

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Solution:
This question is difficult mostly because it takes a little longer to think through the problem and set up the inequalities laid out in the question stem. Let’s tackle this step by step.
We will use the variables c for coffee beans and t for tea. The total weight, in kg, of coffee beans and tea that the
wholesale supplier sends can be expressed as the weight of each package multiplied by the number of each type of
package, which is 18.5c for coffee beans and 10t for tea leaves. Since the supplier will not send shipments that weigh
more than 600 kg, it follows that 18.5c + 10t ≤ 600 expresses the first part of the problem.
Since Yasmine wants to buy at least twice as many packages of coffee beans as packages of tea leaves, the number of
packages of coffee beans should be greater than or equal to two times the number of packages of tea leaves. This
can be expressed by c ≥ 2t.
Thus, the correct answer is choice A.
Why are the other answers wrong? Choice B is incorrect because it misrepresents the relationship between the
numbers of each package that Yasmine wants to buy. Choice C is incorrect because the first inequality of the system
incorrectly doubles the weight per package of coffee beans. The weight of each package of coffee beans is 18.5 kg,
not 37 kg. Choice D is incorrect because it doubles the weight per package of coffee beans and transposes the relationship between the numbers of packages.
ALERT: Beware of the obvious. Don’t be fooled by what look like obvious answers to difficult questions. The answers to
difficult questions require some digging. They never jump out at you.

Be Certain to Answer the Question Being Asked

273

Suppose that you were asked to solve the following problem:

Chapter 7

Example:

MultipleChoice Math

If 5x + 11 = 31, what is the value of x + 4?
A.

4

B.

6

C.

8

D.

10

Solution:
The first step is to solve the equation 5x + 11 = 31.
5 x + 11 = 31 Subtract 11 from both sides.
5 x = 20 Divide both siides by 5.
x=4
Remember that the problem does not ask for the value of x, it asks for the value of x + 4, so the answer is actually 8.
Make certain that the answer you select is the answer to the question that is being asked. The correct answer is
choice C.

Peterson’s SAT® Prep Guide 2017

When Guessing at Hard Questions, You Can Toss Out Easy Answers
Now that you know the difficult questions won’t have easy or obvious answers, use a guessing strategy. (Use all the help you can
get!) When you have less than a clue about a difficult question, scan the answer choices and eliminate the ones that seem easy
or obvious, such as any that just restate the information in the question. Then take your best guess.

Questions of Average Difficulty Won’t Have Trick Answers
Let’s look again at Question 12:
12.

274
Chapter 7
MultipleChoice Math

Line a intersects the x-axis at (3, 0) and the y-axis at (0, –2). Line b passes through the origin and is parallel to
line a. Which of the following is an equation of line b?

A.

y=

3
x
2

B.

y=

2
x
3

C.

3
y=− x
2

D.

2
y=− x
3

This is a bit more difficult than Question 1, but it’s still pretty straightforward. Since we know points on line a, we can calculate
the slope of this line:
y 2 − y1 −2 − 0 2
=
=
x 2 − x1 0 − 3 3
2
. The problem tells us that line b passes through the origin,
3
2
which means that it intercepts the y-axis at 0. Thus, line b can be expressed by the equation y = x . The correct answer is choice B.
3

Since line b is parallel to line a, the two have the same slope, that is,

It’s Smart to Work Backward
Every standard multiple-choice math problem includes four answer choices. One of them has to be correct; the other three are
wrong. This means that it’s always possible to solve a problem by testing each of the answer choices. Just plug each choice into
the problem and sooner or later you’ll find the one that works! Testing answer choices can often be a much easier and surer way
of solving a problem than attempting a lengthy calculation.

When Working Backward, Always Start from the Middle
When working on multiple-choice math questions, remember that all of the numeric answer choices are presented in order—
either smallest to largest, or vice versa. As a result, it’s always best to begin with a middle option, choice B or choice C. This way,
if you start with choice C and it’s too large, you’ll only have to concentrate on the smaller choices. There, you’ve just knocked off
at least two choices in a heartbeat! Now let’s give it a test run!

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Example:
If

8
9
y = , what is the value of y?
9
4

A.

32
81

B.

1
2

C.

2

D.

81
32

Solution:
8
16 9
Start with choice C, because it will be easier to compute with than choice B: (2) =
< .
9
9 4
Since choice C is too small, the only possible answer is choice D.
You can check that

8  81 9
 = .
9  32  4

The correct answer is choice D.
Now try this testing business with a more difficult question:
Example:
In the xy-plane, the line determined by the points (8, c) and (c, 18) passes through the origin. Which of the following
could be the value of c?
A.

10

B.

11

C.

12

D.

13

275
Chapter 7
MultipleChoice Math

Solution:
Start with choice C, because it may be easier to compute with than choice B. The line through (8, 12) and (12, 18)
3
18 − 12
x + b. This equation simplifies to y = x , which is a line through the origin.
2
12 − 8
3
Plug (8, 12) or (12, 18) into y = x , and both sides will be equal.
2
is y =

3
3
12 = (8 ) or 18 = (12)
2
2
12 = 12
18 = 18
The other values of c will not result in an equivalent equation. The correct answer is choice C.

Peterson’s SAT® Prep Guide 2017

It’s Easier to Work with Numbers Than with Letters
Because numbers are more meaningful than letters, try plugging them into equations and formulas in place of variables. This
technique can make problems much easier to solve. Here are some examples:
Example:
If x − 4 is 2 greater than y, then x + 5 is how much greater than y?
A.

3

B.

7

C.

9

D.

11

Solution:
Choose any value for x. Let’s say you decide to make x = 4. All right, 4 − 4 = 0, and 0 is 2 greater than y. So y = −2.
If x = 4, then x + 5 = 4 + 5 = 9, and so x + 5 is 11 more than y. The correct answer is choice D.
Example:
The cost of renting office space in a building is $2.50 per square foot per month. Which of the following represents
the total cost c, in dollars, to rent p square feet of office space each year in the building?

276

A.

c = 2.50(12p)

B.

c = 2.50p + 12

C.

c=

2.50 p
12

D.

c=

12 p
2.50

Chapter 7
MultipleChoice Math

Solution:
Let p = 100, then the rent for one month is $250 and the rent for one year is $3,000. The only equation that will
provide that answer is c = 2.50(12p). The correct answer is choice A.
If a question asks for an odd integer or an even integer, go ahead and pick any odd or even integer you like.

Leave a paper trail! If you need to set up an equation, jot it down in your
test booklet. That way, if you come back to recheck your work, you’ll know
what you were originally thinking.

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Solving for Variables with Restricted Values
When solving problems involving variables, you must pay careful attention to any restrictions on the possible values of the
variables
Consider the following question:
Example:
If x ≥ 2, which of the following is a solution to the equation x(x − 3)(x + 4)(x + 2)(3x − 5) = 0?
A.

2

B.

3

C.

4

D.

5

Solution:
This equation has five solutions, but the problem is looking only for a solution that is at least 2. Set each of the factors
equal to 0 and solve for x. The only answer that is greater than or equal to 2 is 3. The correct answer is choice B.
Now, consider this slightly different version of the same problem.
Example:

277

If x < –2, which of the following is a solution to the equation x(x − 3)(x + 4)(x + 2)(3x − 5) = 0?
A.

–3

Chapter 7

B.

–4

C.

–5

MultipleChoice Math

D.

There is more than one solution.

Solution:
The solutions to the equation can be found by setting each of the factors equal to zero. So x = 0, x – 3 = 0, x + 4 = 0,
x + 2 = 0, and 3x – 5 = 0.
These lead to the solutions x = 0, 3, –4, –2, and

5
respectively.
3

Of these five solutions only –4, choice B, is less than –2. The correct answer is choice B.

The test booklet is yours, so feel free to use it for your scratchwork.
Also, go ahead and mark up any diagrams with length or angle information; it helps. But don’t waste time trying to redraw diagrams; it’s
just not worth it.

Peterson’s SAT® Prep Guide 2017

5
3

Solving Equations in the Three-Statement Format
You may find a three-statement format in certain questions in the multiple-choice math section. The best way to answer this
kind of question is by process of elimination, tackling one statement at a time and marking it as true or false. Here is an example:

Example:

k

p° q°
r°

m
s°

t° v°
w° x°

n

Note: Figure not drawn to scale.

In the figure above, lines k and m intersect at a point, and lines k and n intersect at a different point. If v + s = p + q,
which of the following statements must be true?

278
Chapter 7
MultipleChoice Math

I.

r=w

II.

t=s

III.

q=x

A.

I only

B.

II only

C.

I and II only

D.

I, II, and III

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Solution:
Because v + s = p + q, we know that p + q = 180. Because they both make a straight line, lines m and n must be parallel. Since m and n are parallel, then statements I and II, must be true. While statement III might be true, it is only true
if line k is perpendicular to lines m and n, and that does not have to be true.
The correct answer is choice C.

For multiple-choice math questions, circle what’s being asked so that
you don’t pick a wrong answer by mistake. That way, for example, you
won’t pick an answer that gives perimeter when the question asks for
an area.

Solving Equations Involving Square Roots or Algebraic Fractions
The procedure for solving equations involving square roots or algebraic fractions occasionally results in what are known as
extraneous solutions. An extraneous solution is a number that is correctly obtained from the equation-solving process but doesn’t
actually solve the equation, be sure to check your answer.
Example:
Solve for x:

x + 4 + 15 = 10

A.

−29

B.

−21

C.

21

D.

There are no solutions.

279
Chapter 7

Solution:

MultipleChoice Math

First, solve the equation.
x + 4 + 15 = 10

Subtract 15 from both sides.

x + 4 = −5

(

x+4

)

2

Squaare both sides.

= ( −5)

2

x + 4 = 25
x = 21

It appears that the solution is choice C. However, if you check the solution x = 21 in the original equation, you will see
that it doesn’t solve it.
x + 4 + 15 = 10 ?
21+ 4 + 15 = 10 ?
25 + 15 = 10 ?
5 + 15 ≠ 10.
The correct answer is choice D.

Peterson’s SAT® Prep Guide 2017

Solving Geometry Problems of Measure
When you are asked to find the measure of a side or angle of a figure, using the measure of an angle or a side of another shape
can help you find the measure you need.
Example:
In the figure, what is the length of NP?
A.

8

B.

9

C.

12

D.

15

N
20

M

9

7
P

O

Solution:
This figure is really two right triangles, NMO and NMP. Since NM is a side of both triangles, once you find its length,
you can find the length of NP. The Pythagorean theorem is what you need:
NM2 + MO2 = NO2

280
Chapter 7
MultipleChoice Math

NM2 + (16)2 = (20)2
Note that 16 and 20 are multiples of 4 and 5, respectively, so you now know that this is a 3-4-5 right triangle, which
means that NM = 12.
Since you just found out that triangle NMP has sides of 9 and 12, it’s also a 3-4-5 right triangle, so NP must be 15.
The correct answer is choice D.

Draw a diagram if none is supplied. Drawing a diagram is a great way to
organize information. Mark it up with the information you’re given, and
you’ll have a better idea of what you’re looking for.

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Solving Right Triangles Using the Pythagorean Theorem
The Pythagorean theorem is usually needed to solve problems involving a right triangle for which you are given the lengths
of some of the sides. The Pythagorean theorem enables you to compute the length of the third side of a right triangle if you
know the lengths of the other two sides. It is one of the most useful and common SAT® geometry facts. Consider the problem
below.
Example:
Line segment PQ is tangent to the circle with center O at point T. If T is the midpoint of PQ, OQ = 13, and the radius
of the circle is 5, what is the length of PQ?
A.

10

B.

12

C.

24

D.

26

P
T
O

Q

Solution:
This is a tricky question since, at the moment, it doesn’t appear to involve any triangles at all. However, you are told
that the radius of the circle is 5, and if you draw in radius OT , you will create triangle OTQ. Use the fact that a tangent
line to a circle is perpendicular to the radius at the point of contact to deduce that ∠OTQ is a right angle.

281

P
T

Q

Chapter 7
O

MultipleChoice Math

The diagram now depicts right triangle OTQ, and OT = 5 and OQ = 13. Now, use the Pythagorean theorem to
determine that TQ = 12, as shown here:

OT 2 + TQ 2 = OQ 2
52 + TQ 2 = 132
25 + TQ 2 = 169
TQ 2 = 144
TQ = 12
Finally, since T is the midpoint of line segment PQ, the entire length of the line segment is 12 + 12 = 24.
The correct answer is choice C.

Peterson’s SAT® Prep Guide 2017

Eliminate Answers That Can’t Possibly Be Right
Knowing whether your calculations should produce a number that’s larger or smaller than the quantity you started with can point
you toward the right answer. It’s also an effective way of eliminating wrong answers. Here’s an example:
Example:
Daryl can set up the display for the science fair in 20 minutes. It takes Francisco 30 minutes to set it up. How long will
it take the two boys to complete the setup if they work together?
A.

8 minutes

B.

12 minutes

C.

20 minutes

D.

30 minutes

Solution:
Immediately you can see that choices C and D are impossible because the two boys working together will have to
complete the job in less time than either one of them working alone.

Actual time spentt
Time needed to do entire job alone

282

Daryl
x
20

x
x
+
=1
20 30

Chapter 7
MultipleChoice Math

Multiply by 60 to clear fractions:
3 x + 2 x = 60
5 x = 60
x = 12
The correct answer is choice B.

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Francisco
x
30

Your Eye Is a Good Estimator
Figures in the standard multiple-choice math section are always drawn to scale unless you see the warning “Note: Figure not
drawn to scale.” That means you can sometimes solve a problem just by looking at the picture and estimating the answer. Here’s
how this works:
Example:

4

f (x)

2

–2

0

2

4

6

–2
g (x)
–4

f ( x ) = ( x − 3) − 3
2

283

g ( x ) = −2 ( x − 3) + 3
2

Graphs of the functions f and g are shown in the xy-plane above. For which of the following values of x does
f(x) + g(x) = 0?

Chapter 7

A.

1

MultipleChoice Math

B.

2

C.

3

D.

4

Solution:
The sum of the function values is 0 when the function values for f and g are opposites. That appears to be true at
x = 3. The correct answer is choice C.

If Some Questions Always Give You Trouble, Save Them for Last
You know which little demons haunt your math skills. If you find questions that you know will give you nightmares, save them for
last. They will take up a lot of your time, especially if you’re panicking, and you can use that time to do more of the easier questions.

Peterson’s SAT® Prep Guide 2017

EXERCISES: MULTIPLE-CHOICE MATH
Exercise 1
18 Minutes—15 Questions
For Questions 1–15, solve each problem, choose the best answer from the choices provided, and put a circle around the
correct answer. You may use any available space for scratch work.

ADDITIONAL INFORMATION:

• The use of a calculator is permitted.
• All variables and expressions used represent real numbers unless otherwise indicated.
• Figures provided in this test are drawn to scale unless otherwise indicated.
• All figures lie in a plane unless otherwise indicated.
• Unless otherwise specified, the domain of a given function f is the set of all real numbers x for which f(x) is a real number.

Circle:

Rectangle:

r
C = 2�r
A = �r2

284

l
A = lw

Rectangular
Solid:

MultipleChoice Math

l
V = lwh

h

w

r

c

a
b

x 2

x

b

1
A = bh
2

Cylinder:

w

Chapter 7

Triangle:

x

Cone:

Rectangular-Based
Pyramid:

h
r

V = �r2h

h

h
r

4
V = �r3
3

The number of degrees of arc in a circle is 360.
The number of radians in the arc of a circle is 2 .
The sum of the measures in degrees of the angles of a triangle is 180.

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x

Special Right Triangles

a + b2 = c2
2

Sphere:

2x

x 3

1
V = �r2h
3

l

w
1
3

V = lwh

1.

2.

3.

4.

In a linear function f(x) with a graph that has a slope of
–2.5, if f(5) = 8, what is the value of f(1)?
A.

–2

B.

3

C.

13

D.

18

Which of the following expressions is equal to 0 for some
value of p?
A.

|p – 2| − 2

B.

|2 − p| + 2

C.

|p + 2| + 2

D.

|p – 2| + 2

If f(x) = –3x + 4, what is f(–4x)?
A.

–12x – 4

B.

12x + 4

C.

12x – 4

D.

12x2 – 20x

y
=4
x
5 ( x − 1) = y
If (x, y) is the solution to the system of equations above,
what is the value of x?

5.

SHOW YOUR WORK HERE

A.

–20

B.

–5

C.

5

D.

20

285
Chapter 7
MultipleChoice Math

Which of the following is equivalent to the expression
5(3x – 2)(2x + 1)?
A.

15x

B.

30x2 – 10

C.

30x2 – 5x –10

D.

25x2 – 5

Peterson’s SAT® Prep Guide 2017

6.

7.

Kathleen is preparing to run her first road race. She begins training by doing intervals of running and walking.
The longest interval of running increases by a constant
amount each week. The first week she runs for intervals
of up to 2 minutes at a time. By the ninth week, she runs
for intervals of up to 40 minutes at a time. Which of the
following best describes how the time Kathleen spends
running changes between her first and ninth weeks?
A.

Kathleen increases the time of her longest run by
5 minutes each week.

B.

Kathleen increases the time of her longest run by
4.75 minutes each week.

C.

Kathleen increases the time of her longest run by
4.5 minutes each week.

D.

Kathleen increases the time of her longest run by
4.25 minutes each week.

If

x+y 5
= , which of the following must also be true?
y
9

A.

x
4
=−
y
9

B.

286

x 14
=
y 9

C.

4
x−y
=−
9
y

Chapter 7

D.

x − 2y
13
=−
y
9

MultipleChoice Math

8.

Which of the following shows the solution set to the
equation x − 1 = x − 7 ?
A.

{5, 10}

B.

{1}

C.

{5}

D.

{10}

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SHOW YOUR WORK HERE

9.

A circle whose center is at the origin passes through the
point whose coordinates are (1, 1). What is the area of this
circle?

10.

A.

π

B.

2π

C.

2p

D.

2 2p

SHOW YOUR WORK HERE

In triangle ABC, AB = BC, and AC is extended to D. If
angle BCD measures 100°, find the number of degrees in
angle B.

B

A

11.

C
A.

20

B.

40

C.

50

D.

80

D

Which of the following represents a line that is perpendicular to the line with equation y = 4 – 2x?
A.

y = 2x + 5

B.

y = –2x + 3

C.

y – 2x = 1

D.

2y – x = 6

287
Chapter 7
MultipleChoice Math

x = 3y

12.

( x − 3) + ( y − 1)2 = 10
2

How many ordered pairs satisfy the system of equations
above?
A.

0

B.

1

C.

2

D.

infinitely many

Peterson’s SAT® Prep Guide 2017

13.

14.

15.

If

z+6
= 5 , what is the value of z?
z −5

A.

19
6

B.

35
6

C.

25
4

D.

31
4

If the expression (4 + 3i)(2 – 7i) is rewritten in the form
a + bi, where a and b are real numbers, what is the value
of a? (Note: i = −1 )
A.

29

B.

8

C.

–13

D.

–22

x 2 − 6dx =

e
2

In the quadratic equation above, d and e are
constants. What are the solutions for x?

288
Chapter 7
MultipleChoice Math

A.

x = 3d ±

6 d − 2e
2

B.

x = 3d ±

6 d + 2e
2

C.

x = 3d ±

36d 2 − 2e
2

D.

x = 3d ±

36d 2 + 2e
2

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SHOW YOUR WORK HERE

Exercise 2
12 Minutes—10 Questions
For Questions 1–10, solve each problem, choose the best answer from the choices provided, and put a circle around the
correct answer. You may use any available space for scratch work.

ADDITIONAL INFORMATION:

• The use of a calculator is permitted.
• All variables and expressions used represent real numbers unless otherwise indicated.
• Figures provided in this test are drawn to scale unless otherwise indicated.
• All figures lie in a plane unless otherwise indicated.
• Unless otherwise specified, the domain of a given function f is the set of all real numbers x for which f(x) is a real number.

Circle:

Rectangle:

r
C = 2�r
A = �r2

l
A = lw

r

c

a
b

x 2

x

b

1
A = bh
2

Cylinder:

w
V = lwh

h

w

Rectangular
Solid:

l

Triangle:

x

x

Special Right Triangles

a + b2 = c2
2

Sphere:

2x

x 3

Cone:

Rectangular-Based
Pyramid:

h
r

V = �r2h

h
r

4
3

V = �r3
The number of degrees of arc in a circle is 360.
The number of radians in the arc of a circle is 2 .
The sum of the measures in degrees of the angles of a triangle is 180.

1
3

289

h

V = �r2h

l

w
1
3

V = lwh

Chapter 7
MultipleChoice Math

Peterson’s SAT® Prep Guide 2017

1.

2.

3.

290
Chapter 7
MultipleChoice Math

4.

Mario has made 58 successful 3-point shots in basketball
this season. The school record is 92 3-point shots. If he
makes 3 more 3-point shots each game, which of the
following shows how many 3-point shots he will have
made after g additional games?
A.

3 + 58g

B.

92 + 3g

C.

58 + 3g

D.

58 – 3g

If a = 5b, then
A.

5b
3

B.

3b

C.

3b
5

D.

b
3

3
a=
5

A rectangular door measures 5 feet by 6 feet 8 inches.
What is the distance from one corner of the door to the
diagonally opposite corner?
A.

8 feet 2 inches

B.

8 feet 4 inches

C.

8 feet 8 inches

D.

9 feet 6 inches

Two ships leave from the same port at 11:30 a.m. If one
sails due east at 20 miles per hour and the other due south
at 15 miles per hour, how many miles apart are the ships
at 2:30 p.m.?
A.

25

B.

50

C.

75

D.

80

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SHOW YOUR WORK HERE

5.

6.

7.

8.

The line y = ax – 5, where a is a constant, is graphed in the
xy-plane. If the line contains the point (b, c), which does
not lie on either the x- or y-axis, what is the slope of the
line in terms of b and c?

A.

c +5
b

B.

b+5
c

C.

c −5
b

D.

b−5
c

SHOW YOUR WORK HERE

Ken received grades of 90, 88, and 75 on three tests. What
grade must he receive on the next test so that his average
for these four tests is 85?
A.

87

B.

89

C.

90

D.

92

There is enough food at a picnic to feed 20 adults or 32
children. If there are 15 adults at the picnic, how many
children can still be fed?

291

A.

6

Chapter 7

B.

8

C.

12

MultipleChoice Math

D.

16

In parallelogram ABCD, angle A measures 60°. What is the
sum of angle B and angle D?
A.

180°

B.

240°

C.

280°

D.

300°

Peterson’s SAT® Prep Guide 2017

9.

10.

If 4x – 7 ≤ 1, what is the greatest possible value of 4x + 2?
A.

10

B.

7

C.

3

D.

1

If a train covers 14 miles in 10 minutes, what is the rate of
the train in miles per hour?
A.

64

B.

76

C.

84

D.

98

292
Chapter 7
MultipleChoice Math

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SHOW YOUR WORK HERE

ANSWER KEYS AND EXPLANATIONS
Exercise 1

1.

1. D

4. C

7. A

10. A

13. D

2. A

5. C

8. D

11. D

14. A

3. B

6. B

9. B

12. C

15. D

The correct answer is D. If the slope is –2.5 and
f(5) = 8, then use the slope formula to find f(1).

7.

18 = f (1)

2.

3.

4.

5.

8.

The correct answer is D.
x −1 = x − 7

The correct answer is A. The absolute value of an
expression is always greater than or equal to 0, so
adding a number to the expression will never result in
0. Choice A, which states |p – 2| − 2 = 0 when p = 0 or
4, is correct.

x − 1 = ( x − 7)

2

x − 1 = x 2 − 14 x + 49
0 = x 2 − 15 x + 50

293

Consequently, x = 5 or x = 10. Plug these values into
the original equation to see if they make it true.

Chapter 7

0 = ( x − 5) ( x − 10)

The correct answer is B. If f(x) = –3x + 4, then
f(–4x) = –3(–4x) + 4, and –3(–4x) + 4 = 12x +4.

5 −1 = 5 − 7?
2 ≠ −2

y
= 4 and 5(x – 1) = y, then
x
y = 4x and y = 5x –5. So, 4x = 5x – 5 and x = 5.

The correct answer is C. If

The correct answer is C. Multiply the expression to
find the equivalent one. Use FOIL.

x+y 5
= , then
9
y

x y 5
+ =
y y 9
x y
5
+ − 1= − 1
y y
9
x
4
=−
y
9

8 − f (1)
= −2.5
5 −1
8 − f (1)
= −2.5
4
8 − f (1) = −10
8 + 10 = f (1)

The correct answer is A. If

MultipleChoice Math

10 − 1 = 10 − 7 ?
3=3
9.

The correct answer is B.

5(3 x − 2)(2 x + 1) = 5(6 x 2 + 3 x − 4 x − 2)

(1, 1)

= 5(6 x 2 − x − 2)
= 30 x − 5 x − 10
2

1
6.

The correct answer is B. To find the average increase,
divide the change in increase by the number of weeks:
40 − 2 38
=
= 4.75
9 −1
8

1

12 + 12 = r2
2 = r2
Area = πr2 = 2π

Peterson’s SAT® Prep Guide 2017

10.

The correct answer is A.

14.

(4 + 3i) (2 – 7i) = 8 – 28i + 6i – 21i2

		Angle BCA = Angle BAC = 80°.
		

(4 + 3i) (2 – 7i) = 8 – 22i – 21(–1)

There are 20° left for angle B.
		

11.

The correct answer is D. A line that is perpendicular to
the line with the equation y = 4 – 2x will have a slope
equal to 0.5. If all equations are written in slope–
intercept form, only choice D, 2y – x = 6, will be correct.

The correct answer is A.

15.

(4 + 3i) (2 – 7i) = 29 – 22i
The correct answer is D. Write the equation in
standard form, then plug the numbers into the
quadratic formula and simplify.

12.

The correct answer is C. Graph the system of
equations. Note that the second equation is a circle
that is intersected twice by the line.

e
=0
2
The quadratic formula is:

13.

The correct answer is D.

b 2 − 4 ac
−b
±
2a
2a
So, applyiing that:

z+6
=5
z −5
z + 6 = 5 z − 25
31 = 4 z
31
=z
4

294
Chapter 7
MultipleChoice Math

www.petersons.com

x 2 − 6dx −

x=

x=

− ( −6d )
±
2(1)

x = 3d ±

( −6d ) 2 − 4 (1)  −
2 (1)

2

36d + 2e
2

1 
e
2 

Exercise 2
1. C

3. B

5. A

7. B

9. A

2. B

4. C

6. A

8. B

10. C

1.

The correct answer is C. If Mario already has made 58
shots and will make an additional 3 shots each game,
then he will make an additional 3g 3-point shots in g
games, for a total of 58 + 3g.

6.

The correct answer is A. Ken must score as many
points above 85 as below. So far he has 8 above and 10
below. He needs another 2 above, making 87, choice A,
the correct answer.

2.

The correct answer is B.

7.

The correct answer is B. If 15 adults are fed, 3 of the
4
1
1
food is gone. of the food will feed × 32, or 8,
4
4
children.

8.

The correct answer is B.

3
i 5 b = 3b
5

3.

The correct answer is B.
5 feet = 60 inches

If angle A = 60°, then angle B = 120°.

6 feet 8 inches = 80 inches

Angle B = Angle D. Their sum is 240°.

This is a 6-8-10 triangle, making the diagonal 100
inches, which is 8 feet 4 inches.
4.

5.

9.

If 4x –7 ≤ 1, then 4x + 2 must be at most 10.

The correct answer is C. In 3 hours, one ship went
60 miles, the other 45 miles. This is a 3-4-5 triangle, as
45 = 3(15), and 60 = 4(15). The hypotenuse will be
5(15), or 75.
The correct answer is A. Substitute the point (b, c)
into the equation y = ax – 5 and solve for a to find that
c +5
c = ab – 5 and a =
.
b

The correct answer is A.

4 x − 7 + 9 ≤ 1+ 9
4 x + 2 ≤ 10
10.

The correct answer is C.
10 minutes =

295
Chapter 7

1
hour
6

MultipleChoice Math

In one hour, the train will cover 6(14), or 84, miles.

Peterson’s SAT® Prep Guide 2017

SUMMING IT UP
• Follow the five-step plan for answering basic multiple-choice math questions:
1.

Read the question carefully and determine what’s being asked.

2.

Decide which math principles apply and use them to solve the problem.

3.

Look for your answer among the choices. If it’s there, mark it and go on.

4.	If the answer you found is not there, recheck the question and your calculations.
5.	If you still can’t solve the problem, eliminate obviously wrong answers and take your best guess.

• In the Math Test—Calculator section, use a calculator where it can help the most: on basic arithmetic calculations, when
calculating square roots and percentages, and comparing and converting fractions.

• Always set up your work on paper, then enter the numbers in your calculator; that way, if your calculation becomes confused, you don’t have to try to replicate your setup from memory.

• The question number tells you how hard the question will be, though some questions may be easier for you.
• Work backward from the answer choices. When you do, start with choice B or choice C.
• Try to work with numbers instead of letters. This will help you avoid unnecessary algebraic calculations.
• Figures in the math section are always drawn to scale unless you see a warning. So, if you need to do so, use your eye as an
estimator.

ONLINE
PREP

296
Chapter 7
MultipleChoice Math

Want to Know More?

Access more practice questions, valuable lessons, helpful tips, and expert strategies for the following multiple-choice math topics
in Peterson’s SAT® Online Course:

• Calculator Strategy
• Guessing in Math
• Pacing in Math
• Plugging in Numbers
• Problem Solving
• Working Backwards
To purchase and access the course, go to www.petersons.com/sat.

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Chapter 8:
Grid-In Strategies
OVERVIEW
Why Grid-Ins Are Easier Than You Think
How to Record Your Answers
Guessing on Grid-Ins Can’t Hurt You
Exercises: Grid-Ins
Summing It Up

WHY GRID-INS ARE EASIER THAN YOU THINK
Let’s take a quick break from multiple-choice questions and examine the other kind of question you will see on both the Math
Test—No Calculator and Math Test—Calculator sections: grid-ins. These are officially named “student-produced responses,”
because you have to do the calculations and find the answer on your own; there are no multiple-choice answers from which
to choose.

297
Chapter 8

Many students are intimidated by grid-ins. Don’t be! Grid-in questions test the exact same mathematical concepts as the
multiple-choice questions. The only difference is that there are no answer choices with which to work.

Grid-In
Strategies

The grid-in questions are in a section of their own and arranged in order of difficulty from easy to hard.

Take a Look at a Grid
The special answer grid has some very different sections. There are blank boxes at the top so you can actually write in your answer.
Below the boxes are some circles that have fraction slashes and decimal points. You fill these in if your answer needs them. The
largest section has circles with numbers in them. You have to fill in the circles to correspond to the answer you have written in
the boxes. Yes, it’s a lot to think about, but once you understand how to use the grid-ins, it’s not a big deal.

Peterson’s SAT® Prep Guide 2017

Here is a sample grid:

/
.

/
.

.

0

0

0

0

1

1

1

1

2

2

2

2

3

3

3

3

4
5

4
5

4
5

4
5

6

6

6

6

7
8

7
8

7
8

7
8

9

9

9

9

NOTE: Remember that the student-produced responses will not be negative numbers
and won’t be greater than 9999.

1

ALERT: If the correct answer is a mixed number such as 5 , you must grid it as 21/4 or as 5.25. If you grid it as “51/4,” it will
4
read as “fifty-one fourths,” and it will be marked incorrect.

298
Chapter 8
Grid-In
Strategies

HOW TO RECORD YOUR ANSWERS
On the SAT® exam, each set of grid-in questions starts with directions that look approximately like this:

DIRECTIONS: For these questions, solve the problem and enter your answer in the grid, as described below, on the
answer sheet.

1. Although not required, it is suggested that you write your answer in the boxes at the top of the columns to help you fill in
the circles accurately. You will receive credit only if the circles are filled in correctly.
2. Mark no more than one circle in any column.
3. No question has a negative answer.
4. Some problems may have more than one correct answer. In such cases, grid only one answer.
7
1
5. Mixed numbers such as 3 must be gridded as 3.5 or .
2
2
		
If 3

1
is entered into the grid as
2

www.petersons.com

, it will be interpreted as

1
31
, not 3 .
2
2

6. Decimal answers: If you obtain a decimal answer with more digits than the grid can accommodate, it may be either rounded
or truncated, but it must fill the entire grid.

7
12

Answer:

Answer: 2.5

Write answer
in boxes.

.

Fraction
line
0

Grid in
result.

0

0

0

0
1

1

2

2

3

3

3

4

4

4

5

5

5

6

6

6

6

6

7

7

7

7

7

8

8

8

8

8

8

9

9

9

9

9

9

1

1

2

2

2

1

1

3

3

3

3

3

4

4

4

4

4

5

5

5

5

6

6

6

7

7

8

8

9

9

1

2

Decimal
point

0

Answer: 201
Either position is correct.

0
1

1

0
2

2

0

0

1

1

1

1

2

2

2

2

2

1

0

3

3

3

3

3

3

3

3

4

4

4

4

4

4

4

4

299

2
Acceptable ways to grid are:
3
.

1

0

0

0

1

1

1

2

2

2

1

Chapter 8
.

Grid-In
Strategies

0

0

0

0

0

0

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

3

3

3

3

3

3

3

3

4

4

4

4

4

4

4

4

4

5

5

5

5

5

5

5

5

5

7

7

7

8

8

8

8

8

9

9

9

9

9

3

3

3

4

4

4

5

5

5

6

6

6

6

6

7

7

7

7

7

7

7

7

8

8

8

8

8

8

8

9

9

9

9

9

9

9

6

6

Once you understand the following six rules, you can concentrate just on solving the math problems in this section.

Peterson’s SAT® Prep Guide 2017

1. Write your answer in the boxes at the top of the grid.
2. Mark the corresponding circles, one per column.
3. Start in any column.
4. Work with decimals or fractions.
5. Express mixed numbers as decimals or improper fractions.
6. If more than one answer is possible, grid any one.

NOTE: Don’t use a comma in a number larger than 999. Just fill in the four digits and the corresponding circles.
You only have circles for numbers, decimal points, and fraction slashes; there aren’t any for commas.
Now let’s look at these rules in more detail:
1.

Write your answer in the boxes at the top of the grid. Technically, this isn’t required by the SAT®. Realistically, it gives you
something to follow as you fill in the circles. Do it—it will help you.

2.

Make sure to mark the circles that correspond to the answer you entered in the boxes, one per column. The machine that
scores the test can only read the circles, so if you don’t fill them in, you won’t get credit. Just entering your answer in the
boxes is not enough!

3.

You can start entering your answer in any column, if space permits. Unused columns should be left blank; don’t put in zeroes.
Look at this example:
Here are two ways to enter an answer of “150.”

300

/
.
0

Chapter 8
2

Grid-In
Strategies

4.

/
.

.
0

0

1

1

1

1

2

2

2

2

0

/
.
0
2

/
.

.

0
1

1

2

2

3

3

3

3

3

3

3

3

4
5

4

4
5

4
5

4
5

4
5

4

4
5

6

6

6

6

6

6

6

6

7
8

7
8

7
8

7
8

7
8

7
8

7
8

7
8

9

9

9

9

9

9

9

9

You can write your answer as a decimal or a fraction. For example, an answer can be expressed as 3 or as .75. You don’t have
4
to put a zero in front of a decimal that is less than 1. Just remember that you have only four spaces to work with and that a
decimal point or a fraction slash uses up one of the spaces.

www.petersons.com

ALERT: Take the time to write your answer in the spaces. It will lessen your chances of filling in an incorrect circle.

For decimal answers, be as accurate as possible but keep it within four spaces. Say you get an answer of .1777,
here are your options:

.

.
/
.

0

0

1
2

2

/
.

.

/
.

0

0

0

1

1

1

2

2

2

0
2

/
.

.

0

0

1

1

2

2

3

3

3

3

3

3

3

3

4
5

4
5

4
5

4
5

4
5

4
5

4
5

4
5

6

6

6

6

6

6

6

6

7
8

7
8

8

8

7
8

7
8

8

9

9

9

9

9

9

9

7
9

Answers .177 and .178 would both be marked correct.

Fractions do not have to be simplified unless they don’t fit in the answer grid. For example, you can grid

4
, but you can’t
10

12
3
grid 16 because you’d need five spaces. So, you would simplify it and grid 4 .
5.

301
3

A mixed number has to be expressed as a decimal or as an improper fraction. If you tried to grid 1 , it would be scored as
4
7
13
, which would give you a wrong answer. Instead, you could grid the answer as 1.75 or as .
4
4

Chapter 8
Grid-In
Strategies

.
/

/
.

.

.

/
.

/
.

.

.

0

0

0

0

0

0

0

1

1

1

1

1

1

1

0

2

2

2

2

2

2

2
3

.
0
2

2

/
.

/
.

.
0

0

0

1

1

1

2

2

2

3

3

4
5

4
5

5

3

3

3

3

3

3

3

4
5

4
5

4
5

4

4
5

4
5

4
5

5

4
5

6

6

6

6

6

6

6

6

6

6

6

6

7
8

7
8

8

7
8

7
8

8

7
8

7
8

9

9

9

9

9

9

9

9

7
8

7
8

7
8

7
8

9

9

9

9

The above answers are acceptable.

3

The above answer is unacceptable as

3
4

representing the fraction 1 .

Peterson’s SAT® Prep Guide 2017

6.

Sometimes, the problems in this section will have more than one correct answer. Choose one and grid it.
For example, if a question asks for a prime number between 5 and 13, the answer could be 7 or 11. Grid 7 or grid 11, but
don’t put in both answers.

/
.

/
.

.

0

0

0

0

1

1

2

2

2

2

2

3

3

3

3

3

4
5

4
5

4
5

4
5

4
5

4
5

6

6

6

6

6

6

6

7
8

7
8

8

7
8

7
8

7
8

7
8

9

9

9

9

9

9

9

/
.

/
.

.

.

0

0

0

0

1

1

1

1

2

2

2

3

3

3

4
5

4
5

6
7
8
9

.

Either answer is acceptable but not both.

GUESSING ON GRID-INS CAN’T HURT YOU

302

Unfortunately, you cannot receive partial credit for grid-ins. Your answers are either completely correct or completely wrong. But
no points are deducted for incorrect responses, so guessing is better than leaving a question blank.

Chapter 8
Grid-In
Strategies

www.petersons.com

EXERCISES: GRID-INS
Exercise 1
15 Minutes—10 Questions
DIRECTIONS: For these questions, solve the problem and enter your answer in the grid following each question, as
described below.

1. Although not required, it is suggested that you write your answer in the boxes at the top of the columns to help you fill in
the circles accurately. You will receive credit only if the circles are filled in correctly.
2. Mark no more than one circle in any column.
3. No question has a negative answer.
4. Some problems may have more than one correct answer. In such cases, grid only one answer.
7
1
5. Mixed numbers such as 3 must be gridded as 3.5 or .
2
2
		
If 3

1
is entered into the grid as
2

, it will be interpreted as

1
31
, not 3 .
2
2

6. Decimal answers: If you obtain a decimal answer with more digits than the grid can accommodate, it may be either rounded
or truncated, but it must fill the entire grid.
7
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Answer:

Answer: 2.5

Write answer
in boxes.

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1. If 3x + y = 4 and x + 3y = 5, what is the value of x + y?

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2. Marion is paid $24 for 5 hours of work in the school office.
Janet works 3 hours and makes $10.95. How much more
per hour does Marion make than Janet? (Ignore the dollar
sign when gridding your answer.)

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3. If the outer diameter of a cylindrical oil tank is 54.28 inches
and the inner diameter is 48.7 inches, what is the thickness
of the wall of the tank, in inches?

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4. A car has an average mileage of 30 miles per gallon. If one
gallon of gasoline costs $3.75, how many miles can the car
travel on $20 worth of gasoline?

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5. If r = 25 − s, what is the value of 4(r + s)?

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6. Arc AB is on circle O. If the radius of circle O is 5 centimeters,
and angle AOB measures 30°, what is the length of arc AB
rounded to the nearest tenth of a centimeter?

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7.

SHOW YOUR WORK HERE

4x + 5 = 6
If x is a solution of the equation above, what is the
value of 4x?

8.

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9.

10.

In May, Carter’s Electronics sold 40 smartphones. In
June, because of a special promotion, the store sold 80
smartphones. What is the percent of increase in the
number of smartphones sold?

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ANSWER KEY AND EXPLANATIONS

1.

1. 9/4

3. 2.79

5. 100

7. 31

9. 100

2. 1.15

4. 160

6. 2.6

8. 72.9

10. 18

3x + y = 4
x + 3y = 5
Add the two equations to get:

6.

30
(10p) ≈ 2.6
360

4x + 4y = 9

The correct answer is 2.6.

4( x + y) = 9
x+y=

The circumference of the circle is 2p (5) = 10p .

9
4

4x + 5 = 6
4 x + 5 = 36

7.

The correct answer is 9/4 or 2.25.

4 x = 31
2.

Marion’s hourly wage is

$24
, or $4.80.
5

$10.95
Janet’s hourly wage is
, or $3.65.
3

The correct answer is 31.
8.		 The formula for volume of a cube is s3.
Therefore, 33 = 27.

$4.80 − $3.65 = $1.15

27(2.7) = 72.9

The correct answer is 1.15. (You do not need to grid
in the dollar sign.)
3.

The difference of 5.58 must be divided between
both sides. The thickness on each side is 2.79 inches.

The correct answer is 72.9.
9.

so

Chapter 8
Grid-In
Strategies

40
× 100 = 100% increase.
40

The correct answer is 100.

The correct answer is 160.

4(r + s) = 4(25) = 100

difference
×100. In this case, the difference is
original

80 – 40, which is 40. The original amount sold was 40,

 20 
30 
= 160
 3.75 

5.		r + s = 25

When computing the percent of increase (or decrease),
use

The correct answer is 2.79.
4.

309

10.

(3 2 )(3 2 ) = 9 i 2 = 18
The correct answer is 18.

The correct answer is 100.

Peterson’s SAT® Prep Guide 2017

Exercise 2
15 Minutes—10 Questions
DIRECTIONS: For these questions, solve the problem and enter your answer in the grid following each question, as
described below.

1. Although not required, it is suggested that you write your answer in the boxes at the top of the columns to help you fill in
the circles accurately. You will receive credit only if the circles are filled in correctly.
2. Mark no more than one circle in any column.
3. No question has a negative answer.
4. Some problems may have more than one correct answer. In such cases, grid only one answer.
7
1
5. Mixed numbers such as 3 must be gridded as 3.5 or .
2
2
		
If 3

1
is entered into the grid as
2

, it will be interpreted as

1
31
, not 3 .
2
2

6. Decimal answers: If you obtain a decimal answer with more digits than the grid can accommodate, it may be either rounded
or truncated, but it must fill the entire grid.
7
12

Answer:

Answer: 2.5

Write answer
in boxes.

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Answer: 201
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1. 53 percent of the 2,500 students at Jackson High are girls.
How many boys are there in the school?

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SHOW YOUR WORK HERE

2 centimeters and

6 centimeters. What is the length of the hypotenuse of
the right triangle to the nearest tenth of a centimeter?

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3. In a group of 40 students, 25 applied to Columbia and
30 applied to Cornell. If 3 students applied to neither
Columbia nor Cornell, how many students applied to both
schools?

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5. A gallon of water is added to 6 quarts of a solution that is
50% acid. What percent of the new solution is acid?

6. A gasoline tank is

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1
full. After 10 gallons of gasoline are
4

added to the tank, its gauge indicates that the tank is

2
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full. Find the capacity of the tank in gallons.

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Grid-In
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Peterson’s SAT® Prep Guide 2017

7. A plane flies over Denver at 11:20 a.m. It passes over
Coolidge, 120 miles from Denver, at 11:32 a.m. Find the
rate of the plane in miles per hour.

8.

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x +1
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x −1

		
What value of x is a solution of the equation above?

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9. What is the median of the set of numbers
2, 3, 4, 5, 6, 7, 8, and 9?

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2
10. y = x − 8 x + 7
y = x −1

		
If (x, y) is a solution of the system of equations above, what
is a possible value of x + y?

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Chapter 8
Grid-In
Strategies

Peterson’s SAT® Prep Guide 2017

ANSWER KEY AND EXPLANATIONS
1. 1175

3. 18

5. 30

7. 600

9. 5.5

2. 2.8

4. 5

6. 24

8. 1.5

10. 15 or 1

1. 		 47 percent of 2,500 are boys.

6. 		 10 gallons is

(0.47)(2,500) = 1,175 boys

2 + 6 = (hypotenuse ) 2

2 1 8−3 5
=
− =
3 4
12 12
5
x = 10
12
5 x = 120
x = 24

8 = (hypotenuse ) 2

The correct answer is 24.

The correct answer is 1175. (Do not include the
comma)
2.

2 1
−
of the tank.
3 4

( 2) + ( 6)
2

2

= (hypotenuse ) 2

1
hour.
7. 		 The plane covers 120 miles in 12 minutes, or
5
5
In , or 1 hour, it covers 5(120), or 600 miles.
5

8 = hyypotenuse
2.8 ≈ hypotenuse

The correct answer is 2.8.

The correct answer is 600.

3.

8.

25 – x

x

30 – x

x +1
=5
x −1
x + 1 = 5 ( x − 1)
x + 1= 5x − 5
6 = 4x

316

6
=x
4
x = 1.5

25 − x + x + 30 − x = 37
55 − x = 37
18 = x
18

Chapter 8
Grid-In
Strategies

The correct answer is 1.5.
9.

The correct answer is 18.
4.

The median is equal to the arithmetic mean of the two
numbers in the middle, 5 and 6.

5 + 6 11
= = 5.5
2
2
The correct answer is 5.5.

x 2 − y 2 = ( x − y )( x + y )
100 = 20 ( x + y )
5 = ( x + y)
5

The correct answer is 5.

10.

5.
No. of
quarts

%
acid

Amount
of acid

Original

6

50

3

Added

4

0

0

New

10

3

3
= 30%
10
The correct answer is 30. (Ignore the percent
symbol when griding your answer.)

www.petersons.com

y = x 2 − 8x + 7
y = x −1
x 2 − 8x + 7 = x −1
x 2 − 9x + 8 = 0

( x − 8) ( x − 1) = 0
x = 8 or x = 1
If x = 8, then y = 8 – 1 = 7
If x = 1, then y = 1 – 1 = 0
x + y = 15 or x + y = 1
The correct answer is 15 or 1 (both answers are
acceptable.

SUMMING IT UP
• When you grid answers to student-produced response questions, follow these six rules:
1. Write your answer in the boxes at the top of the grid.
2. Mark the corresponding circles, one per column.
3. Start in any column.
4. Work with decimals or fractions.
5. Express mixed numbers as decimals or improper fractions.
6. If more than one answer is possible, grid any one.

• Remember that grid-ins test the same concepts as multiple-choice math.
• The most important advice for grid-ins? Don’t be intimidated.
•

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topics in Peterson’s SAT® Online Course:

• Grid-Ins
• Problem Solving
• Word Problems
To purchase and access the course, go to www.petersons.com/sat.

317
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Strategies

Peterson’s SAT® Prep Guide 2017

Chapter 9:
Numbers and Operations
OVERVIEW
Operations with Fractions
Test for Divisibility
Exercises: Operations with Fractions
Word Problems Involving Fractions
Exercises: Word Problems Involving Fractions
Complex Numbers
Exercises: Complex Numbers
Direct and Inverse Variation
Exercises: Direct and Inverse Variation
Finding Percents
Exercises: Finding Percents

319

Percent Word Problems

Chapter 9

Exercises: Percent Word Problems

Numbers and
Operations

Summing It Up

Peterson’s SAT® Prep Guide 2017

OPERATIONS WITH FRACTIONS
The four basic arithmetic operations are addition, subtraction, multiplication, and division.

Adding and Subtracting
In adding or subtracting fractions, you must remember that the numbers must have the same (common) denominator.
Example:
Add

1 2 3
+ + .
3 5 4

Solution:
The least number that is divisible by 3, 5, and 4 is 60. Therefore, use 60 as the common denominator. Convert each
fraction to fractions with the common denominator, 60, by multiplying each numerator by the same factor as you
must multiply the denominator by to result in the common denominator of 60.
20 24 45 89
+
+
=
60 60 60 60
29
=1
60

320
Chapter 9
Numbers and
Operations

Fractions should be written in the simplest form. Often, in multiple-choice questions, you may find that the answer
you have correctly computed is not among the choices but an equivalent fraction is. Be careful!
In simplifying fractions involving large numbers, it is helpful to be able to tell whether a factor is common to both
numerator and denominator before a lengthy trial division. Certain tests for divisibility help with this.

TEST FOR DIVISIBILITY
To test if a number is divisible by:

Check to see:

2

if it is even

3

if the sum of the digits is divisible by 3

4

if the number formed by the last two digits is divisible by 4

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5

if its last digit is a 5 or 0

6

if it is even and the sum of the digits is divisible by 3

8

if the number formed by the last three digits is divisible by 8

9

if the sum of the digits is divisible by 9

10

if its last digit is 0

Example:
Simplify:

3, 525
4 , 341

Solution:
This fraction can be simplified by dividing by 3, since the sum of the digits of the numerator is 15 and those of the
denominator add up to 12, both are divisible by 3.

3, 525 1,175
=
4 , 341 1, 447
The resulting fraction meets no further divisibility tests and therefore has no common factor listed above. Larger
divisors would be unlikely on the SAT®.
To add or subtract mixed numbers, it is again important to remember common denominators. In subtraction, you
must borrow in terms of the common denominator.
Addition:

2
6
43 = 43
5
15
1
5
+ 8 =+ 8
3
15
11
51
15

2
6
21
Subtraction: 43 = 43 = 42
5
15
15
2
10
10
−6 = −6 = −6
3
15
15
11
36
15

321
Chapter 9
Numbers and
Operations

Multiplying
To multiply fractions, always try to divide common factors where possible before actually multiplying. In multiplying mixed
numbers, always rename them as improper fractions first.
2

9

2
10
99 18
i
i
=
Multiply:
5
11
110 55
55

Multiply: 4
3

1
2
1
i1 i 5
3
5
2
13

9
5
26
i
i
= 39
2
3
5

Peterson’s SAT® Prep Guide 2017

Dividing
To divide fractions or mixed numbers, remember to multiply by the reciprocal of the divisor (the number after the division sign).
3

2

1 3 9
4
Divide: 4 ÷ =
i
=6
2 4 2
3
25

1
1
125
1
Divide: 62 ÷ 5 =
i
= 12
2
2
2
5
To simplify complex fractions (fractions within fractions), multiply every term by the least common multiple of all denominators
in order to clear all fractions in the given numerator and denominator.
Example:
1 1
+
2 3
1 1
+
4 6

Solution:
The least number that can be used to clear all fractions is 12. Multiplying each term by 12 yields:

1 1 12 12
+
+
2 3 = 2 3 = 6 + 4 = 10 = 2
1 1 12 12 3 + 2 5
+
+
4 6 4 6

322
Chapter 9

Example:

Numbers and
Operations

3 2
+
4 3
1
1−
2
Solution:
Again, multiply by 12.
3 2 36 24
+
+
4 3= 4
3 = 9 + 8 = 17 = 2 5
1
12 12 − 6 6
6
1−
12 −
2
2

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EXERCISES: OPERATIONS WITH FRACTIONS
DIRECTIONS: Work out each problem in the space provided.
Add:
5
3
1
1. 12 + 2 + 21
6
8
4

2.

1 1 1 1 1
+ + + +
2 3 4 5 6

Divide:
1
÷5
7.
5

8. 3 2 ÷ 1 5
3 6

Simplify:
Subtract:
3. 15

9.

3
1
from 10
4
2

5 1
−
6 3
1
2+
5

1
4
10.
1
5−
2
3+

4. 17

2
from 50
3

323
Chapter 9
Numbers and
Operations

Solve:

Multiply:
1
5
5. 5 i 1
4
7

1 1
11. 4  + 
 6 12 

12.
6. 3 i 3 i 3
4 4 4

2  1 1
 + 
5  3 4

13. 4 + 7 − 4
9 18 5

Peterson’s SAT® Prep Guide 2017

ANSWERS AND EXPLANATIONS
1.

2.

324
Chapter 9

5
20
12 = 12
6
24
3
9
2 = 2
8
24
1
6
+21 = 21
4
24
35
11
35 = 36
24
24
1 30
=
2 60
1 20
=
3 60
1 15
=
4 60
1 12
=
5 60
1 10
+ =
6 60
9
87 27
=1 =1
60 20
60

1
3
6
3. 10 = 9 = 9
2
2
4
3
−5
4
3
4
4

7.

1 1 1
i =
5 5 25

8.

11 6
i
=2
3 11

9.

25 − 10 15 5
=
=
60 + 6 66 22

2

Each term was multiplied by 30.
10.

Each term was multiplied by 4.
1 1
2
1
11. 4  +  = 4  + 
 12 12 
 6 12 
 3
= 4 
 12 
 12 
= 
 12 
=1
12.

Numbers and
Operations

4.

49

3
3
2
−17
3
1
32
3
50

3

5.

13.

3

21 12
i
=9
4
7

6. 3 i 3 i 3 = 27
4 4 4 64

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12 + 1 13
=
20 − 2 18

2  1 1 2  4 3 
 + =  + 
5  3 4  5  12 12 
=

21  7 
5  12 6 

=

7
30

4 7 4 8 7 4
+ − = + −
9 18 5 18 18 5
15 4
= −
18 5
5 4
= −
6 5
25 24
=
−
30 30
1
=
30

WORD PROBLEMS INVOLVING FRACTIONS
Fraction problems deal with parts of a whole.
Example:
If a class consists of 12 boys and 18 girls, what part of the class is boys?
Solution:
The class consists of 12 boys out of 30 total students, or

2
12 2
of the class.
= . So, boys represent
5
30 5

Notice that, to find the solution, you must add the boys and girls to find the total number of students. Problems may
require more than one calculation as you can see in this example.
Example:
One-quarter of this year’s seniors have averages above 90. One-half of the remainder have averages between 80 and
90 inclusive. What part of the senior class have averages below 80?
Solution:
We know that

1
have averages above 90.
4

3
1
3
of
, or , have averages between 80 and 90 inclusive.
8
2
4
1 3 2 3 5
+ = + = have averages 80 and above.
4 8 8 8 8

325
Chapter 9

3
Therefore,
of the class have averages below 80.
8

Numbers and
Operations

Example:
14 is

2
of what number?
3

Solution:
14 =
Divide each side of the equation by

2
x
3

2
3
, which is the same as multiplying the reciprocal, or .
2
3
21 = x

Peterson’s SAT® Prep Guide 2017

Example:
If John has p hours of homework and has worked for r hours, what part of his homework is yet to be done?
Solution:

If John had 5 hours of homework and had worked for 3 hours, you would first find he had 5 − 3 hours, or 2 hours, yet
2
of his work. Using letters, his remaining work is represented by p − r .
to do. This represents
5
p

If a problem is given with letters in place of numbers, the same reasoning
must be applied as for numbers. If you are not sure how to proceed, replace
the letters with numbers to determine the steps that must be taken.

326
Chapter 9
Numbers and
Operations

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EXERCISES: WORD PROBLEMS INVOLVING FRACTIONS
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

2.

A team played 30 games, of which it won 24. What part
of the games played did the team lose?
A.

4
5

B.

1
4

C.

1
5

D.

3
4

If Germaine earns $X a week, and he saves $Y, what part
of his weekly salary does he spend?
A.

X
Y

B.

X −Y
X

C.

X −Y
Y

D.

3.

SHOW YOUR WORK HERE

Y−X
X

327
Chapter 9
Numbers and
Operations

What part of an hour elapses between 11:50 a.m. and
12:14 p.m.?
A.

2
5

B.

7
30

C.

17
30

Peterson’s SAT® Prep Guide 2017

D.
4.

5.

1
6

SHOW YOUR WORK HERE

One half of the employees of Acme Co. earn salaries
above $18,000 annually. One third of the remainder earn
salaries between $15,000 and $18,000. What part of the
staff earns below $15,000?
A.

1
6

B.

2
3

C.

1
2

D.

1
3

David received his allowance on Sunday. He spends

1
4

2
of the remainder on
3
Tuesday. What part of his allowance is left for the rest of
the week?
of his allowance on Monday and

328
Chapter 9
Numbers and
Operations

6.

A.

1
13

B.

1
9

C.

1
4

D.

1
3

12 is

3
of what number?
4

A.
B.
C.
D.
7.

16
9
36
20

A piece of fabric is cut into three sections so that the first is
three times as long as the second, and the second section
is three times as long as the third. What part of the entire
piece is the smallest section?
A.

1
13

B.

1
9

C.

1
4

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D.
8.

9.

1
3

SHOW YOUR WORK HERE

If a number is added to both the numerator and denom4
8
inator of , the result is . What is the number?
7
9
A.

2

B.

9

C.

18

D.

20

A factory employs M men and W women. What part of its
employees are women?

10.

11.

A.

M
W

B.

M +W
W

C.

W
M −W

D.

W
M +W

A motion was passed by a vote of 5:3. What part of the
votes cast were in favor of the motion?
A.

5
8

B.

5
3

C.

3
5

D.

3
8

329
Chapter 9
Numbers and
Operations

If the ratio of x:y is 9:7, what is the value of x + y?
A.

2

B.

16

C.

63

Peterson’s SAT® Prep Guide 2017

D.

12.

13.

330

14.

Chapter 9
Numbers and
Operations

15.

It cannot be determined from the information
given.

In a certain class, the ratio of men to women is 3:5. If the
class has 24 people in it, how many are women?
A.

9

B.

12

C.

15

D.

18

1
gallons of water. In the morn2
1
ing, 23 gallons are used. The level of the rain barrel after
4
3
it rains increases by 16 gallons. How many gallons are
4
in the tank after it rains?
A rain barrel contains 56

A.

16

B.

50

C.

64

D.

96

If x is

1
2

1
2

2
3
of y and y is of z, what is the ratio of z:x?
4
3

A.

1:2

B.

1:1

C.

2:1

D.

3:2

Teagan can paint a house in 5 hours and Kevin can paint
a house in 8 hours. How many hours will it take them to
paint a house if they work together?
A.

3

B.

3

1
13

C.

6

1
2

D.

13

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SHOW YOUR WORK HERE

ANSWER KEY AND EXPLANATIONS

1.

2.

1. C

4. D

7. A

10. A

13. B

2. B

5. C

8. D

11. D

14. C

3. A

6. A

9. D

12. C

15. B

The correct answer is C. The team lost 6 games out
6 1
=
of 30.
30 5

8.

4+n 8
= ; 9 ( 4 + n) = 8 (7 + n) ;
7+n 9

The correct answer is B. Germaine spends X − Y
out of X. X − Y
X

3.

4.

36 + 9n = 56 + 8n; n = 20

The correct answer is A. 10 minutes elapse by noon,
and another 14 after noon, making a total of 24
24 2
minutes. There are 60 minutes in an hour.
=
60 5
The correct answer is D. One half earn over $18,000.
1
1
One third of the other , or , earn between $15,000
6
2
and $18,000. This accounts for

1 1
3 1 4 2
+ , or + = =
2 6
6 6 6 3

1
to earn below $15,000.
3
The correct answer is C. David spends 1 on Monday
4
2
3
1
of the other , or , on Tuesday, leaving only
and
3
4
2
1
for the rest of the week.
4
3
The correct answer is A. 12 = x. Multiply each side
4
4
by . 16 = x.
3

9.

The correct answer is D. The factory employs M + W
people, out of which W are women.

10.

The correct answer is A. For every 5 votes in favor, 3
were cast against. 5 out of every 8 votes cast were in
favor of the motion.

11.

The correct answer is D. Remember, a ratio is a
fraction. If x is 18 and y is 14, the ratio x:y is 9:7, but x + y
is 32. The point of this problem is that x and y can take
on many possible values, just as long as the ratio 9:7 is
preserved. Given the multiplicity of possible values, it is
not possible here to establish one definite value for the
sum of x and y.

of the staff, leaving

5.

6.

7.

The correct answer is D. Let n be the number that is
being added.

The correct answer is A. Let the third or shortest
section = x. Then the second section = 3x, and the first
section = 9x. The entire piece of fabric is then 13x, and
x
1
the shortest piece represents
, of the
, or
13
13 x
entire piece.

12.

The correct answer is C. The ratio of women to the
total number of people is 5:8. We can set up a pro5 x
portion. If =
, then x = 15.
8 24

13.

The correct answer is B. To find the water remaining
in the tank, solve:
1
1
3
56 − 23 + 16 .
2
4
4

331
Chapter 9
Numbers and
Operations

2
1
3
The result is 56 − 23 + 16 = 50 gallons.
4
4
4

Peterson’s SAT® Prep Guide 2017

14.

The correct answer is C. There are several ways to
2
2
attack this problem. If x is of y, then x = i y .
3
3
If y is

4
4
3
of z, then z is
of y, or z = i y. Therefore,
3
3
4

z = 2x. The ratio of z:x is 2:1. You could also plug in a
real number and solve. If x is 2, figure out what y and z
would be. Therefore, y would be 3 and z would be 4, so
the ratio of z to x is 2:1.

332
Chapter 9
Numbers and
Operations

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15.

The correct answer is B. To find how long it will take
x x
them to paint the house together solve: + = 1 .
5 8
The result is: 8 x + 5 x = 40
13 x = 40
1
x =3
13

COMPLEX NUMBERS
A complex number is a number made up of a real number and an imaginary number. It can be written in standard form a + bi,
where a and b are real numbers and i is an imaginary unit.

i = −1 , i 2 = −1
For example, in the complex number 2 + 3i the real number is 2 and the imaginary number is 3i.
Complex numbers can be added, subtracted, multiplied, and divided.

Adding Complex Numbers
To add complex numbers, add the real numbers and the imaginary numbers separately.
Sum: (a + bi) + (c + di) = (a + c) + (b + d)i
Example:
Add: (2 + 3i) + (8 + 4i)
Solution:
Add the real numbers and then the imaginary numbers.
(2 + 8) + (3i + 4i) = 10 + 7i

333
Chapter 9

Subtracting Complex Numbers

Numbers and
Operations

To subtract complex numbers, subtract the real numbers and the imaginary numbers separately.
Difference: (a + bi) – (c + di) = (a – c) + (b – d)i
Example:
Subtract: (2 + 3i) – (8 + 4i)
Solution:
Subtract the real numbers and then the imaginary numbers.
(2 + 3i) – (8 + 4i) = (2 – 8) + (3i – 4i) = –6 – i

Peterson’s SAT® Prep Guide 2017

Multiplying Complex Numbers
Multiplying complex numbers is like multiplying polynomials by using the distributive property or the FOIL method.
Product: (a + bi) (c + di) = (ac) + (adi) + (bci) + (bd)i2
Example:
Multiply: 3i (–2 + 9i)
Solution:
Distribute 3i to all of the terms in the parentheses.
3i ( −2 + 9i ) = (3i ) ( −2) + (3i )(9i )
= (3i ) ( −2) + 27i 2
i2 =

( −1)( −1) = −1

= −6i + 27 ( −1)
= −27 − 6i
Example:
Multiply: (2 + 3i) (8 + 4i)

334
Chapter 9
Numbers and
Operations

Solution:
Find the sum of the products of the First terms, the Outer terms, the Inner terms, and the Last terms of the binomials.
The acronym FOIL stands for First Outer Inner Last and will help you to remember how to multiply two binomials.
When simplifying an expression that involves complex numbers, simplify i2 to –1.

(2 + 3i ) (8 + 4i ) = (2 i 8) + (2 i 4i ) + (8 i 3i ) + (3i i 4i )
= 16 + 8i + 24 i + 12i 2
= 16 + 32i + 12 ( −1)
= 4 + 32i
Simplify and write in standard form a + bi.

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Dividing Complex Numbers
Dividing complex numbers is more complicated because the denominator cannot contain a radical. This process is called rationalizing the denominator. In order to make the denominator rational, you must use its complex conjugate. The product of two
complex conjugates is always a real number a2 + b2. The numbers 2 + 8i and 2 – 8i are examples of complex conjugates, and their
product is the real number 22 + 82 = 4 + 64 = 68.
Complex conjugates: (a + bi) and (a – bi)
Product of complex conjugates: (a + bi) (a – bi) = a2 + b2
Example:
Simplify:

8
7i

Solution:
Rationalize the denominator by multiplying the numerator and denominator by i.
8 8 i 8i
8i
8i
= i = 2=
=
7i 7i i 7i
7 ( −1) −7
Example:
Simplify:

4 + 2i
−3 + 5i

335

Solution:
Rationalize the denominator by multiplying the numerator and denominator by the conjugate for the denominator.
Then simplify by combining like terms.

Chapter 9
Numbers and
Operations

4 + 2i
4 + 2i −3 − 5i
=
i
−3 + 5i −3 + 5i −3 − 5i
=

−12 − 20i − 6i − 10

9 + 15i − 15i − 25i 2
−12 − 20i − 6i − 10 ( −1)
=
9 + 15i − 15i − 25 ( −1)
−2 − 26i
9 + 25
−2 − 6i
=
34
−2 26i
−
=
34 34
−1 13i
= −
17 17
=

Recall that i2 = –1.

Peterson’s SAT® Prep Guide 2017

EXERCISES: COMPLEX NUMBERS
DIRECTIONS: Work out each problem. Circle the letter of your choice.
1.

2.

3.

336
Chapter 9
Numbers and
Operations

4.

5.

Add: (8 + 2i) + (2 – 3i)
A.

15i

B.

5 + 4i

C.

6+i

D.

10 – i

Add: (–2 – 5i) + (6 + 9i)
A.

4 + 4i

B.

7+i

C.

8i

D.

8 + 14i

Subtract: (–9 + 4i) – (3 + 7i)
A.

–6 – 3i

B.

–2 + i

C.

–12 – 3i

D.

–i

Subtract: (6 + 3i) – (2 + 9i)
A.

4 – 6i

B.

4 + 12i

C.

8 – 6i

D.

10i

Multiply: (2i)(32i)
A.

64

B.

–64

C.

64i

D.

–64i

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SHOW YOUR WORK HERE

6.

7.

8.

9.

10.

Multiply: (–5i)(12 – 3i)
A.

8 + 17i

B.

–15 – 60i

C.

15 + 60i

D.

–8 + 17i

SHOW YOUR WORK HERE

Multiply: (4 – 6i)(1 – 2i)
A.

4 + 12i

B.

4 – 12i

C.

8 + 14i

D.

–8 – 14i

Multiply: (–5 + 3i)(20 – 7i)
A.

–100 – 4i

B.

100 + 21i

C.

–79 + 95i

D.

121 + 95i

Simplify:

−5i
4 i + 11

A.

–4 – 11i

B.

20 − 55i
137

C.

−20 − 55i
137

D.

4 +11i

Simplify:

337
Chapter 9
Numbers and
Operations

17 − i
3i

A.

1 17
− i
3 3

B.

1 7
+ i
3 3

C.

34 2
+ i
5 5

D.

1 17
− − i
3 3

Peterson’s SAT® Prep Guide 2017

11. Simplify:

22 2
− i
3 3

A.

−

B.

1 7
+ i
3 3

C.

34 2
+ i
5 5

D.

1 17
− − i
3 3

12. Simplify:

338

6 + 14 i
1 + 2i

−3 + 5i
−3 − 4 i

29 27
− i
5
5

A.

−

B.

−11 27
− i
25 25

C.

21 27
− i
25 25

D.

11 27
+ i
7 7

Chapter 9
Numbers and
Operations

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SHOW YOUR WORK HERE

ANSWER KEY AND EXPLANATIONS

1.

1. D

4. A

7. D

9. C

11. C

2. A

5. B

8. C

10. D

12. B

3. C

6. B
9.

The correct answer is D.
(8 + 2i) + (2 – 3i) = (8 + 2) + (2i – 3i) = 10 – i

2.

The correct answer is C.
−5i
−5i
11− 4 i
=
i
11+ 4 i 11+ 4 i 11− 4 i
−5i (11− 4 i )
=
(11+ 4i ) (11− 4i )

The correct answer is A.
(–2 – 5i) + (6 + 9i) = (–2 + 6) + (–5i + 9i) = 4 + 4i

=

3.

121− 44 i + 44 i + 16i 2
−20 − 55i
=
121+ 16
−20 − 55i
=
137

The correct answer is C.

( −9 + 4i ) − (3 + 7i ) = ( −9 − 3) + ( 4i − 7i )
= −12 − 3i
4.

The correct answer is A.

(6 + 3i ) − (2 + 9i ) = (6 − 2) + (3i − 9i )

10.

= 4 − 6i
5.

The correct answer is D.
17 − i 17 − i i
i
=
3i
3i
i
17 − i )(i )
(
=
3i 2

The correct answer is B.
(2i)(32i) = 64i = 642(−1) = −64

6.

−55i + 20i 2

=

The correct answer is B.

17i − i 2
3 ( −1)

1+ 17i
=
−3
1 17
=− − i
3 3

( −5i ) (12 − 3i ) = ( −5i )(12) + ( −5i ) ( −3i )
= −60i + 15i 2

= −60i + 15 ( −1)

339
Chapter 9
Numbers and
Operations

= −15 − 60i

7.

The correct answer is D.

( 4 − 6i ) (1− 2i ) = ( 4) + ( −8i ) + ( −6i ) + (12i 2 )
= 4 + ( −14 i ) + ( −12)
= −8 − 14 i

8.

The correct answer is C.

( −5 + 3i ) (20 − 7i ) = −100 + ( −5) ( −7i ) + (3i )(20) + (3i ) ( −7i )

(

= −100 + 35i + 60i + −21i 2

)

= −100 + 95i + 21
= −79 + 95i

Peterson’s SAT® Prep Guide 2017

11.

The correct answer is C.
6 + 14 i 6 + 14 i 1− 2i
=
i
1 + 2i
1 + 2i 1 − 2i
(6 + 14i ) (1− 2i )
=
(1+ 2i ) (1− 2i )
=

6 − 12i + 14 i − 28i 2

1 − 2i + 2i − 4 i 2
6 − 12i + 14 i − 28 (1)
=
1− 2i + 2i − 4 ( −1)

6 + 2 i + 28
1+ 4
34 + 2i
=
5
34 2
=
+ i
5 5
=

340
Chapter 9
Numbers and
Operations

www.petersons.com

12.

The correct answer is B.
−3 + 5i −3 + 5i −3 + 4 i
i
=
−3 − 4 i −3 − 4 i −3 + 4 i
( −3 + 5i ) ( −3 + 4i )
=
( −3 − 4i ) ( −3 + 4i )
=

9 − 12i − 15i + 20i 2

9 − 12i + 12i − 16i 2
9 − 12i − 15i + 20 ( −1)
=
9 − 12i + 12i − 16 ( −1)
=

9 − 27i + ( −20)
9 + ( −16) ( −1)

−11− 27i
25
−11 27
=
− i
25 25
=

DIRECT AND INVERSE VARIATION
Direct Variation
Two quantities are said to vary directly if as one increases, the other increases, and, as one decreases, the other decreases.
For example, the amount of sugar needed in a recipe varies directly with the amount of butter used. The number of inches between
two cities on a map varies directly with the number of miles between the cities. The equation y = ax represents direct variation
between x and y, and y is said to vary directly with x. The variable a is called the constant of variation. By dividing each side by x,
you can see that the ratio for the variable is the constant a.
Example:
Hooke’s Law states that the distance d a spring stretches varies directly with the force F that is applied to it. Suppose
a spring stretches 15 inches when a force of 9 lbs. is applied. Write an equation that relates d to F, and state the constant of variation.
Solution:
d F
You are comparing the distance that a spring stretches with the force that is applied, so = . Solving for d in terms
15 9
5
5
of F, you get d = F . The constant of variation is .
3
3
Example:
The weight of a person on the Moon varies directly with the weight of a person on Earth. A person who weighs 100
lbs. on Earth weighs 16.6 lbs. on the Moon. How much would a person who weighs 120 lbs. on Earth weigh on the
Moon?

341
Chapter 9
Numbers and
Operations

Solution:
Start with the equation y = ax, where y is the weight of a person on the Moon and x is the weight of a person on
Earth.
y = ax

16.6 = a (100)
0.166 = a
The equation y = 0.166x gives the weight y on the Moon of a person who weighs x pounds on Earth. To solve the
example, substitute x = 120 in the equation to determine the weight of the person on the Moon.
y = 0.166 x
y = 0.166(120 )
y = 19.92
A person who weighs 120 lbs. on Earth would weigh 19.92 lbs. on the Moon.

Peterson’s SAT® Prep Guide 2017

Inverse Variation
Two quantities are said to vary inversely if, as one increases, the other decreases.
For example, the number of workers hired to paint a house varies inversely with the number of days the job will take. A doctor’s
stock of flu vaccine varies inversely with the number of patients injected. The number of days a given supply of cat food lasts
varies inversely with the number of cats being fed.
The equation xy = a, where a ≠ 0, represents inverse variation between x and y, and y is said to vary inversely with x. The variable
a is called the constant of variation.

Whenever two quantities vary directly, you can find a missing term by
setting up a proportion. However, be very careful to compare the same
units, in the same order, on each side of the equal sign.
Example:
The number of songs that can be stored on a hard drive varies inversely with the size of the song. A certain hard
drive can store 3,000 songs when the average size of the song is 3.75 MB. Write an equation that gives the number of
songs y that will fit on the hard drive as a function of the average song size x.
Solution:
First, write an inverse variation equation that relates x and y. Then substitute 3,000 for y and 3.75 for x.

342

xy = a
a
y=
x

Chapter 9

a
3.75
11, 250 = a
3, 000 =

Numbers and
Operations

The inverse variation equation for this situation is y =

www.petersons.com

11, 250
.
x

EXERCISES: DIRECT AND INVERSE VARIATION
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

If 60 feet of uniform wire weighs 80 pounds, what is the
weight of 2 yards of the same wire?

B.

2
2 pounds
3
6 pounds

C.

8 pounds

D.

120 pounds

A.

2.

3.

4.

SHOW YOUR WORK HERE

A gear 50 inches in diameter turns a smaller gear 30 inches
in diameter. If the larger gear makes 15 revolutions, how
many revolutions does the smaller gear make in that
time?
A.

9

B.

12

C.

20

D.

25

The distance a spring stretches varies directly with the
force applied to it. If a 4-lb. weight stretches a spring 65.5
inches, how many inches will the spring stretch if a 12-lb.
weight is applied?
A.

262

B.

1
196
2

C.

21

1
3

D.

16

3
8

343
Chapter 9
Numbers and
Operations

If a furnace uses 40 gallons of oil in a week, how many
gallons, to the nearest gallon, does it use in 10 days?
A.

4

B.

28

C.

57

D.

58

Peterson’s SAT® Prep Guide 2017

5. A recipe requires 13 ounces of sugar and 18 ounces of flour.
If only 10 ounces of sugar are used, how much flour, to the
nearest ounce, should be used?
A.

13

B.

14

C.

15

D.

23

6. If a car can drive 25 miles on two gallons of gasoline, how
many gallons will be needed for a trip of 150 miles?
A.

12

B.

10

C.

7

D.

6

7. A school has enough bread to feed 30 children for 4 days. If
10 more children are added, how many days will the bread
last?

344
Chapter 9
Numbers and
Operations

1
3

A.

1

B.

2

C.

3

D.

5

2
3
1
3

8. The intensity of a sound, I, varies inversely with the square
of the distance, d, from the sound. If the distance is reduced
1
by a factor of , by what factor will the intensity of the
4
sound increase?
A.

4

B.

8

C.

12

D.

16

www.petersons.com

SHOW YOUR WORK HERE

9. If 3 miles are equivalent to 4.83 kilometers, then 11.27
kilometers are equivalent to how many miles?
A.

2

B.

5

C.

6

D.

7

SHOW YOUR WORK HERE

1
3
1
2

10. Suppose the variables x and y vary inversely. If x = –12
and y = 4, find y when x = –2.
A.

48

B.

24

C.

–24

D.

–48

345
Chapter 9
Numbers and
Operations

Peterson’s SAT® Prep Guide 2017

ANSWER KEY AND EXPLANATIONS

1.

1. C

3. B

5. B

7. C

9. D

2. D

4. C

6. A

8. D

10. B

The correct answer is C. You are comparing feet with
pounds. The more feet, the more pounds. This is
DIRECT. Remember to rename yards as feet:
60 6
=
80 x
60 x = 480
x =8

2.

x = 196
4.

8.

5.

The correct answer is B. The more sugar, the more
flour. This is DIRECT.
13 10
=
18 x
13 x = 180
11
x = 13
13

www.petersons.com

The correct answer is D.

( )

I d2 =k
2

 d
I  = k
 4
d2
I  = k
 16 

1
2

The correct answer is C. The more days, the more oil.
This is DIRECT. Remember to rename the week as 7
days.
40 x
=
7 10
7 x = 400
1
x = 57
7

The correct answer is C. The more children, the fewer
days. This is INVERSE.
30 i 4 = 40 i x
120 = 40 x
3= x

4
12
=
65.5 x
4 x = 786

Chapter 9
Numbers and
Operations

7.

The correct answer is B. The more weight that is
applied the farther the spring will stretch.

346

The correct answer is A. The more miles, the more
gasoline. This is DIRECT.
25 150
=
2
x
25 x = 300
x = 12

The correct answer is D. The larger a gear, the fewer
times it revolves in a given period of time. This is
INVERSE.
50 i 15 = 30 i x
750 = 30 x
25 = x

3.

6.

( )

I d 2 = 16k

9.

The correct answer is D. The more miles, the more
kilometers. This is DIRECT.
3
x
=
4.83 11.27
4.83 x = 33.81
x =7

10.

The correct answer is B.
xy = k

( −12)( 4) = k
−48 = k
−2 y = −48
y = 24

FINDING PERCENTS
Percent Equivalents
“Percent” means “out of 100.” If you understand this concept, it becomes very easy to rename a percent as an equivalent decimal
or fraction.
5% =

2.6% =

c% =

5
= 0.05
100
2.6
= 0.026
100

c
1
i c = 0.01c
=
100 100

1
1 1
1
1
2
%=
=
i =
i 0.5 = 0.005
100 100 2 100
2

Certain fractional equivalents of common percents occur frequently enough that they should be memorized. Learning the values
in the following table will make your work with percent problems much easier.

Percent-Fraction Equivalents
50% =

1
2

1
25% =
4

347

9
10

1
1
12 % =
2
8

1
1
33 % =
3
3

1
3
37 % =
2
8

90% =

75% =

3
4

20% =

1
5

1
5
62 % =
2
8

10% =

1
10

40% =

2
5

1
7
87 % =
2
8

30% =

3
10

60% =

3
5

2
1
16 % =
3
6

70% =

7
10

80% =

4
5

1
5
83 % =
3
6

Chapter 9
Numbers and
Operations

Most percentage problems can be solved by using the following proportion:
part
%
=
100 whole

Peterson’s SAT® Prep Guide 2017

Although this method works, it often yields unnecessarily large numbers that are difficult to compute. Following are examples
of the four basic types of percent problems and different methods for solving them.

How to Solve Percentage Problems

• To change a % to a decimal, remove the % sign and divide by 100. This has the effect of moving
the decimal point two places to the LEFT.

• To change a decimal to a %, add the % sign and multiply by 100. This has the effect of moving
the decimal point two places to the RIGHT.

• To change a % to a fraction, remove the % sign and divide by 100. This has the effect of putting
the % over 100 and simplifying the resulting fraction.

• To change a fraction to a %, add the % sign and multiply by 100.

To Find a Percent of a Number
Example:
Find 27% of 92.
Solution:

348

Proportional Method

Chapter 9

Rename the percent as its decimal or fraction equivalent and
multiply. Use fractions only when they are among the familiar
ones given in the previous chart.

Numbers and
Operations

27
x
=
100 92
100 x = 2, 428
x = 24.84

www.petersons.com

Shorter Method

92
× 0.27
644
184
24.84

Example:
1
Find 12 % of 96.
2
Solution:

Proportional Method

Decimal Method

1
2= x
100 96
100 x = 1, 200

0.125
× 96

12

x = 12

Fractional Method

1
i 96 = 12
8

750
1125
12.000

Which method is easiest? It really pays to memorize those fractional equivalents.

To Find a Number When a Percent of It Is Given
Example:
7 is 5% of what number?

349

Solution:

Chapter 9

Proportional Method

Shorter Method

Numbers and
Operations

Translate the problem into an algebraic equation. In
doing this, the percent must be written as a fraction or
decimal.
5
7
=
100 x
5 x = 700
x = 140

7 = 0.05 x
700 = 5 x
140 = x

Peterson’s SAT® Prep Guide 2017

Example:
1
20 is 33 of what number?
2
Solution:

Proportional Method

Shorter Method

1
2 = 20
x
100
1
33 x = 2, 000
3
100
x = 2, 000
3
100 x = 6 , 000
x = 60
33

1
x
3
60 = x

20 =

1
1
Just think of the time you will save and the number of extra problems you will solve if you know that 33 % = .
3
3

To Find What Percent One Number Is of Another

350
Chapter 9
Numbers and
Operations

Example:
90 is what percent of 1,500?
Solution:

Proportional Method

Shorter Method

x
90
=
100 1, 500
1, 500 x = 9 , 000
15 x = 90
x =6

Put the part over the whole. Simplify the fraction and
multiply by 100.

Example:
7 is what percent of 35?

www.petersons.com

90
9
3
=
=
i 100 = 6
1, 500 150 50

Solution:

Proportional Method

Shorter Method

7
x
=
100 35
35 x = 700
x = 20

7 1
= = 20%
35 5

Example:
18 is what percent of 108?
Solution:

Proportional Method

Shorter Method

18
x
=
100 108
108 x = 1, 800

18
9 1
2
=
= = 16 %
108 54 6
3

Time-consuming long division is necessary to get:
x = 16

2
3

Once again, if you know the fraction equivalents of
common percents, computation can be done in a
few seconds.

351
Chapter 9

To Find a Percent Over 100

Numbers and
Operations

Example:
Find 125% of 64.
Solution:

Proportional Method

125 x
=
100 64
100 x = 8 , 000
x = 80

Decimal Method
64
× 1.25
320
128
64
80.00

Fractional Method
1
1 i 64
4
5
i 64 = 80
4

Peterson’s SAT® Prep Guide 2017

Example:
36 is 150% of what number?
Solution:

Proportional Method
150 36
=
100 x
150 x = 3, 600
15 x = 360
x = 24

Decimal Method

Fractional Method

36 = 1.50 x
360 = 15 x
24 = x

1
36 = 1 x
2
3
36 = x
2
72 = 3 x
24 = x

Example:
60 is what percent of 50?
Solution:

Proportional Method

352
Chapter 9
Numbers and
Operations

www.petersons.com

60
x
=
100 50
50 x = 6 , 000
5 x = 600
x = 120

Fractional Method

1
60 6
= = 1 = 120%
50 5 5

EXERCISES: FINDING PERCENTS
DIRECTIONS: Work out each problem. Circle the letter of your choice.
1. Write 0.2% as a decimal.

2.

A.

0.002

B.

0.02

C.

0.2

D.

20

Write 3.4% as a fraction.
A.

3.

4.

5.

SHOW YOUR WORK HERE

34
1, 000

B.

34
10

C.

34
100

D.

340
100

3
Write % as a decimal.
4
A.

0.75

B.

0.075

C.

0.0075

D.

0.00075

353
Chapter 9
Numbers and
Operations

Find 60% of 70.
A.

4,200

B.

420

C.

42

D.

4.2

What is 175% of 16?
A.

28

B.

24

C.

22

D.

12

Peterson’s SAT® Prep Guide 2017

6.

7.

8.

354

9.

What percent of 40 is 16?
A.

20%

B.

1
2 %
2

C.

1
33 %
3

D.

40%

What percent of 16 is 40?
A.

20%

B.

1
2 %
2

C.

200%

D.

250%

$4 is 20% of what?
A.

$5

B.

$20

C.

$200

D.

$500

12 is 150% of what number?

Chapter 9

A.

18

Numbers and
Operations

B.

15

C.

9

D.

8

10.

1
How many sixteenths are there in 87 % ?
2
A.

7

B.

13

C.

14

D.

15

www.petersons.com

SHOW YOUR WORK HERE

ANSWER KEY AND EXPLANATIONS

1.

1. A

3. C

5. A

7. D

9. D

2. A

4. C

6. D

8. B

10. C

7.

The correct answer is A.

40 5
1
= = 2 = 250%
16 2
2

0.2% = 0.002
The decimal point moves to the
LEFT two places.
8.
2.

The correct answer is A.

3.4% =

3.

3.4
34
=
100 1, 000

9.

The correct answer is D.
150% = 1

3
5

3
i 70 = 42
5
10.

The correct answer is A.
7
i 16 = 28
4

The correct answer is D.

1
2

3
x = 12
2
3 x = 24
x =8

The correct answer is C.

3
175% = 1
4
6.

1
1 , so
4= x
5
5
20 = x

The correct answer is C.

60% =

5.

The correct answer is B.
20% =

3
% = 0.75% = 0.0075
4
4.

The correct answer is D.

The correct answer is C.

355

1
7 14
87 % = =
2
8 16

Chapter 9
Numbers and
Operations

16 2
= = 40%
40 5

Peterson’s SAT® Prep Guide 2017

PERCENT WORD PROBLEMS
Certain types of business situations are excellent applications of percent.

Percent of Increase or Decrease
The percent of increase or decrease is found by putting the amount of increase or decrease over the original amount and renaming
this fraction as a percent.

Example:
Over a five-year period, the enrollment at South High dropped from 1,000 students to 800. Find the percent of
decrease.
Solution:

1, 000 − 800 200
20
=
=
= 20%
1, 000
1, 000 100
Example:

356

A company normally employs 100 people. During a slow spell, the company fired 20% of its employees. By what
percent must the company now increase its staff to return to full capacity?
Solution:

Chapter 9
20% =

Numbers and
Operations

1
5

1
i 100 = 20%
5

20 1
The company now has 100 − 20 = 80 employees. If it then increases its staff by 20, the percent of increase is
= ,
80 4
or 25%.

In word problems, of can usually be interpreted to mean times (in other
words, multiply).

www.petersons.com

Discount
A discount is usually expressed as a percent of the marked price that will be deducted from the marked price to determine the
sale price.
Example:
Bill’s Hardware offers a 20% discount on all appliances during a sale week. If they take advantage of the sale, how
much must the Russells pay for a washing machine marked at $280?
Solution:

Long Method
20% =

1
5

1
i 280 = $56 discount
5

Shorter Method
If there is a 20% discount, the Russells will pay 80% of the
marked price.

$280 − $56 = $224 sale price
The danger inherent in this method is that $56 is sure to
be among the multiple-choice answers.

80% =

4
5

4
i 280 = $224 sale price
5

357

Example:
A store offers a television set marked at $340 less discounts of 10% and 5%. Another store offers the same television
set also marked at $340 with a single discount of 15%. How much does the buyer save by buying at the better price?

Chapter 9
Numbers and
Operations

Solution:
9
, of $340, which is $306. The additional 5%
10
discount means the buyer pays 95% of $306, or $290.70. Note that the second discount must be figured on the

In the first store, the initial discount means the buyer pays 90%, or

first sale price. Taking 5% off $306 is a smaller amount than taking the additional 5% off $340. The second store will
therefore have a lower sale price. In the second store, the buyer will pay 85% of $340, or $289, making the price $1.70
less than in the first store.

Peterson’s SAT® Prep Guide 2017

Commission
Many salespeople earn money on a commission basis. In order to encourage sales, they are paid a percentage of the value of
goods sold. This amount is called a commission.
Example:
A salesperson at Brown’s Department Store is paid $80 per week in salary plus a 4% commission on all her sales. How
much will that salesperson earn in a week in which she sells $4,032 worth of merchandise?
Solution:
Find 4% of $4,032 and add this amount to $80.

4032
× 0.04
$161.28 + $80 = $241.28
Example:
Bill Olson delivers frozen food for a delivery service and keeps 8% of all money collected. One month he was able to
keep $16. How much did he forward to the delivery service?
Solution:

358

First, determine how much he collected by finding the number that 16 is 8% of.
16 = 0.08 x
1, 600 = 8 x
200 = x

Chapter 9
Numbers and
Operations

If Bill collected $200 and kept $16, he gave the delivery service $200 − $16, or $184.

www.petersons.com

Taxes
Taxes are a percent of money spent or money earned.
Example:
Noname County collects a 7% sales tax on automobiles. If the price of a car is $8,532 before taxes, what will this car
cost once sales tax is added in?
Solution:
Find 7% of $8,532 to determine tax and then add it to $8,532. This can be done in one step by finding 107% of
$8,532.
$8532
× 1.07
59724
85320
$9129.24
Example:
If the tax rate in Anytown is $3.10 per $100, what is the annual real estate tax on a house assessed at $47,200?
Solution:
annual tax = tax rate • assessed value

 $3.10 
=
( 47, 200 )
 $100 
= (0.031)( 47, 200 )
= $1, 463.20

359
Chapter 9
Numbers and
Operations

Peterson’s SAT® Prep Guide 2017

EXERCISES: PERCENT WORD PROBLEMS
DIRECTIONS: Work out each problem. Circle the letter of your choice.
1.

2.

3.

A suit marked at $80 is sold for $68. What is the rate of
discount?
A.

12%

B.

15%

C.

20%

D.

85%

What was the original price of a phone that sold for $70
during a 20%-off sale?
A.

$56

B.

$84

C.

$87.50

D.

$90

How many dollars does a salesperson earn on a sale of
$800 at a commission of 2.5%?

360

A.

20

B.

200

Chapter 9

C.

2,000

Numbers and
Operations

D.

20,000

4.

5.

At a selling price of $273, a refrigerator yields a 30%
profit on the cost. What selling price will yield a 10%
profit on the cost?
A.

$210

B.

$221

C.

$231

D.

$235

What single discount is equivalent to two successive
discounts of 10% and 15%?
A.

25%

B.

24.5%

C.

24%

D.

23.5%

www.petersons.com

SHOW YOUR WORK HERE

6.

7.

8.

The net price of a certain article is $306 after successive
discounts of 15% and 10% off the marked price. What is
the marked price?
A.

$234.09

B.

$382.50

C.

$400

D.

$408

SHOW YOUR WORK HERE

If a merchant makes a profit of 20% based on the selling
price of an article, what percent does the merchant make
on the cost?
A.

20

B.

25

C.

40

D.

80

A certain radio costs a merchant $72. At what price must
the merchant sell this radio in order to make a profit of
20% of the selling price?
A.

$86.40

B.

$90

C.

$92

D.

$144

361
Chapter 9

9.

10.

A baseball team has won 40 games out of 60 played. It
has 32 more games to play. How many of these must the
team win to make its record 75% for the season?
A.

26

B.

28

C.

29

D.

30

Numbers and
Operations

If prices are reduced 25% and sales increase 20%, what
is the net effect on gross receipts?
A.

They increase by 5%.

B.

They decrease by 5%.

C.

They increase by 10%.

D.

They decrease by 10%.

Peterson’s SAT® Prep Guide 2017

11.

12.

13.

362
Chapter 9
Numbers and
Operations

14.

A salesperson earns 5% on all sales between $200 and
$600, and 8% on the part of the sales over $600. What is
her commission in a week in which her sales total $800?
A.

$20

B.

$36

C.

$46

D.

$78

If the enrollment at State U. was 3,000 in 1998 and 12,000
in 2008, what was the percent of increase in enrollment?
A.

400%

B.

300%

C.

25%

D.

3%

2
If 6 students, representing 16 % of the class, failed
3
algebra, how many students passed the course?
A.

30

B.

32

C.

36

D.

42

If 95% of the residents of Coral Estates live in private
homes and 40% of these live in air-conditioned homes,
what percent of the residents of Coral Estates live in
air-conditioned homes?
A.

3%

B.

3.8%

C.

30%

D.

38%

15. A salesperson receives a salary of $100 a week and a commission of 5% on all sales. What must be the amount of
sales for a week in which the salesperson’s total weekly
income is $360?
A.

$6,200

B.

$5,200

C.

$2,600

D.

$720

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SHOW YOUR WORK HERE

ANSWER KEY AND EXPLANATIONS

1.

4. C

7. B

10. D

13. A

2. C

5. D

8. B

11. C

14. D

3. A

6. C

9. C

12. B

15. B

The correct answer is B. The amount of discount is
$12. Rate of discount is figured on the original price.
12 3
=
80 20

2.

1. B

6.

If marked price = m, first sale price = 0.85m, and
net price = 0.90(0.85m) = 0.765m

3
i 100 = 15%
20

0.765m = 306
m = 400

The correct answer is C. $70 represents 80% of the
original price.

In this case, it would be easy to work from the answer
choices.

70 = 0.80 x
700 = 8 x
$87.50 = x
3.

The commission is
4.

15% of $400 is $60, making a first sale price of $340.
10% of this price is $34, making the net price $306.
Choices A, B, and D would not give a final answer in
whole dollars.

The correct answer is A.
2.5% =

2.5
100

7.

2.5
(800) = $20.
100

8.

The new price will add 10% of cost, or $21, for profit.
9.

Final sale price is 85% of $90, or $76.50
Total discount is $100 − $76.50 = $23.50
23.50
= 23.5%
100

Numbers and
Operations

3
The correct answer is C. The team must win 75%, or ,
4
of the games played during the entire season. With 60
games played and 32 more to play, the team must win
3
3
of 92 games, and i 92 = 69. Since 40 games have
4
4
already been won, the team must win 29 additional
games.

First sale price is 90% of $100, or $90

Percent of discount =

Chapter 9

90 = x

New price = $231
The correct answer is D. Work with a simple figure,
such as 100.

The correct answer is B. If the profit is to be 20% of
the selling price, the cost must be 80% of the selling
price.

363

72 = 0.80 x
720 = 8 x

$273 represents 130% of the cost.

5.

The correct answer is B. Use an easy amount of $100
for the selling price. If the profit is 20% of the selling
price, or $20, the cost is $80. Profit based on cost is
20 1
= = 25%
80 4

The correct answer is C.
1.30 x = 273
13 x = 2, 730
x = $210 = cost

The correct answer is C.

10.

The correct answer is D. Let original price = p, and
original sales = s. Therefore, original gross receipts = ps.
Let new price = 0.75p, and new sales = 1.20s. Therefore,
new gross receipts = 0.90ps. Gross receipts are only
90% of what they were.

Peterson’s SAT® Prep Guide 2017

11.

12.

The correct answer is C. Five percent of sales between
$200 and $600 is 0.05(600) = $30. Then, 8% of sales
over $600 is 0.08(200) = $16. Total commission =
$30 + $16 = $46.
The correct answer is B. There was an increase of
9,000 students. To determine the percent of this
increase in enrollment:
9 , 000
= 3 = 300%
3, 000

13.

The correct answer is A.
2
1
16 % =
3
6
1
6= x
6
36 = x
36 students in class: 6 failed, 30 passed

364
Chapter 9
Numbers and
Operations

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14.

The correct answer is D.
40% =

2
5

2
of 95% = 38%
5
15.

The correct answer is B.
Let s = sales
$100 + 0.05s = 360
0.05s = 260
5s = 26 , 000
s = $5, 200

SUMMING IT UP
• If the arithmetic looks complex, try to simplify it first.
• If a problem is given with letters in place of numbers, the same reasoning must be applied as for numbers. If you are not
sure how to proceed, replace the letters with numbers to determine the steps that must be taken.

• Fractions should be written in the simplest form. Often, in multiple-choice questions, you may find that the answer you
have correctly computed is not among the choices but an equivalent fraction is. Be careful!

• Whenever two quantities vary directly, you can find a missing term by setting up a proportion. However, be very careful to
compare the same units, in the same order, on each side of the equal sign.

• When solving percentage problems, remember the following:
ºº To change a % to a decimal, remove the % sign and divide by 100. This has the effect of moving the decimal point two
places to the LEFT.
ºº To change a decimal to a %, add the % sign and multiply by 100. This has the effect of moving the decimal point two
places to the RIGHT.
ºº To change a % to a fraction, remove the % sign and divide by 100. This has the effect of putting the % over 100 and
simplifying the resulting fraction.
ºº To change a fraction to a %, add the % sign and multiply by 100.

• In problems dealing with percent, you may be presented with certain types of business situations, such as taxes or commissions. For problems asking about the percent of increase or decrease, put the amount of increase or decrease over the
original amount and rename that fraction as a percent. A discount is usually expressed as a percent of the marked price that
will be deducted from the marked price to determine the sale price.

365

•

ONLINE
PREP

Want to Know More?

Chapter 9
Access additional practice questions, helpful lessons, valuable tips, and top-notch strategies for the following numbers and
operations review topics in Peterson’s SAT® Online Course:

Numbers and
Operations

• Arithmetic Strategy
• Hard Arithmetic
• Percent Word Problems
• Proportions and Rates
• Word Problems
• Working Backwards
To purchase and access the course, go to www.petersons.com/sat.

Peterson’s SAT® Prep Guide 2017

Chapter 10:
Basic Algebra
OVERVIEW
Signed Numbers

Roots and Radicals

Exercises: Signed
Numbers

Exercises: Roots and
Radicals

Linear Equations

Monomials and
Polynomials

Exercises: Linear
Equations
Simultaneous Equations
Exercises: Simultaneous
Equations
Exponents
Exercises: Exponents
Quadratic Equations
Exercises: Quadratic
Equations
Literal Equations

Exercises: Monomials and
Polynomials
Problem Solving in
Algebra
Exercises: Problem
Solving in Algebra
Inequalities

367

Exercises:
Inequalities

Chapter 10
Basic Algebra

Summing It Up

Exercises: Literal
Equations

SIGNED NUMBERS
To solve algebra problems, you must be able to compute accurately with signed numbers.
Addition: To add signed numbers with the same sign, add the magnitudes of the numbers and keep the same sign. To add signed
numbers with different signs, subtract the magnitudes of the numbers and use the sign of the number with the greater magnitude.
Subtraction: When subtracting a positive number from a negative number, add the magnitudes and make the difference (the
answer) negative. When subtracting a negative number from a positive number, add the magnitude and make the difference
positive. When asked to find the difference or a distance between a negative number and a positive number, the answer will
always be positive.

Peterson’s SAT® Prep Guide 2017

Multiplication: If there is an odd number of negative signs, the product is negative. An even number of negative signs gives a
positive product.
Division: If the signs are the same, the quotient is positive. If the signs are different, the quotient is negative.
Practicing these basic operations with signed numbers will help you on the more difficult problems on the SAT® in which these
and more complex skills are tested.

EXERCISES: SIGNED NUMBERS
DIRECTIONS: Work out each problem. Circle the letter next to your choice.

1.

2.

368
Chapter 10
Basic Algebra

3.

4.

When +3 is added to −5, what is the sum?
A.

−8

B.

+8

C.

−2

D.

+2

When −4 and −5 are added, what is the sum?
A.

−9

B.

+9

C.

−1

D.

+1

Subtract −6 from +3.
A.

−3

B.

+3

C.

+18

D.

+9

When −5 is subtracted from +10, what is the result?
A.

+5

B.

+15

C.

−5

D.

−15

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SHOW YOUR WORK HERE

5.

6.

7.

8.

(−6)(−3) equals
A.

−18

B.

+18

C.

+2

D.

−9

SHOW YOUR WORK HERE

 1
What is the product of ( −6)  +  ( −10) ?
 2
A.

−15

1
2

B.

+15

1
2

C.

−30

D.

+30

When the product of (−4) and (+3) is divided by (−2), the
quotient is

1
2

A.

+

B.

+15

C.

+6

D.

−

1
2

1
2

Last winter the meteorology class recorded the daily
temperatures. The coldest recorded temperature was
–37 degrees Fahrenheit, and the warmest was 38 degrees
Fahrenheit. How many degrees warmer was the warmest
day than the coldest day?
A.

–75

B.

–1

C.

1

D.

75

369
Chapter 10
Basic Algebra

Peterson’s SAT® Prep Guide 2017

9.

10.

The highest point in California is Mt. Whitney with an
elevation of 14,494 feet. The lowest point is Death Valley
with an elevation of –294 feet. How much higher is the
base of a tree at the top of Mt. Whitney than a person
standing at the lowest point in Death Valley?
A.

294 feet

B.

14,200 feet

C.

14,494 feet

D.

14,788 feet

A submarine started at an elevation of –1,250 feet, or
1,250 feet below sea level, and it submerged another 25
feet per second. It continued to submerge for 10 seconds.
What was its new elevation?
A.

–1,500 feet

B.

–1,000 feet

C.

250 feet

D.

1,500 feet

370
Chapter 10
Basic Algebra

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SHOW YOUR WORK HERE

ANSWER KEY AND EXPLANATIONS
1. C

3. D

5. B

7. C

9. D

2. A

4. B

6. D

8. D

10. A

1.

The correct answer is C. In adding numbers with
opposite signs, subtract their magnitudes (5 − 3 = 2)
and use the sign of the number with the greater
magnitude (negative).

2.

The correct answer is A. In adding numbers with the
same sign, add their magnitudes (4 + 5 = 9) and keep
the same sign.

3.

The correct answer is D. Change the sign of the
second number and follow the rules for addition.

8.

The correct answer is D. When subtracting a positive
number from a negative number, you add the
magnitude and make the difference positive. This is
like a distance question. It’s really asking what the
distance is on the number line from –37 to +38. Try to
picture the number line. Add the magnitudes and
make the answer positive:
–37 – 38 = –75

+ 3
+ 6
+ 9
4.

+ 15

6.

The correct answer is B. The product of two negative
numbers is a positive number.
The correct answer is D. The product of an even
number of negative numbers is positive.
6  1
  (10) = 30
1  2

7.

9.

The correct answer is D. This question is really asking
about distance between two things, so the answer will
be positive. You can figure it is 14,494 feet from the top
of Mt. Whitney to sea level, and then another 294 feet
to get to the lowest point of Death Valley. Add the two
numbers: 14,494 + 294 = 14,788.

10.

The correct answer is A. Find the additional distance
submerged. Multiply –25 by 10. When multiplying a
negative number by a positive, the product is negative.

The correct answer is B. Change the sign of the
second number and follow the rules for addition.
+ 10
+ 5

5.

The difference is 75 degrees.

–25 × 10 = –250
Subtract 250 from –1,250. When subtracting a positive number from a negative number, add the magnitude and make the difference (the answer) negative:

371
Chapter 10
Basic Algebra

1,250 + 250 = 1,500
The answer is –1,500.

The correct answer is C.
(−4)(+3) = −12
Dividing a negative number by a negative number
gives a positive quotient.
−12
= +6
−2

Peterson’s SAT® Prep Guide 2017

LINEAR EQUATIONS
The next step in solving algebra problems is mastering linear equations. Whether an equation involves numbers or only variables,
the basic steps are the same.

Four-Step Strategy
1.

If there are fractions or decimals, remove them by multiplication.

2.

Collect all terms containing the unknown for which you are solving on the same side of the equation. Remember that
whenever a term crosses the equal sign from one side of the equation to the other, it must “pay a toll.” That is, it must
change its sign.

3.

Determine the coefficient of the unknown by combining similar terms or factoring when terms cannot be combined.

4.

Divide both sides of the equation by this coefficient.

If you have a string of multiplications and divisions to do and the
number of negative factors is even, the result will be positive; if the
number of negative factors is odd, the result will be negative.

Example:
Solve for x: 5x − 3 = 3x + 5

372

Solution:

Chapter 10

2x = 8
x=4

Basic Algebra

Example:
Solve for x:

3
2
x +2= x +3
4
3

Solution:
Multiply by 12:
9 x + 24 = 8 x + 36
x = 12

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Example:
Solve for x: 0.7x + 0.04 = 2.49
Solution:
Multiply by 100:
70 x + 4 = 249
70 x = 245
x = 3.5
An equation that is true for every value of the variable is called an identity. It has infinitely many solutions. An
equation has no solution if no value for the variable will make the equation true.
Example:
Solve for x: 0.5(6x + 4) = 3x + 2
Solution:
First, simplify each side of the equation. Then solve for x:
3x + 2 = 3x + 2
3x = 3x
x=x

373

Since x = x is always true, the original equation has infinitely many solutions and is called an identity.

Chapter 10
Basic Algebra

Example:
Solve for x: 9x + 7 = x + 2(4x + 3)
Solution:
First, simplify each side of the equation. Then solve for x:
9x + 7 = x + 8x + 6
9x + 7 = 9x + 6
7=6
Since 7 ≠ 6, the original equation has no solution.
If you eliminate the variable in the process of solving the equation, then you will have either infinitely many solutions
or no solution.

Peterson’s SAT® Prep Guide 2017

Example:
In the equation ax + 7 = 2x + 3, for which values of a will the equation have no solutions?

Solution:
If a = 2, the original equation becomes 2x + 7 = 2x + 3. Since 2x + 7 ≠ 2x + 3 there are no solutions that make the
equation true.

Real-World Linear Equations
A literal equation is an equation that involves two or more variables. You can solve for one variable in terms of the others using
the properties of equalities. A formula is a literal equation that defines a relationship among quantities. For example, the perimeter
of a rectangle can be found using the formula P = 2l + 2w, where P = perimeter, l = length, and w = width.
Example:
What is the width of a rectangle with perimeter 42 and length 8?
Solution:
Using the formula P = 2l + 2w, where P = perimeter, l = length, and w = width, solve for the variable w.

374

Substitute the values that you know into the formula.

Chapter 10

42 = 2 (8) + 2w

Basic Algebra

42 = 16 + 2w
42 − 16 = 2w
26 = 2w
26
=w
2
13 = w

The width of the rectangle is 13 units.

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Example:
The drama club sold student and adult tickets to its spring play. The adult tickets cost $15 each, and the student
tickets cost $10 each. Tickets sales were $4,050. If 120 adult tickets were sold, how many student tickets were sold?
Solution:
Using the formula R = 15a + 10s, where R = revenue from ticket sales, a = adult tickets, and s = student tickets, solve it
for the variable s.
Substitute the values that you know into the formula.
4 , 050 = 15 (120) + 10 s
4 , 050 = 1, 800 + 10 s
4 , 050 − 1, 800 = 10 s
2, 250 = 10 s
2, 250
=s
10
225 = s

The drama club sold 225 student tickets.

EXERCISES: LINEAR EQUATIONS

375

DIRECTIONS: Work out each problem. Circle the letter of your choice.
1.

2.

If 5x + 6 = 10, then x equals
A.

16
5

B.

5
16

C.

−

D.

4
5

SHOW YOUR WORK HERE

Chapter 10
Basic Algebra

5
4

Solve for x: 2x – 3 = 4x – 15
A.

–6

B.

–3

C.

3

D.

6

Peterson’s SAT® Prep Guide 2017

3.

Solve for k:
A.
B.
C.
D.

4.

5.

376

6.

Chapter 10
Basic Algebra

7.

k k
+ =1
3 4

8
11
7
12
12
7
1
7

If x + y = 12 and x – y = 2, what is the value of xy?
A.

10

B.

12

C.

20

D.

35

If 7x = 3x + 12, then 2x + 5 =
A.

10

B.

11

C.

12

D.

13

Solve for x: 6x + 7 = 2(3x – 5)
A.

0

B.

4

C.

No solution

D.

Infinitely many solutions

Solve for x:

2x + 9 3x + 8
=
2
7

A.

No solution

B.

Infinitely many solutions

C.

17
38

D.

−

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38
17

SHOW YOUR WORK HERE

8.

9.

10.

11.

12.

Solve for x:

1
1
(2 x + 8) = 4  x + 1
2
4

A.

Infinitely many solutions

B.

No solution

C.

0

D.

8

SHOW YOUR WORK HERE

1
The formula for the area of a triangle is A = bh . Find the
2
base of the triangle if the area is 28 in.2 and its height is
7 in.
A.

2 in.

B.

8 in.

C.

98 in.

D.

196 in.

For what values of n is n + 5 equal to n − 5?
A.

No value

B.

0

C.

All positive values

D.

All values

Membership to Iron Gym costs $40 per month plus a $30
registration fee. The monthly cost is deducted automatically from your account. If your starting balance is $350,
how many months will you be able to go to the gym
before you have to add money to your account?
A.

6

B.

7

C.

8

D.

9

377
Chapter 10
Basic Algebra

Your cell phone plan costs $45 per month plus $0.08 per
text message. Your latest bill is $66.04. How many text
messages were included in this bill?
A.

825

B.

562

C.

420

D.

263

Peterson’s SAT® Prep Guide 2017

ANSWER KEY AND EXPLANATIONS

1.

1. D

4. D

7. D

10. A

2. D

5. B

8. A

11. C

3. C

6. C

9. B

12. D

The correct answer is D.

6.

5x = 4
4
x=
5
2.

6 x + 7 = 6 x − 10
7 = −10
Since 7 ≠ –10, there are no values of x that will satisfy
the equation. There is no solution.
7.

4 x + 18 = 21x + 56
−17 x = 38
38
x=−
17

4 k + 3k = 12
7k = 12
12
k=
7
4.

Basic Algebra

The correct answer is D. Add the two equations to
cancel the y-variable and solve for x.

8.

7 + y = 12
y =5
xy = 35
5.

The correct answer is B. Solve for x:
4 x = 12
x =3

2 x + 5 = 3 (3) + 5 = 11

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The correct answer is A.
1
1
(2 x + 8) = 4  x + 1
2
4
x+4= x+4
x=x

x + y = 12
x−y =2
2 x = 14
x =7
Substitute the value of x into the addition equation to
find the value of y.

The correct answer is D.
2x + 9 3x + 8
=
2
7
2 (2 x + 9 ) = 7 (3 x + 8 )

The correct answer is C. Multiply by 12:

378
Chapter 10

6 x + 7 = 2 (3 x − 5)

The correct answer is D.
2 x − 3 = 4 x − 15
−2 x − 3 = −15
−2 x = −12
x =6

3.

The correct answer is C.

All values of x will make this equation true, so there
are infinitely many solutions.
9.

The correct answer is B. Solve the formula for b. Then
substitute in the values of the variables.
1
A = bh
2
2A
=b
h
2 (28)
=b
7
8=b

10.

11.

The correct answer is A. There is no number such
that, when 5 is added, you get the same result as when
5 is subtracted. Do not confuse choices A and B. Choice
B would mean that the number 0 satisfies the
equation, which it does not.
The correct answer is C.
40 x + 30 = 350
40 x = 320
x =8

12.

The correct answer is D. Write an equation to model
your monthly cell phone bill. Use the variable B for your
total bill and t for the number of text messages. Then
plug in the given values and solve for t.
B = 45 + 0.08t
B − 45
=t
0.08
66.04 − 45
=t
0.08
263 = t
Your latest bill included charges for 263 text
messages.

379
Chapter 10
Basic Algebra

Peterson’s SAT® Prep Guide 2017

SIMULTANEOUS EQUATIONS
In solving equations with two unknowns, you must work with two equations simultaneously. The object is to eliminate one of
the two unknowns and solve for the resulting single unknown.
Example:
Solve for x: 2 x − 4 y = 2
3 x + 5 y = 14
Solution:
Multiply the first equation by 5:
10x − 20y = 10
Multiply the second equation by 4:
12x + 20y = 56
Since the y-terms now have the same numerical coefficients, but with opposite signs, you can eliminate them by
adding the two equations. If they had the same signs, you would eliminate them by subtracting the equations.
Add the equations:
10 x − 20 y = 10
12 x + 20 y = 56
22x = 66

380

x =3
Chapter 10
Basic Algebra

Since you were only asked to solve for x, stop here. If you were asked to solve for both x and y, you would now substitute 3 for x in either equation and solve the resulting equation for y.
3 (3) + 5 y = 14
9 + 5 y = 14
5y = 5
y =1
In the previous example, the system of equations has exactly one solution (3, 1). It is also possible for a system of
equations to have no solution or infinitely many solutions.
Example:
Solve the system: −12 x + 8 y = 2
3x − 2y = 7

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Solution:
Multiply the second equation by 4, so that the x-terms and y-terms have the same numerical coefficients:
−12 x + 8 y = 2
12 x − 8 y = 28
Now add the equations:
−12 x + 8 y = 2
12 x − 8 y = 28
0 = 30
Since 0 ≠ 30, there is no solution to the system.
Example:
Solve the system: 3 x − 2 y = −15
2
x − y = −5
3
Solution:
Multiply the second equation by 3, so that the x- and y-terms have the same numerical coefficients:
3 x − 2 y = −15
3 x − 2 y = −15

381
Chapter 10

Now subtract the equations:

Basic Algebra

3 x − 2 y = −15
3 x − 2 y = −15
0 = 0
Since 0 = 0 is an identity, there are infinitely many solutions to the system.

Peterson’s SAT® Prep Guide 2017

Example:
For which value of a will the system

3 x − 4 y = 12
have infinitely many solutions?
ax − 3 y = 9

Solution:
Consider solving each equation for y:
3
x −3
4
a
y=
3x − 3
y=

To have infinitely many solutions,
9
So a = .
4

3 a
= .
4 3

Example:
Solve the system: x 2 + y = 9

x − y = −3
Solution:

382

Add the equations to eliminate the y-variable:

Chapter 10

x2 + y = 9
x − y = −3

Basic Algebra

x2 + x = 6
Set the equation equal to 0 and solve for x:
x2 + x – 6 = 0
Factor to solve for x:
(x – 2)(x + 3) = 0
x = 2 or x = –3
Find the corresponding y-values by substituting each value of x into the linear equation.
2 – y = –3; y = 5 and
–3 – y = –3; y = 0

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There are 2 possible solutions for the system: (2, 5) and (–3, 0). Check each in both equations.

( )

1st equation: x 2 + y = 9 and

( 2 )2 + 5 = 9

( x )2 + y = 9
( −3)2 + 0 = 9
9=9

9=9
2nd equation: ( x ) − y = −3

(2) − 5 = −3

( x ) − y = −3
( −3) − 0 = −3

−3 = −3

−3 = −3

and

The solutions to the system are (2, 5) and (–3, 0).

Applications of Systems
Systems of equations can be used to solve many real-life problems.
Example:
Computer Connect, Inc. makes and sells computer parts. The material for each part costs $3.00 and sells for $12.75
each. The company spends $1,200 on additional expenses each month. How many computer parts must the
company sell each month in order to break even?

Solution:
The break-even point is when the income equals the expenses. The first equation 12.75x = y represents the income.
The second equation 3x + 1,200 = y represents expenses.
y = 12.75 x
y = 3.00 x + 1, 200

383
Chapter 10
Basic Algebra

Solve the system by subtracting the second equation from the first.
y = 12.75 x
y = 3.00 x + 1, 200
0 = 9.75 x − 1,2
200
−9.75 x = −1, 200
−1, 200
x=
− 9.75
x ≈ 123.08
To break even, the company would have to sell at least 124 computer parts.

Peterson’s SAT® Prep Guide 2017

Example:
Jordan and Alex are planning a vacation. They plan to spend some of the time in Naples, Florida, and the rest of time
in Key West. They estimate that it will cost $250 per day in Naples and $325 per day in Key West. If they plan to
vacation a total of 8 days and have a budget of $2,375. How many days should they spend in each city?

Solution
To write the equations, let x = the number of days in Naples and y = the number of days in Key West. The first
equation x + y = 8 represents the total number of days on vacation. The second equation, 250x + 325y = 2,375,
represents total cost.
The system of equations is:

x+y =8
250 x + 325 y = 2, 375

Solve the system by multiplying the first equation by 325 and then subtract the second equation from the first.
325 x + 325 y = 2, 600
250 x + 325 y = 2, 375
75 x = 225
x=
3
Since x + y = 8, if x = 3 then y = 5. The couple can spend 3 days in Naples and 5 days in Key West.

384
Chapter 10

Solving Systems of Inequalities by Graphing
Systems of inequalities are solved using the same methods as systems of equations. Recall that you must reverse the sign of the
inequality if you multiply or divide by a negative value.

Basic Algebra

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You can graph a system of linear inequalities in the coordinate plane. The solution of the system is where the graphs of the
inequalities overlap. Recall, that an inequality with a < or > sign, is graphed as a dashed line, while an inequality with a ≤ or ≥ is
graphed with a solid line. A solid line shows that answers along the line are included in the solution set for that inequality.
Example:
What system of inequalities is represented by the graph shown?

4

2

–4

–2

0

2

4

–2
–2

–
4
–4

385
Chapter 10

Solution

Basic Algebra

First write the inequality that represents the region bounded by the solid line, using two points along the line and
the slope-intercept formula y = mx + b, replacing the equal sign with a comparison symbol.
y ≥ –0.5x + 2
Then write the inequality that represents the region bounded by the dashed line.
y < 2x + 1
The graph shows the intersection of the system y ≥ –0.5x + 2
					
y < 2x + 1

Peterson’s SAT® Prep Guide 2017

Example:
Solve the system of inequalities by graphing: x + 2 y ≤ 8
3x − y ≥ 3
Solution
First solve each inequality for y to rewrite in slope-intercept form:
			

1st inequality: x + 2 y ≤ 8
2y ≤ − x + 8
1
y≤− x+4
2

2nd inequality: 3 x − y ≥ 3
− y ≥ −3 x + 3
y ≤ 3x − 3

Then graph each inequality.

10
8
6
4
2

386

–1
10
–10

–8
–8

–6
–6

4
–4

2
–2

–
2
–2
4
–4

Chapter 10

–6
–6

Basic Algebra

–8
–8
–10
–10
–1

The solution is the region where the graphs overlap.

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2

4

6

8

10
10

EXERCISES: SIMULTANEOUS EQUATIONS
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

Solve the system for x:

SHOW YOUR WORK HERE

− x + 2y = 7
3 x − 2 y = −3

2.

A.

4.5

B.

2

C.

–2

D.

–11

Solve the system for y:
2x − 9y = 4
−x + y = 5

3.

A.

7

B.

2

C.

–2

D.

–7

Solve the system:
2 y = x + 16
1
− x + y = −2
2

4.

387

A.

No solution

B.

Infinitely many solutions

C.

0

D.

All positive values

Chapter 10
Basic Algebra

Solve the system for y:
x − y2 = 2
2x + 5y = 7
A.

y = 11, 2.25

B.

y = 0.5, –3

C.

y = 1.75, 17

D.

Infinitely many solutions

Peterson’s SAT® Prep Guide 2017

5.

SHOW YOUR WORK HERE

Solve the system:
14 y = −2 x + 23

2 ( x + 7 y ) = 23

6.

A.

0

B.

All negative values

C.

No solution

D.

Infinitely many solutions

What system of inequalities is shown in the graph?
y

5
4
3
2
1
–5

–4

–3 –2

–1
–1

1

2

3

4

5

x

–2
–3
–4
–5

388
Chapter 10

A.

y < 4 x −1
2x + y ≥ 5

B.

y > 4 x −1
−2 x + y ≥ 5

C.

y > 4x +1
2x − y ≥ 5

D.

y ≥ 4 x −1
2 x + y ≥ −5

Basic Algebra

7.

Speedy Rent-A-Car charges $45 a day plus $0.60 per mile
driven to rent a car. Zippy Rental charges $40 a day plus
$0.70 per mile driven to rent a car. After how many miles
would it cost the same amount to rent a car from either
Speedy Rent-A-Car or Zippy Rental?
A.

25 miles

B.

50 miles

C.

75 miles

D.

100 miles

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8.

A used bookstore sells hardcover books for $5 each
and paperback books for $1 each. You buy 13 books for
summer reading, and you spend $45. How many more
hardcover books than paperback books did you buy?
A.

8

B.

5

C.

4

D.

3

SHOW YOUR WORK HERE

389
Chapter 10
Basic Algebra

Peterson’s SAT® Prep Guide 2017

ANSWER KEY AND EXPLANATIONS

1.

1. B

3. A

5. D

7. B

2. C

4. B

6. A

8. D

The correct answer is B. Add the equations to
eliminate y.

4.

The correct answer is B. Eliminate the x-variable and
solve for y.

− x + 2y = 7
3 x − 2 y = −3

x − y2 = 2
2x + 5y = 7

2x = 4
x= 2
2.

2x − 2y2 = 4
2xx + 5 y = 7
2y2 + 5y = 3

The correct answer is C. Rewrite the equations and
multiply the second equation by 2. Then add the
equations to eliminate x.

2y + 5y − 3 = 0
2

(2 y − 1) ( y + 3) = 0
1
y = , −3
2

2x − 9y = 4
−x + y = 5
2x − 9y = 4
−2 x + 2 y = 10

5.

14 y = −2 x + 23

− 7 y = 14
y = −2

390

3.

2 ( x + 7 y ) = 23
2 x + 14 y = 23
2 x + 14 y = 23

The correct answer is A. Multiply the second equation
by 2 and add to eliminate a variable.
2 y = x + 16
1
− x + y = −2
2
− x + 2 y = 16
x − 2y = 4

Chapter 10
Basic Algebra

0 = 20
Since 0 ≠ 20, there is no solution.

The correct answer is D.

0=0

Since 0 = 0 is always true, there are infinitely many
solutions.
6.

The correct answer is A. First write the inequality that
represents the region bounded by the dashed line
using two points and the slope intercept formula.
y < 4x –1
Then write the inequality that represents the region
bounded by the solid line.
y ≥ –2x + 5
The graph shows the intersection of the system:
y < 4x –1
2x + y ≥ 5

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7.

The correct answer is B. The system of equations that
represents the cost of renting a car from the places is:
y = 45 + 0.6x
y = 40 + 0.7x
Solve this system by subtracting the second equation
from the first. The solution is 50 miles.
Rewrite the equations so that you can eliminate the y
variable. So:
40 = −0.7 x + y
−45 = 0.6 x − y
Add the second equation to the first to eliminate the
y variable: –5 = –0.1x. When simplified, x = 50. Therefore, 50 is the number of miles it would take for the
cost (y) of both rental companies to be equal.

8.

The correct answer is D. The system of equations that
represents this situation is:
x + y = 13
5 x + y = 45
Where x represents the number of hardcover books
purchased and y represents the number of paperback
books purchased. Solve this system by subtracting
the first equation from the second.
5 x + y = 45
− x − y = −13
You are left with 4x = 32 or x = 8. If 8 hardcover
books are purchased, then 5 paperback books are
purchased. This means that you purchased 3 more
hardcover books than paperback books.

391
Chapter 10
Basic Algebra

Peterson’s SAT® Prep Guide 2017

EXPONENTS
An exponent is a mathematical notation indicating that a number, called the base, has been multiplied one or more times by
itself. For example, in the term 23, the 2 is the base and the 3 is the exponent. This term means “two times two times two” and is
read “two to the third power.” The word power tells how many times the base number appears in the multiplication.
x 3 = x times x times x
x 2 = x times x
x1 = x
x0 = 1

Commit to memory small powers of small numbers that come up in many
questions. For example, the powers of 2: 2, 4, 8, 16, 32, . . . the powers of
3: 3, 9, 27, 81, . . . and so on.

The Five Rules of Exponents
1.

To multiply powers of the same base, add the exponents.
x2 times x3 = x2 + 3 = x5
x5 times x4 = x5 + 4 = x9

2.

To divide powers of the same base, subtract the exponent of the divisor from the exponent of the dividend.

x6
= x6 − 2 = x4
x2
x 10
= x 10 − 3 = x 7
x3

392
Chapter 10
Basic Algebra

3.

Negative powers represent reciprocals.
x −1 =
 x3 
 y 5 

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−2

1
x
2

 y5 
y 10
= 3 = 6
x
x 

4.

To find the power of a power, multiply the exponents.
(x2)3 = x(2)(3) = x6
(x3y5)2 = x(3)(2)y(5)(2) = x6y10
A variable base with an even exponent has two values, one positive and one negative.
x2 = 25; x could be positive 5 or negative 5.
A variable base can be zero (unless otherwise stated in the problem). In that case, no matter what the exponent, the value
of the term is zero.
Is x4 always greater than x2? No; if x is zero, then x4 and x2 are equal.
When the base is a fraction between 0 and 1, the larger the exponent, the smaller the value of the term.
2
1
 37 
 37 
 37 
 37 
Which is greater,   or   ? The correct answer is   because   is almost ,
2
 73 
 73 
 73 
 73 
2
 37 
1
while   is about .
73
4

5.

Fractional exponents represent roots.
1

x2 = x
1

x3 =3 x
2

x5 =

( x)
5

5

= x2

2

393
Chapter 10
Basic Algebra

Peterson’s SAT® Prep Guide 2017

EXERCISES: EXPONENTS
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

2.

3.

394
Chapter 10

p8 × q4 × p4 × q8 =
A.

p12q12

B.

p4q4

C.

p32q32

D.

p64q64

(x2y3)4 =
A.

x6y7

B.

x8y12

C.

x12y8

D.

x2y

x 16 y 6
=
x4 y2
A.

x20y8

B.

x4y3

C.

x12y3

D.

x12y4

Basic Algebra

4.

If x4 = 16 and y2 = 36, then the maximum possible value
for x − y is
A.

–14

B.

–4

C.

8

D.

20

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SHOW YOUR WORK HERE

5.

a −4 × a−3 =
1

A.

a 12
1

B.

6.

SHOW YOUR WORK HERE

a7

C.

a7

D.

a12
2

1

x 3x 2 =
A.

3

x

B.

5

x3

C.

( x)

D.

5

6

7

1
x2

395
Chapter 10
Basic Algebra

Peterson’s SAT® Prep Guide 2017

ANSWER KEY AND EXPLANATIONS

1.

1. A

3. D

5. B

2. B

4. C

6. C

The correct answer is A. The multiplication signs do
not change the fact that this is the multiplication of
terms with a common base and different exponents.
Solve this kind of problem by adding the exponents.
p

8+4

4+8

×q

5.

a −4 × a −3 = a −4

12 12

=p q

2.

The correct answer is B. To raise a power to a power,
multiply the exponents. x(2)(4)y(3)(4) = x8y12

3.

The correct answer is D. All fractions are implied
division. When dividing terms with a common base
and different exponents, subtract the exponents.
Therefore, 16 − 4 = 12 and 6 − 2 = 4.

The correct answer is B. To multiply, add the exponents. The resulting exponent will be negative, which
means you need to take the reciprocal.

= a −7
1
= 7
a

6.

The correct answer is C. To multiply, add the exponents.
Then, use the rules for rational exponents to convert to
root form.

x
4.

396

The correct answer is C. First, determine the value for
x and the value for y, and you will find that x could be 2
or –2, and y could be 6 or –6. Therefore, the four
possible values for x − y are as follows:
2 − 6 = −4

2 − ( −6) = 8

Chapter 10

−2 − 6 = −8

−2 − ( −6) = 4

Basic Algebra

The maximum value would be 8.

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+ ( −3)

2 1
3x 2

=x

2 1
+
3 2

=x

7
6

=

( x)
6

7

QUADRATIC EQUATIONS
Roots and Factoring
In solving quadratic equations, remember that there will always be two roots, even though these roots may be equal. A complete
quadratic equation is of the form ax2 + bx + c = 0.

ALERT: Don’t forget: In working with any equation, if you move a term from one side of the equal sign to the other, you must
change its sign.

Example:
Factor: x2 + 7x + 12 = 0
Solution:
(x + 3)(x + 4) = 0
The last term of the equation is positive; therefore, both factors must have the same sign, since the last two terms
multiply to a positive product. The middle term is also positive; therefore, both factors must be positive, since they
also add to a positive sum.
(x + 4)(x + 3) = 0
If the product of two factors is 0, each factor may be set equal to 0, yielding the values for x of −4 or −3.

397

Example:
Chapter 10
Factor: x2 + 7x − 18 = 0

Basic Algebra

Solution:
(x + 9)(x − 2) = 0
Now you are looking for two numbers with a product of −18; therefore, they must have opposite signs. To yield + 7 as
a middle coefficient, the numbers must be + 9 and −2.
(x + 9)(x − 2) = 0
This equation gives the roots −9 and + 2.
Incomplete quadratic equations are those in which b or c is equal to 0.

Peterson’s SAT® Prep Guide 2017

Example:
Solve for x: x2 − 16 = 0
Solution:
x2 = 16
x = ±4
Remember, there must be two roots.
Example:
Solve for x: 4x2 − 9 = 0
Solution:

4 x2 = 9
9
x2 =
4
x=±

3
2

Example:

398
Chapter 10

Solve for x: x2 + 4x = 0
Solution:

Basic Algebra

Never divide through an equation by the unknown, as this would yield an equation of lower degree having fewer
roots than the original equation. Always factor this type of equation.
x(x + 4) = 0
The roots are 0 and −4.

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Example:
Solve for x: 4x2 − 9x = 0
Solution:
x(4x − 9) = 0
The roots are 0 and

9
.
4

The quadratic formula can also be used to find the solutions to quadratic equations ax2 + bx + c = 0 where a, b, and c are real
numbers and a ≠ 0.

x=

−b ± b 2 − 4 ac
2a

Example:
Use the quadratic formula to solve for x: 2x2 + 7x = 3
Solution:
Write the equation in standard form: 2x2 + 7x – 3 = 0. Then, identify the values of a, b, and c and substitute them into
the quadratic formula.
a = 2, b = 7, and c = −3
−b ± b 2 − 4 ac
x=
2a
x=
x=

The solutions are x =

−7 ±

399
Chapter 10
Basic Algebra

(7) 2 − 4 (2) ( −3)
2 ( 2)

−7 ± 73
4

−7 − 73
−7 + 73
≈ −3.886
≈ 0.386 or x =
4
4

Peterson’s SAT® Prep Guide 2017

EXERCISES: QUADRATIC EQUATIONS
DIRECTIONS: Work out each problem. Circle the letter of your choice.
1.

2.

3.

Solve for x: x2 − 2x − 15 = 0
A.

+5 or −3

B.

−5 or +3

C.

−5 or −3

D.

+5 or +3

Solve for x: x2 + 12 = 8x
A.

+6 or −2

B.

−6 or +2

C.

−6 or −2

D.

+6 or +2

Solve for x: 4x2 = 12
A.

400

B.

Chapter 10
Basic Algebra

4.

3
3 or −3

C.

3 or − 3

D.

3 or

−3

Solve for x: 3x2 = 4x
A.

4
3

B.

4
or 0
3

C.

4
− or 0
3

D.

4
4
or −
3
3

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SHOW YOUR WORK HERE

5.

Solve for x: 2x2 + 3x = 7
A.

x=

3 − 47
3 + 47
,x=
4
4

B.

x=

3 + 65
3 − 65
,x=
4
4

C.

x=

−3 + 47
−3 − 47
,x=
4
4

D.

x=

−3 + 47
−3 − 47
,x=
4
4

SHOW YOUR WORK HERE

401
Chapter 10
Basic Algebra

Peterson’s SAT® Prep Guide 2017

ANSWER KEY AND EXPLANATIONS
1. A

1.

2. D

The correct answer is A.

5.

( x − 5) ( x + 3) =
x = 5 or − 3
2.

3. C

The correct answer is D.

4. B

5. D

The correct answer is D. Write the equation in
standard form:
2x2 + 3x – 7 = 0.
a = 2, b = 3, and c = –7.

x 2 − 8 x + 12 = 0

( x − 6 ) ( x − 2) = 0

x = 6 or 2
3.

The correct answer is C.
x2 = 3

The correct answer is B.
3x 2 − 4 x = 0

402
Chapter 10
Basic Algebra

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−3 ± (3)2 − 4(2)( −7)
2(2)

x=

−3 ± 65
4

The solutions are x =

x = 3 or − 3
4.

x=

x (3 x − 4 ) = 0
x = 0 or

4
3

−3 − 65
−3 + 65
.
or x =
4
4

LITERAL EQUATIONS
There are many equations that can be used to represent common problems, such as distance. The equation to find distance, given
rate and time, is written as d = rt. This type of equation is called a literal equation in which the equation is solved for one variable
in particular, such as d in the distance formula. Here are some examples:
Example:
Solve for l in the equation P = 2l + 2w.
Solution:
Subtract 2w from both sides of the equation. This will eliminate it from the right side of the equation.
P – 2w = 2l
Divide both sides of the equation to solve for l.
P − 2w P
; −w = l
2
2

Example:
Solve for c in the equation E = mc2.
Solution:

Divide both sides by m. This will eliminate it from the right side of the equation.
E
= c2
m

403
Chapter 10
Basic Algebra

Take the square root of both sides to solve for c.
E
=c
m
Example:
1
The volume of a cone can be found using the formula V = pr 2h . Solve for r.
3
Solution:

Divide both sides by h:
Divide both sides by pi:

v 1 2
= pr
h 3
v

ph

1
= r2
3

Multiply both sides by 3: 3 ×

v

ph

1
= r2 × 3
3

v
Take the square root of both sides: 3 × ph = r

Peterson’s SAT® Prep Guide 2017

EXERCISES: LITERAL EQUATIONS
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

2.

404

Solve for x in the equation z = 5x – 25xy
A.

z
=x
−20 y

B.

z
=x
1− 5 y

C.

z
=x
5 − 25 y

D.

z
=x
30 y

A bakery orders vanilla beans at a cost of $12.45 for a
package of 10. There is a shipping cost of $6.00 for all sizes
of shipments. Which of the following shows the equation
solved for p, the total number of packages purchased,
where c is the total cost of the order.
A.

p = 6c + 12.45

B.

p = 12.45c + 6

C.

p=

c
−6
12.45

D.

p=

c −6
12.45

Chapter 10
Basic Algebra

3.

The distance between two points is determined by the
equation below, where x1 and x2 are the x-coordinates of
two points, and y1 and y2 are the y-coordinates of the two
points.
d=

( x2 − x1)2 + ( y2 − y1)2

Which shows the equation solved for y2?
A.

y 2 = d 2 − ( x 2 − x1 ) + y1

B.

y 2 = d − ( x 2 − x1 ) + y1

C.

y 2 = d − ( x 2 − x1 ) + y1

D.

y 2 = d 2 − ( x 2 − x1 ) + y1

2

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2

SHOW YOUR WORK HERE

4.

5.

The surface area of a cylinder is found using the formula
SA = 2πrh + 2πr2. Which shows the equation solved for
h, the height of the cylinder, in simplest form?

A.

SA − 2pr 2
=h
2pr

B.

SA
-r =h
2pr

C.

SA
=h
2pr

D.

SA
=h
4 pr

SHOW YOUR WORK HERE

The formula for converting degrees Fahrenheit to degrees
Celsius is shown below.
C=

5
(F − 32)
9

Which equation shows the formula correctly solved for F?

A.

9
(C + 32) = F
5

C
B. 		 − 32 = F
9
5

6.

C.

9
C − 32 = F
5

D.

9
C + 32 = F
5

Solve for f:

405
Chapter 10
Basic Algebra

1 1 1
= +
f a b

A.

f = a+ b

B.

f=

C.

f = ab

D.

f=

ab
a+b

ba
b−a

Peterson’s SAT® Prep Guide 2017

ANSWER KEY AND EXPLANATIONS

1.

1. C

3. A

5. D

2. D

4. B

6. B

The correct answer is C. Factor out an x from both terms
on the right side of the equation: z = x(5 – 25y)

5.

Divide both sides by 5 – 25y:

The correct answer is D. To solve for F, divide both sides
5
9
by or multiply both sides by .
9
5
9
C = F − 32
5

z
=x
5 − 25 y
2.

by the equation 12.45p + 6 = c. To solve for p, subtract
6 from both sides. Then, divide both sides by 12.45 to
isolate p.
3.

Add 32 to both sides to isolate F.

The correct answer is D. The total cost, c, is determined

The correct answer is A. Solve for y2 by squaring both
sides to remove the square root from the right side.

9
C + 32 = F
5
6.

The correct answer is B. To solve for f, first multiply both
sides of the equation by f to move it into the numerator.

1=

d2 = (x2 – x1)2 + (y2 – y1)2
Subtract the term (x2 – x1)2 from both sides.

406

Multiply by b and then by a to write the fractions with
a common denominator, ab.

d2 – (x2 – x1)2 = (y2 – y1)2
1=

Take the square root of both sides to eliminate the
exponent on the right side.

Chapter 10

f f
+
a b

fb fa
+
ab ab

Add the fractions.

Basic Algebra

d 2 − ( x 2 − x1 ) = y 2 − y1
2

Add y1 to both sides of the equation.
d 2 − ( x 2 − x1 ) + y1 = y 2
2

4.

The correct answer is B. To solve for the height, h, factor
out 2πr from the terms on the right side of the equation.
SA = 2πr(h + r)
Divide both sides of the equation by 2πr.
SA
= h+r
2pr
Subtract r from both sides of the equation.
SA
−r =h
2pr

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1=

fb + fa
ab

Multiply both sides of the equation by ab.
ab = fb + fa
Factor f out of the terms on the right side of the
equation.
ab = f(b + a)
Divide both sides of the equation by (b + a).
ab
=f
a+b

ROOTS AND RADICALS
Adding and Subtracting
Rules for adding and subtracting radicals are much the same as for adding and subtracting variables. Radicals must be exactly
the same if they are to be added or subtracted, and they merely serve as a label that does not change.
4 2 +3 2 =7 2
2 +2 2 =3 2
2 + 3 cannot be added
Sometimes, when radicals are not the same, simplification of one or more radicals will make them the same. Remember that
radicals are simplified by factoring out any perfect square factors.
27 + 75
9 i 3 + 25 i 3
3 3 +5 3 =8 3

Multiplying and Dividing
In multiplying and dividing, treat radicals in the same way as you treat variables. They are factors and must be handled as such.

2i 3= 6

407

2 5 i 3 7 = 6 35

(2 3 )

2

= 2 3 i 2 3 = 4 i 3 = 12

75
= 25 = 5
3

Chapter 10
Basic Algebra

10 3
=2
5 3

Simplifying
To simplify radicals that contain a sum or difference under the radical sign, add or subtract first, then take the square root.
16 x 2 + 9 x 2
25 x 2 5 x
x2 x2
+
=
=
=
9 16
144
144
12

If you take the square root of each term before combining, you would have
clearly not the same answer. Remember that

x x
7x
+ , or
, which is
12
3 4

25 is 5. However, if you write that

25 as

16 + 9 , you cannot say it is

4 + 3, or 7. Always combine the quantities within a radical sign into a single term before taking the square root.

Peterson’s SAT® Prep Guide 2017

Radicals
In solving equations containing radicals, always get the radical alone on one side of the equation; then square both sides to
remove the radical and solve. Remember that all solutions to radical equations must be checked, as squaring both sides may
sometimes result in extraneous roots.
Example:
Solve for x:

x +5 =7

Solution:
x + 5 = 49
x = 44
Checking, we have 49 = 7 , which is true.
Example:
Solve for x: x = − 6
Solution:
You may have written the answer: x = 36.

408
Chapter 10
Basic Algebra

Checking, we have 36 = − 6, which is not true, as the radical sign means the positive, or principal, square root only.
This equation has no solution because 36 = 6 , not −6.
Example:
Solve for x:

x2+ 6 − 3 = x

Solution:

x 2 +6 −3= x
x2 +6 = x +3
x 2 + 6 = x 2 + 6x + 9
6 = 6x + 9
−3 = 6 x
1
− =x
2

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Checking, we have

6

1
1
−3= −
4
2

25
1
−3= −
4
2
5
1
−3= −
2
2
1
1
2 −3= −
2
2
1
1
− =−
2
2
This is a true statement. Therefore,

1
is a true root.
2

Roots as Fractional Powers
Roots can also be written as fractional exponents to make them easier for you to work with. The square root of 4 ( " 4 " ) , for example,
1

1

can also be written as 4 2 . The value of a square root includes both the positive and negative root, so

4 = ±2 and 4 2 = ± 2.

To write an nth root as a fractional exponent, use the root as the denominator of the fraction under 1 or

n

1

x = x n.

Use the laws of exponents to solve radical expressions and equations.
Example:
Simplify:

3x i

A.

9x2

B.

6x

C.

3x

D.

3x

409

3x

Chapter 10
2

Basic Algebra

Solution:

3x i

1

1

3 x = (3 x ) 2 i (3 x ) 2
1

= (3 x ) 2

+

1
2

= 3x

Peterson’s SAT® Prep Guide 2017

EXERCISES: ROOTS AND RADICALS
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

2.

3.

410
Chapter 10
Basic Algebra

4.

5.

What is the sum of 12 + 27 ?
A.

29

B.

3 5

C.

13 3

D.

5 3

What is the difference between 150 and 54 ?
A.

2 6

B.

16 6

C.

96

D.

6 2

What is the product of 18x and 2x , where x is greater
than 0?
A.

6x2

B.

6x

C.

36x2

D.

6 x

If

1
= 0.25 , what does x equal?
x

A.

2

B.

0.5

C.

0.2

D.

20

Solve for x: 5 2 x 2 = 500
A.

10

B.

100

C.

2 5

D.

10

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SHOW YOUR WORK HERE

6.

A.

–2

B.

7.

2

D.

4

Which expression is equivalent to
A.

5

B.

10 x 5

C.

5

5

4x + 5 6x ?

10 x
1

10 + 2 x

(4 x )

1
5

+ (6 x )

1
5

Divide 6 45 by 3 5.
A.

6

B.

9

C.

15

D.

30
y2 y2
+
=
25 16

9.

A.

2y
9

B.

9y
20

C.

y
9

D.
10.

2

C.

D.
8.

SHOW YOUR WORK HERE

Solve for x: 8 x + 7 = 3 x + 17

y

411
Chapter 10
Basic Algebra

41
20

Which expression shows
form?
A.

3
2

B.

0
2

C.

4 − 13
2

D.

2 − 23
2 2

5

32 − 3 2
written in simplest
8

1

1

Peterson’s SAT® Prep Guide 2017

ANSWER KEY AND EXPLANATIONS

1.

1. D

3. B

5. D

7. D

9. D

2. A

4. A

6. D

8. A

10. D

The correct answer is D.

6.

12 = 4 3 = 2 3

Subtract 3 x from each side: 5 x = 10.

27 = 9 3 = 3 3
2 3 +3 3 =5 3
2.

Divide each side by 5:

x = 2, x = 4 .

7.

The correct answer is D. The terms under the roots
cannot be added because they are not like terms. The
terms must remain as 4x and 6x. The fifth root can be
1
rewritten as an exponent of .
5

8.

The correct answer is A.

The correct answer is A.
150 = 25 6 = 5 6
54 = 9 6 = 3 6

The correct answer is D. Subtract 7 from each side of
the equation: 8 x = 3 x + 10 .

5 6 −3 6 =2 6
3.

The correct answer is B.
18 x i 2 x = 36 x 2 = 6 x

412

4.

9.

Chapter 10
Basic Algebra

The correct answer is D.

The correct answer is D.
16 y 2 + 25 y 2
y2 y2
+
=
25 16
400

The correct answer is A.
0.25 = 0.5
1
= 0.5
x
1 = 0.5 x
10 = 5 x
2= x

5.

6 45
=2 9 =2i 3=6
3 5

=

10.

y 41
41y 2
=
400
20

The correct answer is D. The fifth root of 32 is 2, because
2 × 2 × 2 × 2 × 2 = 32.
2−3 2
8

Factor 500 = 100 5 = 10 5 so you get:

The third root of 2 can be rewritten as

5 2 x 2 = 10 5.

The square root of 8 can be simplified:

Factor 5 from each side of the equation:
2x 2 = 2 5
Square each side:
2x2 = 4 × 5 = 20.   x2 = 10, so x = 10 .

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8 = 4×2 =2 2
1

2−23
2 2

1
23

1

2−23 .
:
8

MONOMIALS AND POLYNOMIALS
When we add a collection of expressions together, each expression is called a term. Monomial means one term. For example, we
might say that 2x + 3y2 + 7 is the sum of three terms, or three monomials. When we talk about a monomial, we generally mean
a term that is just the product of constants and variables, possibly raised to various powers. Examples might be 7, 2x, −3y2, and
4x2z5. The constant factor is called the coefficient of the variable factor. Thus, in −3y2, −3 is the coefficient of y2.
If we restrict our attention to monomials of the form Axn, the sums of such terms are called polynomials (in one variable). Expressions
like 3x + 5, 2x2 − 5x + 8, and x4 − 7x5 − 11 are all examples of polynomials. The highest power of the variable that appears is called
the degree of the polynomial. The three examples just given are of degree 1, 2, and 5, respectively.
In evaluating monomials and polynomials for negative values of the variable, the greatest pitfall is keeping track of the minus
signs. Always remember that in an expression like −x2, the power 2 is applied to the x, and the minus sign in front should be
thought of as (−1) times the expression. If you want to have the power apply to −x, you must write (−x)2.

Combining Monomials
Monomials with identical variable factors can be added together by adding their coefficients. So 3x2 + 4x2 = 7x2. Of course, subtraction is handled the same way, thus:
3x4 − 9x4 = −6x4
Monomials are multiplied by taking the product of their coefficients and taking the product of the variable part by adding exponents of factors with like bases. So, (3xy2)(2xy3) = 6x2y5.
Monomial fractions can be simplified to simplest form by dividing out common factors of the coefficients and then using the
usual rules for subtraction of exponents in division. An example might be:
6x 3y 5
2x 4 y 3

=

3y 2
x

413
Chapter 10
Basic Algebra

Example:
3
Combine into a single monomial: 8 x − 6 x
2

4x

Solution:
The fraction simplifies to 2x, and 2x − 6x = −4x.

Peterson’s SAT® Prep Guide 2017

Combining Polynomials and Monomials
Polynomials are added or subtracted by just combining like monomial terms in the appropriate manner. Thus,
(3x2 − 3x − 4) + (2x2 + 5x − 11)
is summed by removing the parentheses and combining like terms, to yield
5x2 + 2x − 15.
In subtraction, when you remove the parentheses with a minus sign in front, be careful to change the signs of all the terms within
the parentheses. So:

(3 x

2

) (

)

− 3 x − 4 − 2 x 2 + 5 x − 11 = 3 x 2 − 3 x − 4 − 2 x 2 − 5 x + 11
= x 2 − 8x + 7

(Did you notice that 3x2 − 2x2 = 1x2 but the “1” is not shown?)
To multiply a polynomial by a monomial, use the distributive property to multiply each term in the polynomial by the monomial
factor. For example, 2x(2x2 + 5x − 11) = 4x3 + 10x2 − 22x.
When multiplying a polynomial by a polynomial, you are actually repeatedly applying the distributive property to form all
possible products of the terms in the first polynomial with the terms in the second polynomial. The most common use of this is
in multiplying two binomials (polynomials with two terms), such as (x + 3)(x − 5). In this case, there are four terms in the result,
x • x = x 2; x(−5) = −5x; 3 • x = 3x; and 3 • (−5) = −15; but the two middle terms are added together to give −2x. Thus, the product
is x2 − 2x − 15.

414

This process is usually remembered as the FOIL method. That is, form the products of First, Outer, Inner, Last, as shown in the
figure below.

Chapter 10

O
F

Basic Algebra

(x + 3)(x – 5) = x2 + (–5x + 3x) – 15
I
L

Example:
If d is an integer, and (x + 2)(x + d) = x2 − kx −10, what is the value of k + d?
Solution:
The product of the two last terms, 2d, must be −10. Therefore, d = −5. If d = −5, then the sum of the outer and inner
products becomes −5x + 2x = −3x, which equals −kx. Hence, k = 3, and k + d = 3 + (−5) = −2.

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Factoring Monomials
Factoring a monomial simply involves reversing the distributive property. For example, if you are looking at 4x2 + 12xy, you should
see that 4x is a factor of both terms. Hence, you could just as well write this as 4x(x + 3y). Multiplication using the distributive
property will restore the original formulation.
Example:
If 3x − 4y = −2, what is the value of 9x − 12y?
Solution:
Although you seem to have one equation in two unknowns, you can still solve the problem, because you do not need to know
the values of the individual variables. Just rewrite:
9x − 12y = 3(3x − 4y).
Since 3x − 4y = −2, 9x − 12y is 3 times −2, or −6.

415
Chapter 10
Basic Algebra

Peterson’s SAT® Prep Guide 2017

EXERCISES: MONOMIALS AND POLYNOMIALS
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

2.

3.

416
Chapter 10
Basic Algebra

4.

SHOW YOUR WORK HERE

6x3(x 2)3 =
A.

6x7

B.

6x8

C.

6x9

D.

6x10

(3w  2y  3)4 =
A.

18w 8y12

B.

18w6y7

C.

81w6y7

D.

81w8y12

5x3(−3x8) =
A.

15x11

B.

−15x11

C.

−15x24

D.

15x24

3
2
Simplify: 3 x y − 9 x
2
xy

A.

−6 xy

B.

3 x ( xy − 3)

C.

3x 2 − 9 x
y

D.

3x2 – 9x

xy 2

y2

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5.

6.

7.

8.

(a3 + 4a2 – 11a + 4) – (8a2 + 2a + 4) =
A.

–3a2 – 13a

B.

a3 – 12a2 – 13a – 8

C.

a3 – 4a2 – 9a + 8

D.

a3 – 4a2 – 13a

SHOW YOUR WORK HERE

( 4 x 3 − 2 x + 5) + (8 x 2 − 2 x − 10 )

A.

4x3 + 8x2 – 4x – 5

B.

12x2 – 4x – 5

C.

12x3 – 5

D.

4x3 + 8x2 + 15

What is the product of 2x2 and 3x3 – x + 4?
A.

6x6 + 6x2

B.

6x5 – 2x3 + 8x2

C.

5x5 – x2 + 4

D.

6x5 + 2x3 + 8x2

Multiply: (x2 – 1)(2x3 + 5)

417

A.

2x5 + 5x2

B.

–2x3 – 5

C.

2x5 – 2x3 + 5x2 – 5

Chapter 10

D.

2x6 – 5x2 – 2x3 – 5

Basic Algebra

Peterson’s SAT® Prep Guide 2017

ANSWER KEY AND EXPLANATIONS

1.

1. C

3. B

5. D

7. B

2. D

4. B

6. A

8. C

The correct answer is C.

5.

The correct answer is D. The term a3 will remain because there are no other like terms to subtract. Subtract
the squared terms: 4a2 – 8a2 = –4a2; the single variable
terms –11a – 2a = –13a; and the integer terms –4 – 4 = 0.
Combining these terms with a3 gives us a3 – 4a2 – 13a.

6.

The correct answer is A. The only like terms that may be
combined are

( ) = (6 i x ) i ( x )
3

6x 3 x 2

2 3

3

= 6 i x 3 i x (2 i 3)
=6i x3 i x6
=6i x9
= 6x 9

2.

The correct answer is D.

(3w y )

2 3 4

(–2x) + (–2x) = –4x and +5 + (–10) = 5 – 10 = –5.

= 81 i w 8 i y 12

7.

= 81w 8 y 12

3.

Combine all the terms to get 4x3 + 8x2 – 4x – 5.

4
= (3) i w (2 i 4) i y (3 i 4)

The correct answer is B.

418

(

2x 2 i 3x 3 = 6 x 5
2 x 2 i ( − x ) = −2 x 3

)

5 x 3 −3 x 8 = 5 i x 3 i ( −3) i x 8
3

= 5 i ( −3) i x i x

Chapter 10

= −15 i x 3+ 8

Basic Algebra

= −15 x 11

4.

The correct answer is B. Multiply each term in the polynomial by 2x2:

2x 2 i 4 = 8x 2

8

Combine the products to get 6x5 – 2x3 + 8x2.
8.

The correct answer is B. The variable x can be
factored out of the all the terms in the numerator and
denominator of the fraction:

The correct answer is C. Use the FOIL method to multiply:
First: (x2)(2x3) = 2x5
Outer: (x2)(5) = 5x2
Inner: (–1)(2x3) = –2x3

3

3x y − 9 x
xy 2

2

=
=

2

3x y − 9 x

1

(1) y 2

3x 2 y − 9 x
y2

Factor the term 3x out of the expression in the
numerator: 3x(xy – 3).
Because y appears in only one of the terms in the
numerator, it cannot be factored out of either the
numerator or the denominator.

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Last: (–1)(5) = –5
Combine the terms to get 2x5 – 2x3 + 5x2 – 5.

PROBLEM SOLVING IN ALGEBRA
When you are working with algebraic word problems, remember that before you begin solving the problem you should be
absolutely certain that you understand precisely what you need to answer. Once this is done, show what you are looking for
algebraically. Write an equation that translates the words of the problem to the symbols of mathematics. Then solve that equation
by using the techniques you just learned.
This section reviews the types of algebra problems most frequently encountered on the SAT® exam. Thoroughly familiarizing
yourself with the problems that follow will help you to translate and solve all kinds of word problems.

Solving Two Linear Equations in Two Unknowns
Many word problems lead to equations in two unknowns. Usually, one needs two equations to solve for both unknowns, although
there are exceptions. There are two generally used methods to solve two equations in two unknowns. They are the method of
substitution and the method of elimination by addition and subtraction.
We’ll illustrate both methods via example. Here is one that uses the method of substitution.
Example:
Mr. Green took his four children to the local craft fair. The total cost of their admission tickets was $14. Mr. and Mrs.
Molina and their six children had to pay $23. What was the cost of an adult ticket to the craft fair, and what was the
cost of a child’s ticket?
Solution:

419

Expressing all amounts in dollars, let x = cost of an adult ticket and let y = cost of a child’s ticket.
Chapter 10

For the Greens: x + 4y = 14

Basic Algebra

For the Molinas: 2x + 6y = 23
The idea of the method of substitution is to solve one equation for one variable in terms of the other and then
substitute that solution into the second equation. So we solve the first equation for x, because that is the simplest
one to isolate:
x = 14 − 4y
and substitute into the second equation:
2(14 − 4y) + 6y = 23
This gives us one equation in one unknown that we can solve:
28 − 8y + 6y = 23
−2y = −5; y = 2.5

Peterson’s SAT® Prep Guide 2017

Now that we know y = 2.5, we substitute this into x = 14 − 4y to get:
x = 14 − 4(2.5) = 4
Thus, the adult tickets were $4 each, and the children’s tickets were $2.50 each.

Here is an example using the method of elimination.
Example:
Paul and Denise both have after-school jobs. Two weeks ago, Paul worked 6 hours, Denise worked 3 hours, and they
earned a total of $39. Last week, Paul worked 12 hours, Denise worked 5 hours, and they earned a total of $75. What
is each one’s hourly wage?
Solution:
Again, let us express all amounts in dollars. Let x = Paul’s hourly wage, and let y = Denise’s hourly wage.
For the first week: 6x + 3y = 39
For the second week: 12x + 5y = 75

420

The idea of the method of elimination is that adding equal quantities to equal quantities gives a true result. So we
want to add some multiple of one equation to the other one so that if we add the two equations together, one
variable will be eliminated. In this case, it is not hard to see that if we multiply the first equation by −2, the coefficient
of x will become −12. Now when we add the two equations, x will be eliminated. Hence,
−12 x − 6 y = −78

Chapter 10

12 x + 5 y = 75
− y = −3

Basic Algebra

Thus, y = 3. We now substitute this into either of the two equations. Let’s use the first:
6x + (3)(3) = 39; x = 5.
Thus, Denise makes only $3 per hour, while Paul gets $5.

Word Problems in One or Two Unknowns
Word problems can be broken down into a number of categories. To do consecutive integer problems, you need to remember
that consecutive integers differ by 1, so a string of such numbers can be represented as n, n + 1, n + 2 . . .
Rate-time-distance problems require you to know the formula d = rt. That is, distance equals rate times time.
Here are some examples of several types of word problems.

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Example:
Movie tickets are $13.50 for adults and $8 for senior citizens. On Saturday night, a total of 436 adults and senior citizens attended, and the movie theater collected $4,885 from these adults and senior citizens. How many senior citizens were at the movie theater?
Solution:
To solve the problem, you must write two different equations using the data in the question. Let a be the number of
adults and s represent senior citizens.
a + s = 436
You also know that a × $13.50 can be used to find the total amount the movie theater collected for adult tickets and
s × $8 to find the total amount collected for senior citizen tickets. Together these dollar amounts total $4,885.
13.5a + 8s = 4,885
Solve using elimination, by multiplying the first equation by –8 to eliminate one of the variables during addition of
the equations:
(–8)(a + s) = (–8)(436)
–8a + –8s = –3,488
Add the equations:
− 8a + −8 s = −3, 488
+13.5a + 8 s = 4 , 885

421

5.5a = 1, 397
a = 254

Chapter 10

254 adults attended, so 436 – 254 = 182 senior citizens were at the movie theater.

Basic Algebra

Example:
A supermarket places two orders for regular and extra-large packages of paper towels. The first order had 48 regular
and 120 extra-large packages and cost $644.40. The second order had 60 regular and 40 extra-large and cost $338.
What is the difference in cost between a regular and extra-large package of paper towels?
Solution:
Write two equations using the data given in the question. Use r to represent a regular package of paper towels and e
to represent an extra-large package.
48r + 120e = 644.40
60r + 40e = 338
Multiply the bottom equation by –3 to eliminate e:

( −3) (60r + 40e) = ( −3)(338)
−180r − 120e = −1, 014

Peterson’s SAT® Prep Guide 2017

Add the 2 equations:
48r + 120e = 644.4
+ − 180r − 120e = −1, 014
− 132r = −369.6
r = 2.8

The price of a regular package of paper towels is $2.80. To find the price of an extra-large package, substitute 2.8 into
one of the equations:

( 48)(2.8) + 120e = 644.4
120e = 510
e = 4.25
The difference in cost between an extra-large package and a regular package of paper towels is:
$4.25 – $2.80 = $1.45
Example:
It took Andrew 15 minutes to drive downtown at 28 miles per hour to get a pizza. How fast did he have to drive back
in order to be home in 10 minutes?
Solution:

422
Chapter 10

1
of an hour. Hence, going 28 miles per hour, the distance to the pizza parlor can be computed using
4
1
1
 1
the formula d = rt; d = (28)   = 7 miles. Since 10 minutes is of an hour, we have the equation 7 = r . Multiplying
 4
6
6
by 6, r = 42 mph.

15 minutes is

Basic Algebra

Fraction Problems
2
, it does not mean the numerator must be 2 and the
3
denominator 3. The numerator and denominator could be 4 and 6, respectively, or 1 and 1.5, or 30 and 45, or any of infinitely
many combinations. All you know is that the ratio of numerator to denominator will be 2:3. Therefore, the numerator may be
2x
represented by 2x, the denominator by 3x, and the fraction by
.
3x

A fraction is a ratio between two numbers. If the value of a fraction is

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Example:
3
The value of a fraction is . If 3 is subtracted from the numerator and added to the denominator, the value of the
4
2
fraction is . Find the original fraction.
5
Solution:
Let the original fraction be represented by
the new fraction becomes

3x
. If 3 is subtracted from the numerator and added to the denominator,
4x

3x − 3
.
4x + 3

2
We know that the value of the new fraction is .
5
3x − 3 2
=
4x + 3 5
Cross-multiply to eliminate fractions.
15 x − 15 = 8 x + 6
7 x = 21
x =3

Therefore, the original fraction is

3x 9
= .
4 x 12

423
Chapter 10
Basic Algebra

Peterson’s SAT® Prep Guide 2017

EXERCISES: PROBLEM SOLVING IN ALGEBRA
DIRECTIONS: Work out each problem. Circle the letter of your choice.
1.

2.

424

A train with a heavy load travels from Albany to Binghamton at 15 miles per hour. After unloading the load,
it travels back from Binghamton to Albany at 20 miles
per hour. The trip from Albany to Binghamton took 1.5
hours longer than the return trip. Which of the following
equations can be used to calculate the time, t, it took for
the train to go from Albany to Binghamton?
A.

1.5t = 20t

B.

15t = 20t – 30

C.

t = 20t – 30

D.

15t = 20 + 30

If a fleet of m buses uses g gallons of gasoline every two
days, how many gallons of gasoline will be used by 4
buses every five days?

A.

10g
m

B.

10gm

C.

10m
g

D.

20g
m

Chapter 10
Basic Algebra

3.

A faucet is dripping at a constant rate. If, at noon on
Sunday, 3 ounces of water have dripped from the faucet
into a holding tank and, at 5 p.m. on Sunday, a total of
7 ounces have dripped into the tank, how many ounces
will have dripped into the tank by 2:00 a.m. on Monday?
A.

10

B.

51
5

C.

12

D.

71
5

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SHOW YOUR WORK HERE

4.

Prior to the 2016 Summer Olympics, the world record
in 800-meter freestyle swimming was approximately 8
minutes and 15 seconds. Marta is recording her times in
seconds, s, for the 800-meter freestyle competition at her
school. Which expression below could be used to calculate Marta’s time as a percentage of that world record?
A.

s
× 100
8.25

B.

 s

 ÷ 8.15 × 100
60

C.

 s

 ÷ 495 × 100
60

D.

SHOW YOUR WORK HERE

s
× 100
495

425
Chapter 10
Basic Algebra

Peterson’s SAT® Prep Guide 2017

ANSWER KEY AND EXPLANATIONS
1. B

1.

2. A

The correct answer is B. Use the distance formula
d = rt. The train’s rate from Albany to Binghamton is
15mph. Its time can be expressed as t. The train’s rate
from Binghamton to Albany is 20mph. Its time can be
expressed as t – 1.5.

3. D

3.

For the trip from Binghamton to Albany:

7+

d = 20(t – 1.5) = 20t – 30

15t = 20t – 30
2.

426

The correct answer is A. Running m buses for two
days is the same as running one bus for 2m days. If we
g
use g gallons of gasoline, each bus uses
gallons
2m
each day. So if you multiply the number of gallons per
day used by each bus by the number of buses and the
number of days, you should get total gasoline usage.

Chapter 10
Basic Algebra

That is,

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10 g
g
.
× ( 4 )(5) =
2m
m

The correct answer is D. In 5 hours, 4 ounces (7 − 3)
4
have dripped. Therefore, the drip rate is of an ounce
5
per hour.
From 5:00 p.m. on Sunday until 2:00 a.m. on Monday
is 9 hours, which means the total will be:

For the trip from Albany to Binghamton: d = 15t

Since the distance from Albany to Binghamton is
equal to the distance from Binghamton to Albany, we
can set these two expressions equal to each other:

4. D

4.

4
36 71
×9=7 =
5
5
5

The correct answer is D. Convert the world record
of 8 minutes and 15 seconds to seconds by multiplying 8 minutes × 60 sec/min = 480 seconds. Add 15:
480 + 15 = 495 seconds.
s
495
and multiply by 100 to express it as a percent. So the
s
correct answer is
× 100.
495

Divide Marta’s time, s, by the world record, 495:

INEQUALITIES
Algebraic inequality statements are solved just as equations are solved. However, you must remember that whenever you multiply
or divide by a negative number, the order of the inequality, that is, the inequality symbol, must be reversed.
Example:
Solve for x: 3 − 5x > 18
Solution:
Add −3 to both sides:
−5x > 15
Divide by −5, remembering to reverse the inequality:
x < −3
Example:
Solve for x: 5x − 4 > 6x − 6
Solution:
Collect all x terms on the left and numerical terms on the right. As with equations, remember that if a term crosses
the inequality symbol, the term changes sign.

427

−x > −2

Chapter 10

Divide (or multiply) by −1:

Basic Algebra

x<2

Inequality Symbols
> greater than

< less than

≥ greater than or equal to

≤ less than or equal to

You can solve inequalities in the same way you solve linear equations, except in cases where you multiply or divide by a negative
number. In these situations, you must reverse the direction of the inequality symbol.

Peterson’s SAT® Prep Guide 2017

Properties of Inequalities
In the properties below, assume that a, b, and c are real numbers.

Addition Property of Inequality

Subtraction Property of Inequality

If a > b, then a + c > b + c

If a > b, then a – c > b – c

If a < b, then a + c < b + c

If a < b, then a – c < b – c

Multiplication Property of Inequality

Division Property of Inequality

c>0

c>0
If a > b, then ac > bc
If a < b, then ac < bc

If a > b, then

a b
>
c c

If a < b, then

a b
<
c c

c<0

c<0

a b
<
c c
a b
If a < b, then >
c c

If a > b, then
If a > b, then ac < bc
If a < b, then ac > bc

428

These properties are also true for inequalities that include the ≤ and ≥ symbols.
Example:

Chapter 10
Basic Algebra

The meeting room can hold at most 250 people. If 182 people have already been admitted, how many more people
can be allowed into the meeting room?
Solution:
Let x = number of people that can be admitted.
182 + x ≤ 250
182 − 182 + x ≤ 250 − 182
x ≤ 68
There can be at most 68 more people admitted to the meeting room.

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Example:
To stay healthy, an adult’s goal should be to walk at least 10,000 steps each day. A fitness tracker showed that a
person had walked 6,349 steps. What is the minimum number of steps this person can take to reach this goal?
Solution:
Let x = number of steps.
6 , 349 + x ≥ 10 , 000
6 , 349 − 6 , 349 + x ≥ 10 , 000 − 6 , 349
x ≥ 3, 651

The person must walk at least 3,651 more steps.
Example:
Alexander plans on mowing lawns this summer. He charges $20 per lawn. How many lawns will he have to mow in
order to earn at least $575?
Solution:
Let x = number of lawns.
20 x ≥ 575
20 x 575
≥
20
20
x ≥ 28.75

429
Chapter 10
Basic Algebra

Alexander must mow at least 29 lawns to earn at least $575.

Peterson’s SAT® Prep Guide 2017

Use properties of inequality to solve multi-step inequalities.

Example:
The marching band members are making a banner for the pep rally. The banner is in the shape of a rectangle that is
12 ft. long. They have no more than 32 ft. of fringe for the banner. What are the possible widths of the banner?
Solution:
Let w = width of the banner and l = length of the banner. Use the perimeter formula (2l + 2w ≤ P) to find the possible
allowable widths.
2 (12) + 2w ≤ 32
24 + 2w ≤ 32
2w ≤ 8
2w 8
≤
2 2
x≤4
The width of the banner can be 4 ft. or less.
Example:

430
Chapter 10
Basic Algebra

A prepaid cell phone has a balance of $40. Calls are $0.04 per minute, and texts are $0.07 per text. If 342 texts were
used, how many minutes can be used for calls?
Solution:
Let x = number of minutes used on calls and y = number of texts.

0.04 x + 0.07 y ≤ 40

0.04 x + 0.07 (342) ≤ 40
0.04 x + 23.94 ≤ 40
0.04 x ≤ 16.06
0.04 x 16.06
≤
0.04
0.04
x ≤ 401.5

The number of minutes that can be used for calls is 401 or less.

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EXERCISES: INEQUALITIES
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

2.

3.

4.

If 3x – 8 ≥ 4x, then
A.

–8 ≤ x

B.

–8 ≥ x

C.

–8 < x

D.

–8 > x

SHOW YOUR WORK HERE

If –9x + 7 < 43, then
A.

x ≥ –4

B.

x ≤ –4

C.

x > –4

D.

x < –4

The admission to the local carnival is $6. You want to play
games that cost $1.50 per game. If you have $30, how
many games can you play?
A.

At least 24

B.

No more than 20

C.

Less than or equal to 16

D.

Greater than or equal 20

You order a bouquet of flowers that contains roses and
carnations. You have only $48 to spend. Roses cost $3
each, and carnations cost $1.25 each. How many roses
can be in the bouquet if there are 18 carnations in the
bouquet? Let r represent the number of roses.
A.

r≥8

B.

r≤9

C.

r≥9

D.

r≤8

431
Chapter 10
Basic Algebra

Peterson’s SAT® Prep Guide 2017

5.

6.

Bike rentals cost $15 for the first 2 hours and $6 for any
additional hours. Ali wants to bike for more than 2 hours,
but she only has $35. Which inequality represents the
situation, where h is the total number of hours Ali can
bike?
A.

6 +15h ≤ 35

B.

6(h – 2) +15 ≤ 35

C.

6h +15 ≤ 35

D.

6(h + 2) +15 ≤ 35

Selena is adding trim to a rectangular tablecloth that
measures 108 in. long. If she has 330 in. of trim, what is
the greatest possible width of the tablecloth?
A.

At most 54 in.

B.

At most 57 in.

C.

At most 222 in.

D.

At most 114 in.

432
Chapter 10
Basic Algebra

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SHOW YOUR WORK HERE

ANSWER KEY AND EXPLANATIONS
1. B

3. C

5. B

2. C

4. D

6. B

1.

The correct answer is B. Subtracting 3x from both sides
leaves –8 ≥ x.

2.

The correct answer is C.
−9 x + 7 < 43
−9 x < 36
x > −4
By dividing by –9, you reverse the inequality sign.

3.

The correct answer is C.

5.

The correct answer is B. The cost for the first 2 hours is
$15. Subtract the 2 hours from the total number of hours
(h), and multiply by $6 an hour to compute the cost for
the number of hours over 2. Be sure to add the initial $15
for the first 2 hours:
6(h – 2) + 15 ≤ 35

6.

The correct answer is B.

2l + 2w ≤ 330

2 (108) + 2w ≤ 330

6 + 1.5 x ≤ 30
1.5 x ≤ 24
1.5 x 24
≤
1.5 1.5
x ≤ 16
4.

The correct answer is D.
3r + 1.25c ≤ 48

3r + 1.25 (18) ≤ 48

3r + 22.5 ≤ 48
3r ≤ 25.5
3r 25.5
≤
3
3
r ≤ 8.5

216 + 2w ≤ 330
2w ≤ 114
2w 114
≤
2
2
w ≤ 57

The tablecloth can be at most 57 in. wide.

433
Chapter 10
Basic Algebra

There can only be at most 8 roses in the bouquet.

Peterson’s SAT® Prep Guide 2017

SUMMING IT UP
• In complex questions, don’t look for easy solutions.
• Always keep in mind what is being asked.
• Keep the negatives and positives straight when you’re doing polynomial math.
• Don’t be distracted by strange symbols.
• An equation can have one solution, infinitely many solutions, or no solutions.
• Let a, b, and c be real numbers.
ºº If a > b, then a + c > b + c
ºº If a < b, then a + c < b + c
ºº If a > b, then a – c > b – c
ºº If a < b, then a – c < b – c
ºº If c > 0, then if a > b, then ac > bc and if a < b, then ac < bc
ºº If c < 0, then if a > b, then ac < bc and if a < b, then ac > bc

• You can graph a system of linear inequalities in the coordinate plane. The solution of the system is where the graphs of the
inequalities overlap.

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PREP

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Access more practice questions, valuable lessons, helpful tips, and expert strategies for the following basic algebra review topics
in Peterson’s SAT® Online Course:

Chapter 10

• Algebra Strategy

Basic Algebra

• Inequalities
• Linear Equations
• Polynomials
• Quadratics
• Solving Equations
To purchase and access the course, go to www.petersons.com/sat.

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Chapter 11:
Geometry
OVERVIEW
Geometric Notation
Angle Measurement
Intersecting Lines
Area
Circles
Volume
Triangles
Parallel Lines
Coordinate Geometry

435

Exercises: Geometry
Answer Key and Explanations

Chapter 11

Summing It Up

Geometry

GEOMETRIC NOTATION
1.

A point is represented by a dot and denoted by a capital letter.
·P
Point P

2.

A line can be denoted in two different ways. First, a small letter can be placed next to the line. For example, the diagram
below depicts line l. The arrowheads on both ends of the line indicate that lines extend infinitely in both directions.
l

3.

A line can also be denoted by placing a small double-headed arrow over two of its points. The diagram below depicts

line AB.

A

B

Peterson’s SAT® Prep Guide 2017

4.

A line segment is the part of a line between two of its points, which are called the endpoints of the line segment. A line segment
is denoted by placing a small line segment over the two endpoints. The diagram below depicts the line segment AB.

A

5.

B

The length of a line segment is denoted by placing its two endpoints next to each other. In the diagram below, CD = 7.
C

D
7

6.

Two line segments that have the same length are said to be congruent. The symbol for congruence is ≅ . Thus, if AB = 12 and
EF = 12, then AB is congruent to EF , or AB ≅ EF .

7.

A ray is the part of a line beginning at one point, called the endpoint, and extending infinitely in one direction. A ray is denoted
by placing a small one-headed arrow over its endpoint and another point on the ray. The first diagram below depicts the
ray AB , and the second diagram depicts the ray AC .
A

B

A

C

8.

436

Two lines that cross each other are said to intersect. Two lines that do not intersect are said to be parallel. The symbol || is
used to represent parallel lines. In the diagrams below, line k intersects line l at point P, while lines m and n are parallel, that
is, m || n.
k

Chapter 11
Geometry
P

l

m

n

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ANGLE MEASUREMENT
1.

When two rays share a common endpoint, they form angles. The point at which the rays intersect is called the vertex of the
angle, and the rays themselves are called the sides of the angle.
Side

Vertex

2.

Side

The symbol for angle is ∠. Angles can be denoted in several different ways, as shown in the diagram below. The most common
way to denote an angle is to name a point on one side, then the vertex, and then a point on the other side as shown in the
diagram.
Angles can also be denoted by writing a letter or a number within the angle, as shown below.
If there is no ambiguity, an angle can be named by simply naming the vertex.

A

B

437

C
ABC or CBA

Chapter 11
Geometry
2

a
a

3.

2

A

A

The size of an angle is measured in degrees. The symbol for degree is °. A full circle contains 360°, and all other angles can
be measured as a fractional part of a full circle. Typically, the measure of an angle is written in the interior of the angle, near
the vertex.

120°

60°

30°

Peterson’s SAT® Prep Guide 2017

4.

A straight angle is an angle that measures 180°.

180°

5.

A right angle is an angle that measures 90°. Note, as shown in the diagram below, a “box” is used to represent a right angle.

90°

438

6.

The measure of an angle is denoted by writing the letter m followed by the name of the angle. For example, m∠ABC = 45°
tells that angle ABC has a measure of 45°.

7.

Two angles that have the same number of degrees are said to be congruent. Thus, if m∠P = m∠Q, then ∠P ≅ ∠Q.

8.

Two angles whose measures add up to 180° are said to be supplementary. Two angles whose measures add up to 90° are
said to be complementary.

Chapter 11
Geometry
E
G

C

A

B

D

F

ABC and CBD
are supplementary.

9.

To convert degrees to radians, multiply the degree measure by

www.petersons.com

H

EFG and GFH
are complementary.

p radians .
180°

Example:
Convert 85° to radians:
Solution:
 p radians  17p
85° = 85 
=
 180°  36

To convert from radians to degrees, multiply the radian measure by

180°

p radians

.

Example:
Convert

π
radians to degrees:
6

Solution:

p  p radians   180° 
=
= 30°
 
6 
6
p radians 

439

INTERSECTING LINES
1.

When two lines intersect, four angles are formed. The angles opposite each other are congruent.

Chapter 11
Geometry

∠1 ≅ ∠3 and ∠2 ≅ ∠4

1
2

4
3

Peterson’s SAT® Prep Guide 2017

2.

When two lines intersect, the angles adjacent to each other are supplementary.
m∠5 + m∠6 = 180°
m∠6 + m∠7 = 180°
m∠7 + m∠8 = 180°
m∠ 8 + m∠5 = 180°
8

5

6

7

If you know the measure of any one of the four angles formed when two lines intersect, you can determine the measures of
the other three. For example, if m∠1 = 45°, then m∠3 = 45°, and m∠2 = m∠4 = 180° − 45° = 135°.
3.

Two lines that intersect at right angles are said to be perpendicular. In the figure below, AB is perpendicular to CD . This can
be denoted as AB ⊥ CD .

C

440
Chapter 11

B

A

Geometry

D

Note that all four of the angles in the diagram above are right angles.

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AREA
Many of the geometry problems you will encounter on the SAT® will ask about the perimeter or area of a polygon. A polygon is
a closed plane figure that consists of straight line segments called sides. The perimeter of a closed plane figure is the distance
around the figure and is computed by adding all the lengths of the segments/sides that form its outer boundary. The area of a
two-dimensional plane figure is the number of unit squares needed to cover it. The units of area measure are square inches, square
feet, square yards, square centimeters, square meters, and so on. Following are useful area formulas for common plane figures:

1.

Rectangle = bh

3

Area = 6 i 3 = 18

2.

6

Parallelogram = bh

5

4

Area = 8 i 4 = 32
8

3.

Square = s2
Area = 62 = 36

6

441
Chapter 11
Geometry

1
Square = d 2 (d = diagonal)
2
1
Area = (10)(10) = 50
2

4.

Triangle =
Area =

1
bh
2

1
(12)( 4) = 24
2

10

4

12

Peterson’s SAT® Prep Guide 2017

6

5.

Trapezoid = 1 h (b1 + b2 )
2
1
Area = (5)(16) = 40
2

5

10

CIRCLES
1.

A circle is a closed flat figure formed by a set of points all of which are the same distance from a point called the center. The
boundary of the circle is called the circumference, and the distance from the center to any point on the circumference is
called the radius. A circle is denoted by naming the point at its center, that is, the circle whose center is at point P is called
circle P.

P

2.

442

A diameter of a circle is a line segment that passes through the center of the circle, and whose endpoints lie on the circle.
The diameter of a circle is twice as long as its radius. Typically, the letter r is used to represent the radius of a circle, and the
letter d is used to represent the diameter.
2r = d

r

Chapter 11
d

Geometry

3.

A chord of a circle is a line segment both of whose endpoints lie on the circumference of the circle. The chords of a circle
have different lengths, and the length of the longest chord is equal to the diameter.
AB, CD, and EF are chords of circle O. EF is also a diameter.
E

B

A

O

F D

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C

4.

A tangent is a line that intersects the circle at exactly one point. A radius drawn to the point of intersection is perpendicular
to the tangent line.
OQ ⊥ CD

O

C

5.

Q

D

A central angle is an angle that is formed by two radii of a circle. As the diagram below shows, the vertex of a central angle
is the center of the circle.

A

O
B

443

Central angle ∠ AOB

6.

A central angle is equal in degrees to the measure of the arc that it intercepts. That is, a 40° central angle intercepts a 40° arc,
and a 90° central angle intercepts a 90° arc.

Chapter 11
Geometry

90°
40°

40°

Peterson’s SAT® Prep Guide 2017

7.

An arc is a piece of the circumference of a circle. The symbol  placed on top of the two endpoints is used to denote an
 is indicated in the figure below.
arc. For example, MN
M


MN

N

8.

The measure of an arc is in degrees. The arc’s length depends on the size of the circle because it represents a fraction of the
circumference.


Length of AB


mAB
2pr
360

If the radius of a circle is 5 cm and the measure of the arc is 45°, then the length of the arc is
45°
5p
i 2p (5) =
cm
360°
4
= 1.25p cm
≈ 3.93 cm

444
Chapter 11
Geometry

9.

The area of a circle is measured using the formula pr 2. If the diameter of the circle is 12, its radius is 6:
Area of a circle = πr2
12

Area = π(6)2 = 36π

10. The area of a sector of a circle is the product of the ratio measure of the arc and the area of the circle. If the radius of a circle
360°
is 3 cm and the measure of the arc is 60°, then the area of the sector is
Area of sector AOB =


mAB
• pr 2
360

60°
i
360°

p (3)2 =

9 p 3p 2
=
cm .
6
2

A

O

60◦
3

B

When an SAT® exam problem asks you to find the area of a shaded region, you probably
won’t be able to calculate it directly. That’s OK— instead, think of the shaded region as being
whatever is left over when one region is subtracted or removed from a larger region. Use
the formulas you know to find those two regions and perform the necessary subtraction.

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 measure of the central angle 
11. The length of a chord of a circle is the product of 2r and sin 
 . If the radius of a circle is

2
 60° 
4 inches and the measure of the central angle is 60°, then the length of the chord is 2 ( 4 ) sin 
= 8 sin (30°) = 4 inches.
 2 
(In a right-angled triangle, the sine of an angle is the length of the opposite side divided by the length of the hypotenuse.
The abbreviation is sin.)

VOLUME
1.

The volume of a right rectangular prism is equal to the product of its length, width, and height.
V = lwh				

4

V = (6)(2)(4) = 48

2
6

2.

The volume of a cube is equal to the cube of an edge.
V = e3				
V = (5)3 = 125

445

5

Chapter 11
Geometry

5
5

3.

The volume of a right circular cylinder is equal to π times the square of the radius of the base times the height.
V = πr 2h
V = π(5)2(3) = 75π

3
5

Peterson’s SAT® Prep Guide 2017

4.

The volume of a sphere is equal to

4
p times the cube of the radius.
3

4 3
pr
3
4
3 4
v = p (3) = p (27) = 36 p
3
3

V=

5.

3

The volume of a right square pyramid is equal to

1
times the product of the square of the side and the height.
3

1
V = s2h
3
1 2
V = ( 2) ( 6 )
3
V =8

6

2
6.

1
3

The volume of a right circular cone is equal to p times the product of the square of the radius and the height.
1 2
pr h
3
1
2
V = p (6) (9)
3
V = 108 p
V=

446

1
9

Chapter 11
Geometry

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6

TRIANGLES
1.

A triangle is a polygon with three sides. A vertex of a triangle is a point at which two of its sides meet. The symbol for a triangle
is ∆, and a triangle can be named by writing its three vertices in any order.

∆ ABC contains sides AB , BC, and AC, and angles ∠A, ∠B, and ∠C.
B

A

C

2.

The sum of the measures of the angles in a triangle is 180°. Therefore, if the measures of any two of the angles in a triangle
are known, the measure of the third angle can be determined.

3.

In any triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle. In the
triangle below, if a° > b° > c°, then BC > AC > AB.
c°

C

447
a°

b°

B

Chapter 11

A

Geometry

4.

If two sides of a triangle are congruent, the angles opposite these sides are also congruent.
If AB ≅ AC , then ∠B ≅ ∠C.

A

B

C

Peterson’s SAT® Prep Guide 2017

5.

If two angles of a triangle are congruent, the sides opposite these angles are also congruent.
If ∠B ≅ ∠C, then AB ≅ AC .

A

C

B
6.

In the following diagram, ∠1 is called an exterior angle. The measure of an exterior angle of a triangle is equal to the sum of
the measures of the two remote interior angles, that is, the two interior angles that are the farthest away from the exterior
angle.
A

m ∠1 = 115°

B

448
Chapter 11
Geometry

70°

7.

1

45°

C

D

If two angles of one triangle are congruent to two angles of a second triangle, the third angles are also congruent.
∠D ≅ ∠A
D

A

B

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C

E

F

Right Triangles
1.

Pythagorean theorem: The Pythagorean theorem states that the square of a hypotenuse (the leg opposite the right angle)
of a right triangle is equal to the sum of the squares of the other two legs a2 + b2 = c2.

(leg)2 + (leg)2 = (hypotenuse)2
4 2 + 52 = x 2
16 + 25 = x 2
41 = x 2
41 = x
5

x

4

Remember that in a right triangle, you can use the two legs (the two sides
other than the hypotenuse) as base and altitude.

2.

Pythagorean triples: These are sets of whole numbers that satisfy the Pythagorean theorem. When a given set of numbers,
such as 3-4-5, forms a Pythagorean triple (32 + 42 = 52), any multiples of this set, such as 6-8-10 or 15-20-25, also form a
Pythagorean triple. The most common Pythagorean triples that should be memorized are:

449
Chapter 11
Geometry

3-4-5
5-12-13
8-15-17
7-24-25

39

x

15

Peterson’s SAT® Prep Guide 2017

Squaring the numbers 15 and 39 in order to apply the Pythagorean theorem would take too much time. Instead, recognize
the hypotenuse as 3(13). Suspect a 5-12-13 triangle. Since the given leg is 3(5), the missing leg must be 3(12), or 36, with no
computation and a great saving of time.

Remember the well-known Pythagorean triples, that is, sets of whole numbers such as
3-4-5 for which a2 + b2 = c2. (Also watch for their multiples, such as 6-8-10 or 15-20-25.)
On the SAT®, you’ll encounter right triangles with sides whose lengths correspond to
these values. Other Pythagorean triples are: 5-12-13, 8-15-17, and 7-24-25.

3.

The 30°-60°-90° triangle is a special right triangle whose sides are in the ratio of x : 3 : 2 x.
1
i hypotenuse .
2
1
b. The leg opposite the 60° angle is i hypotenuse i 3 .
2
a. The leg opposite the 30° angle is

c. An altitude in an equilateral triangle forms a 30°–60°–90° triangle and is therefore equal to

30°

8
x

60°

4.

Geometry

30°

30°

10

5

x=4

450
Chapter 11

y

60°

8

z

1
i hypotenuse i
2

x

60°

The 45°–45°–90° triangle (isosceles right triangle) is a special right triangle whose sides are in the ratio of x: x : 2
a. Each leg is

1
i hypotenuse i
2

b. The hypotenuse is leg i

2.

2.

c. The diagonal in a square forms a 45°–45°–90° triangle and is therefore equal to the length of one side i

45°

45°

3 2

8

w

45°

x

45°
w=6

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2.

y

45°
x=8 2

6

10

z

45°
y=5 2

z=6 2

3.

5.

One way to solve SAT® questions that involve 30°–60°–90° triangles and 45°–45°–90° triangles is to use right triangle
trigonometric relationships. In the triangle below, side AB is called the side adjacent to ∠A, side BC is called the side
opposite ∠A, and side AC is the hypotenuse. Relative to ∠A, the trigonometric ratios sine, cosine, and tangent are defined as
shown.

opposite
sin e a° = sin a° =
hypotenuse
adjacent
cosine a° = cos a° =
hypotenuse
opposite
tangent a° = tan a° =
adjacent

C

		

p

Hy

se

nu

e
ot

Opposite

a°

A

B

Adjacent

The table below shows the values of the sine, cosine, and tangent for 30°, 45°, and 60°.

Angle a

sin a°

cos a°

tan a°

30°

1
2

3
2

3
3

45°

2
2

2
2

1

60°

3
2

1
2

3

451
Chapter 11

By using the values of sine, cosine, and tangent above, problems involving 30°–60°–90° triangles and 45°–45°–90° triangles
can also be solved. For example, consider the 30°-60°–90° triangle with hypotenuse of length 8. The computations that follow
show how to determine the lengths of the other two sides.

Geometry

Peterson’s SAT® Prep Guide 2017

30°

sine 60° =
or y =

8

y

opposite
y
y
3
= . Also, since , then =
. Cross-multiply to get 2 y = 8 3 ,
hypotenuse 8
8
2

8 3
= 4 3. Note that this is the same answer that would be obtained using the
2

properties of 30°-60°-90° triangles.
cos 60° =

1
adjacent
x
x 1
= . Also, since cos 60°= , then = . Cross-multiply to get
2
hypotenuse 8
8 2

2x = 8, or x = 4. This, again, is the same answer that would be obtained using the
60°

x

properties of 30°–60°–90° triangles above.

Here are two sample problems:

Example:
Suppose a building has a handicap ramp that rises 2.5 ft. and forms a 3° angle with the ground. How far from the
base of the building is the start of the ramp? Round your answer to the nearest tenth.
Solution:

452
Chapter 11
Geometry

www.petersons.com

2.5
x
2.5
x=
tan 3°
x = 47.7 feet

tan 3°=

Example:
Suppose a 26-ft. wire is attached to the top of a pole and staked in the ground at a 40-degree angle. Estimate the
height of the pole.

B

A

40°

C

Solution:
x
26
26 sin 40° = x
16.7 ft. ≈ x
sin 40° =

To find how far the stake is from the ground, you can use the Pythagorean theorem:
a2 + b 2 = c 2
16.72 + b 2 = 262
19.9 ft. ≈ x

453

PARALLEL LINES

Chapter 11
Geometry

1.

If two parallel lines are cut by a transversal, the alternate interior angles are congruent.
If AB ⊥ CD , then
1 2

A

∠ 1 ≅ ∠3, and
∠2 ≅ ∠4.

B

4 3
C

2.

D

If two parallel lines are cut by a transversal, the corresponding angles are congruent.
If AB || CD , then
∠ 1 ≅ ∠ 5,

A

∠ 2 ≅ ∠ 6,
∠ 3 ≅ ∠ 7, and

C

1 2
4 3

5 6
8 7

B
D

∠ 4 ≅ ∠ 8.

Peterson’s SAT® Prep Guide 2017

3.

If two parallel lines are cut by a transversal, interior angles on the same side of the transversal are supplementary.

If AB || CD , then
1

A

∠ 1 is supplementary to ∠ 4, and
∠ 2 is supplementary to∠ 3.

4

2

B

3

C

D

COORDINATE GEOMETRY
1.

Lines and other geometric figures can be positioned on a plane by means of the rectangular coordinate system. The rectangular coordinate system consists of two number lines that are perpendicular and cross each other at their origins (0 on each
of the number lines). The horizontal number line is called the x-axis, and the vertical number line is called the y-axis.
y
6
5
4
3
2

454

–6 –5 –4 –3 –2 –1
–1
–2

Chapter 11

–3

Geometry

–4
–5
–6

www.petersons.com

x

1
1

2

3

4

5

6

2.

Any point on the plane can be designated by a pair of numbers. The first number is called the x-coordinate and indicates how
far to move to the left (negative) or to the right (positive) on the x-axis, and the second number is called the y-coordinate
and tells how far to move up (positive) or down (negative) on the y-axis. Generically, a point on the plane can be written as
(x, y). When two points need to be expressed generically, they are typically written as (x1, y1) and (x2, y2).
The points (2, 3), (−4, 1), (−5, −2), and (2 –4) are graphed on a coordinate system as shown below.
y
6
5
4

( 2, 3 )

3
2

( – 4, 1 )

1

–6 –5 –4 –3 –2 –1
–1
–2
( – 5, – 2 )
–3

x
1

–4

2

3

4

5

6

( 2, – 4)

–5
–6

3.

The slope of a straight line is a number that measures how steep the line is. Traditionally, the variable m is used to stand for
the slope of a line. By convention, a line that increases from left to right has a positive slope, and a line that decreases from
left to right has a negative slope. A horizontal line has a slope of 0 since it is “flat,” and a vertical line has an undefined slope.

455
Chapter 11
Geometry

y

y

x

Positive Slope

4.

y

x

Negative Slope

y

x

Zero Slope

If (x1, y1) and (x2, y2) are any two points on a line, the slope is given by the formula m =

x

Undefined Slope

( y 2 − y1 ) .
( x 2 − x1 )

3
7−4 3
= . A slope of represents
2
5−3 2
the fact that for every 2 units moved horizontally along the x-axis, the line rises vertically 3 units.

Therefore, for example, if a line contains the points (5, 7) and (3, 4), the slope would be m =

5.

An equation of degree one that contains the variables x and/or y raised to the first power, but no higher, will always have a
straight line as its graph. A very convenient way to write the equation of a line is in the slope-intercept form, y = mx + b. In
this form, m represents the slope of the line, and b is the y-intercept, that is, the point where the graph crosses the y-axis.

Peterson’s SAT® Prep Guide 2017

Example:
Consider the line represented by the equation 2x + 5y = 12.
Solution:
Begin by writing this equation in slope-intercept form.
2 x + 5 y = 12
5 y = −2 x + 12

Subtract 2x from both sides.
Divide both sides by 5.

2
12
y=− x+
5
5

2
12
Therefore, the slope of the line is − , and the y-intercept is . Here is the graph of this line.
5
5
y
6
5
4
3
2

456

1
–6

–5

–4

–3

–2

–1–1
–2

Chapter 11
Geometry

–3
–4
–5
–6

www.petersons.com

x
1

2

3

4

5

6

The following are sample problems:
Example:
Write the equation of the line containing the point (2, 1) and having slope 5.
Solution:
Begin by taking the slope-intercept form y = mx + b, and substituting m = 5, to obtain y = 5x + b. To determine the
value of the y-intercept b, substitute the coordinates of the point (2, 1) into the equation.
y = 5 x + b Substitute (2,1).
1 = 5(2) + b Solve for b.
1 = 10 + b
−9 = b
Therefore, the equation of the line is y = 5x − 9.
Example:
Graph the function f(x) = 2x + 5.
Solution:
Begin by recognizing that the slope is 2 and the y-intercept is (0, 5). To graph this function, first graph the point (0, 5).
Then move up 2 units and then to the right 1 unit. This location is (1, 7). Starting at (0, 5) again, go down 2 units and
then to the left 1 unit. This location is (–1, 3). Connect these points to form the graph of the function f(x) = 2x + 5.

457
Chapter 11
Geometry

7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
–1
–2
–3
–4
–5
–6
–7

Peterson’s SAT® Prep Guide 2017

You can write an equation of a line from a graph given any two points on the line. First, use the two points to find the
slope. Then use the point–slope form of an equation of a line: y – y1 = m(x – x1). As an example, consider the graph
shown below.
y

4
(0, 3)

2
(–1, 1)

–4

–2

0

2

4

x

–2

–4

You need to first find the slope using two points on the line.
The slope formula is m =

458
Chapter 11

y 2 − y1
. Use the points (0, 3) and (–1, 1) to find the slope.
x 2 − x1
m=

y 2 − y1 1− 3 −2
=
=
=2
x 2 − x1 −1− 0 −1

Use the point–slope form and either given point.
y − y 1 = m ( x − x 1)

Geometry

y − 1 = 2 ( x − ( −1))
y − 1 = 2 ( x + 1)

To write this equation in slope–intercept form, solve for y.
y − 1 = 2 ( x + 1)

y = 2 ( x + 1) + 1
y = 2x + 2 +1
y = 2x + 3

Here is a sample question of how to write an equation of a line from a word problem and then graph the equation
using the intercepts.

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Example:
You decide to purchase holiday gift cards for your family. You have $300 to spend on the cards, and you purchase
cards for either $20 or $30. What are three combinations of cards that you can purchase?
Solution:
To write an equation that represents this situation, first define your variables.
Let x = number of $20 gift cards purchased and y = number of $30 gift cards purchased.
Now write an equation to represent the situation.
20x + 30y = 300
Use the intercepts to draw the graph.
20 x + 30 y = 300

20 x + 30 y = 300

20 (0) + 30 y = 300

20 x + 30 (0) = 300

30 y = 300
y = 10

20 x = 300
x = 15

11

# of $30

10
9

459

8
7

Chapter 11

6

Geometry

5
4
3
2
1
1

2

3

4

5

6

7

8

9

10

11

12

13

# of $20

You cannot purchase a fraction of a card, so only the integer combinations can be solutions. You can purchase fifteen $20 cards
and zero $30 cards, six $20 cards and six $30 cards, or zero $20 cards and ten $30 cards. (Although the question only asked for
three answers, the other possible combinations are twelve $20 gift cards and two $30 gift cards, nine $20 gift cards and four $30
gift cards, and three $20 gift cards and eight $30 gift cards.)

Peterson’s SAT® Prep Guide 2017

6.

The standard form of the equation of a circle with center at (h, k) and radius r is (x – h)2 + (y – k)2 = r2. For example, to graph
y2 = –x2 + 64, first rewrite the equation in standard form: x2 + y2 = 64. Then identify the radius, r = 8, and the center, (0, 0).
Now plot points that are on the circle: (0, 8), (0, –8), (–8, 0), and (8, 0). Connect the points to draw the circle.
y
10
8
6
4
2
–10

–8

–6

–4

–2

2

4

6

8

10

x

–2
–4
–6
–8
–10

To graph (x – 1)2 + (y + 2)2 = 4, first identify the radius, r = 2, and the center, (1, –2). Next, plot points that are on the circle. The
easiest way to do this is to find points 2 units above, below, left, and right of the center: (1, 0), (1, –4), (–1, –2), and (3, –2). Connect
the points to draw the circle.

460

y
4

Chapter 11

3

Geometry

2
1
–4

–3

–2

–1

1
–1
–2
–3
–4

www.petersons.com

2

3

4

x

7.

Parallel lines have the same slope. Therefore, one way to tell whether two lines are parallel or not is to write them in slope-intercept form and compare the slopes. To write the equation of the line that is parallel to the line y = 3x + 7 and contains
the point (5, 2), begin by noting that the equation of the line we are looking for must have a slope of 3, just like the line
y = 3x + 7. Thus, it must be of the form y = 3x + b.

y = 3x + b
y = 3x + b
2 = 3(5) + b
2 = 15 + b
−13 = b

Substitute (5, 2).
Solve for b.

Therefore, the equation of the line is y = 3x − 13.
8.

a
, then the slope of
b
5
b
2
the perpendicular line would be − . Thus, the line perpendicular to the line with slope
would have a slope of − .
2
a
5
The slopes of perpendicular lines are negative reciprocals of each other. That is, if a line has a slope of

To write the equation of the line that is perpendicular to the line y =

1
x − 7 and contains the point (4, −3), begin by
2

noting that the equation of the line to be determined has a slope of −2. Thus, the equation must be of the form
y = −2x + b.
y = −2 x + b

Substitute ( 4 , −3).

−3 = −2 ( 4 ) + b Solve for b.
−3 = −8 + b
b=5

461

Therefore, the equation of the line is y = −2x + 5.
Chapter 11
Geometry

Peterson’s SAT® Prep Guide 2017

EXERCISES: GEOMETRY
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

2.

3.

If the angles of a triangle are in the ratio 2:3:7, what is the
measure of the smallest angle?
A.

15°

B.

30°

C.

45°

D.

105°

What is the volume of a square pyramid with height 34.2
feet and base edges of 25 feet?
A.

285 ft.3

B.

2,565 ft.3

C.

7,125 ft.3

D.

81,225 ft.3

If the radius of a circle is decreased by 10%, by what
percent is its area decreased?

462

A.

10

B.

19

Chapter 11

C.

21

Geometry

D.

81

4.

5.

A spotlight is mounted on the ceiling 5 feet from one wall
of a room and 10 feet from the adjacent wall. How many
feet is it from the intersection of the two walls?
A.

15

B.

5 2

C.

5 5

D.

10 2

A sphere has a diameter of 3 cm, and a cone has a height
of 10 cm and a radius of 1.5 cm. How much bigger is the
volume of the cone than the volume of the sphere?
A.

0.5π

B.

π

C.

3π

D.

4.5π

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SHOW YOUR WORK HERE

6.

7.

In parallelogram PQRS, angle P is 4 times angle Q. What
is the measure in degrees of angle P?
A.

36

B.

72

C.

125

D.

144

SHOW YOUR WORK HERE

If PQ ≅ QS , QR ≅ RS and the measure of angle
PRS = 100°, what is the measure, in degrees, of
angle QPS?
S

P

Q

R

Note: Figure not drawn to scale

8.

9.

A.

10

B.

15

C.

20

D.

25

A circle has a radius of 6 cm. What is the area of a sector
bounded by a 60° minor arc?
A.

p

B.

2p

C.

6p

D.

20p

463
Chapter 11
Geometry

A rectangular box with a square base contains 6 cubic
feet. If the height of the box is 18 inches, how many feet
are there in each side of the base?
A.

1

B.

2

C.

3

D.

3
3

Peterson’s SAT® Prep Guide 2017

10.

11.

12.

464

The surface area of a cube is 150 square feet. How many
cubic feet are there in the volume of the cube?
A.

30

B.

50

C.

100

D.

125

Peter lives 12 miles west of his school, and Bill lives north
of the school. Peter finds that the direct distance from his
house to Bill’s is 6 miles shorter than the distance by way
of his school. How many miles north of the school does
Bill live?
A.

6

B.

9

C.

10

D.

6 2

A square is inscribed in a circle of area 18 p. Find a side of
the square.
A.

3

B.

6

C.

3 2

D.

6 2

Chapter 11
Geometry

13.

14.

A carpet is y yards long and f feet wide. How many dollars
will it cost if the carpet sells for x cents per square foot?
A.

3xyf

B.

xyf
3

C.

0.03 yf
x

D.

0.03xyf

If a triangle of base 6 has the same area as a circle of
radius 6, what is the altitude of the triangle?
A.

6p

B.

8p

C.

10p

D.

12p

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SHOW YOUR WORK HERE

15.

16.

17.

18.

The vertex angle of an isosceles triangle is p°. How many
degrees are there in one of the base angles?
A.

180 − p

B.

90 −

C.

180 − 2p

D.

180 −

SHOW YOUR WORK HERE

p
2
p
2

In a circle with center O, the measure of arc RS = 132°.
How many degrees are there in angle RSO?
A.

66°

B.

48°

C.

24°

D.

22°

The ice compartment of a refrigerator is 8 inches long,
4 inches wide, and 5 inches high. How many ice cubes will
it hold if each cube is 2 inches on an edge?
A.

8

B.

10

C.

12

D.

16

465

In the figure, PQ is a straight line and RS is perpendicular to ST. If the measure of angle RSQ = 48°, how many

Chapter 11
Geometry

degrees are there in angle PST?

R

P

S

48°

Q
T

A.

48°

B.

90°

C.

132°

D.

138°

Peterson’s SAT® Prep Guide 2017

19.

A cylindrical pail has a radius of 7 inches and a height of
9 inches. If there are 231 cubic inches to a gallon, approximately how many gallons will this pail hold?

B.

12
7
6

C.

7.5

D.

8.2

A.

20.

In triangle PQR, QS and SR are angle bisectors (meaning
∠ PQS is congruent to ∠ SQR, and ∠ PRS is congruent
to ∠ SRQ), and the measure of angle P = 80°. How many
degrees are there in angle QSR?

P
80°

S

R

Q

466
Chapter 11
Geometry

21.

A.

115°

B.

120°

C.

125°

D.

130°

If the m∠a is six less than twice the m∠b, what is the
value of a?

O
a°

A
A.

28

B.

32

C.

48

D.

58

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b°

B

SHOW YOUR WORK HERE

22.

23.

24.

25.

One end of a wire is attached to the top of a 70-foothigh pole, and the other end is attached to a stake in the
ground. If the wire makes a 30° angle with the ground,
how long is the wire?
A.

70 feet

B.

70 3 feet

C.

140 feet

D.

140 2 feet

SHOW YOUR WORK HERE

What is the equation of the line containing the points
(4, 6) and (3, 8)?
A.

y = –2x + 14

B.

y = 2x + 14

C.

y = –2x + 2

D.

y = 2x – 2

Which of the following represents the equation of a line
with a slope of −7 and a y-intercept of 12?
A.

y = 7x − 12

B.

y = −7x + 12

C.

y = 7x + 12

D.

y = 12x − 7

Which of the following represents the equation of a line
parallel to the line y = 7x – 6 and containing the point
(1, 7)?
A.

y = 7x

B.

y = 7x − 7

C.

y = −7x + 7

D.

y = −7x

467
Chapter 11
Geometry

Peterson’s SAT® Prep Guide 2017

ANSWER KEY AND EXPLANATIONS

1.

1. B

6. D

11. B

16. C

21. D

2. C

7. C

12. B

17. D

22. C

3. B

8. C

13. D

18. D

23. A

4. C

9. B

14. D

19. B

24. B

5. C

10. D

15. B

20. D

25. A

The correct answer is B. Represent the angles as 2x,
3x, and 7x.

5.

2 x + 3 x + 7 x = 180°
12 x = 180°
x = 15°

V=

Chapter 11

V=

The correct answer is C.
1
V = s 2h
3
1
2
V = (25) (34.2)
3

468

V = 7,125 ft. 3

3.

Geometry

4.

The correct answer is B. If the radii of the two circles
have a ratio of 10:9, the areas have a ratio of 100:81.
Therefore, the decrease is 19 out of 100, or 19%.
The correct answer is C.
5 2 + 10 2 = x 2
25 + 100 = x 2
x 2 = 125
x = 125 = 25 5 = 5 5

www.petersons.com

4
p (1.5) 3 = 4.5p
3

The volume of the cone is calculated using the
1
formula V = pr 2h :
3

The angles are 30°, 45°, and 105°. The smallest angle
is 30°.
2.

The correct answer is C. The volume of the sphere is
4
calculated using the formula V = pr 3 :
3

1
p (1.5) 2 (10) = 7.5p
3

Since 7.5p − 4.5p = 3p, the volume of the cone is 3p
bigger than the volume of the sphere.
6.

The correct answer is D. The consecutive angles of a
parallelogram are supplementary, so
x + 4 x = 180°
5 x = 180°
x = 36°
Angle P is 4(36), or 144°.

7.

The correct answer is C.

11.

The correct answer is B.

S
Bill

40°

x

P

140° 40°

100°

Q

R

Since QR ≅ RS, ∠RQS ≅ ∠RSQ. There are 80° left in
the triangle, so each of these angles is 40°.

Peter

School

12

∠SQP is supplementary to ∠SQR, making it 140°.
The direct distance from Peter’s house to Bill’s can be
represented by means of the Pythagorean theorem

Since QP ≅ QS, ∠QPS ≅ ∠QSP. There are 40° left in
the triangle, so each of these angles is 20°.
8.

as

The correct answer is C. The area of a sector of a circle
measure of the arc
and the
is the product of the ratio
360°
area of the circle. Since the radius of the circle is 6 cm

144 + x 2 = (12 + x ) − 6
144 + x 2 = x + 6
Square both sides.

and the measure of the arc is 60°, the area of the sector
60°
2
is
i p (6) = 6 p.
360°
9.

144 + x 2 = x 2 + 12 x + 36
144 = 12 x + 36
108 = 12 x
9= x

The correct answer is B. Change 18 inches to 1.5 feet.
Letting each side of the base be x, the volume is 1.5x2.
1.5 x 2 = 6
15 x 2 = 60

144 + x 2 . Then:

12.

The correct answer is B.

Chapter 11

2

x =4
x =2

10.

469
Geometry

The correct answer is D. The surface area of a cube is
made up of 6 equal squares. If each edge of the cube is
x, then
6 x 2 = 150
x 2 = 25
x =5

Volume = (edge)3 = 53 = 125
The diagonal of the square will be a diameter of the
circle.

pr 2 = 18p
r 2 = 18
r = 18 = 9 2 = 3 2

The diameter is 6 2 and, since the triangles are
45°–45°–90°, a side of the square is 6.

Peterson’s SAT® Prep Guide 2017

13.

The correct answer is D. To find the area in square
feet, change y yards to 3y feet. The area is then (3y)(f),
or 3yf square feet. If each square foot costs x cents,
change this to dollars by dividing x by 100. Thus, each
square foot costs x dollars. The cost of 3yf square
100
feet will then be

17.

The correct answer is D. The compartment will hold 2
layers, each of which contains 2 rows of 4 cubes each.
This leaves a height of 1 inch on top empty. Therefore,
the compartment can hold 16 cubes.

3 xyf
3
. Since
= 0.03, the correct
100
100
5"

answer is choice D.
14.

The correct answer is D. The area of the circle is (6)2 p ,
or 36 p . In the triangle,
18.

1
(6)(h) = 36p
2
3h = 36 p
h = 12p

15.

16.

8"

48°

138°

Q

42°

T
Since ∠RST is a right angle, 42° are left for ∠QST.
Since ∠PSQ is a straight angle of 180°, ∠PST contains
138°.

°

132

S

19.
R
O

Geometry

S

P

The correct answer is C.

Chapter 11

The correct answer is D.

R

The correct answer is B. There are (180 − p) degrees
left, which must be divided between 2 congruent
p
180 − p
angles. Each angle will contain
, or 90 −
2
2
degrees.

470

4"

The correct answer is B. The volume of the pail is
found using the formula V = pr 2h. Since the answers
are not in terms of p, it is best to use 22 as a value for
7
p because the 7 will divide out r  2:
V=

P

22 7
i 49 i 9
7

Rather than multiply this out, which will take unnecessary time, divide by 231 and cancel wherever
possible.

By extending SO until it hits the circle at P, arc PRS is
a semicircle. Therefore, the measure of arc PR = 48°,
and the measure of the inscribed angle RSO = 24°.

2
3
22 i 7 i 9
=6
231

21
3

20.

www.petersons.com

The correct answer is D. If m ∠P = 80°, there are 100°
left between ∠PQR and ∠PRQ. If they are both
bisected, there will be 50° between ∠SQR and ∠SRQ,
leaving 130° for ∠QSR.

21.

The correct answer is D. Since the sum of the three
angles in a triangle is 180°, the measures of the missing
angles must add to 90°. The m ∠a = 2(m∠b) – 6, so

23.

line is m =

( y 2 − y 1) , where (x , y ) and (x , y ) are any two
1 1
2 2
( x 2 − x 1)

points on the line. In this problem, the two points are
( y 2 − y 1) = 8 − 6 = 2 = −2
.
(4, 6) and (3, 8), so m =
( x 2 − x 1) 3 − 4 −1

b + (2b − 6 ) = 90
3b − 6 = 90
3b = 96
b = 32 .

Use either point and the slope to complete the
equation.

Then, to find the m ∠a:
a = 2(32) − 6
a = 58
22.

The correct answer is A. The formula for the slope of a

y = mx + b

6 = ( −2)( 4 ) + b
6 = −8 + b
14 = b

The correct answer is C. Begin by making a diagram
of the situation described in the problem.

So the equation of the line is y = –2x + 14.

70ft.

w

30°

Note: Figure not drawn to scale.

There are two ways to solve this problem. The first is
to notice that the triangle is a 30°-60°-90° triangle,
and in such a triangle, the length of the side opposite the 30° angle is half of the length of the hypotenuse. Thus, the length of the wire, which represents
by the hypotenuse in the triangle, would be
2 × 70 feet = 140 feet.
The problem can also be solved by using trigonometry. Let w stand for the length of the wire, then write
70 . Since sine 30° = 1 , the equation is
sine 30° =
2
w
70 1
= . Cross-multiply: w = 140 feet.
w 2

24.

The correct answer is B. In the formula y = mx + b,
m represents the slope of the line, and b represents the
y-intercept. Thus, the equation y = − 7x + 12 represents
the equation of a line with a slope of −7 and y-intercept
of 12.

25.

The correct answer is A. Since parallel lines have the
same slope, the slope of the line to be determined is the
same as the slope of y = 7x − 6, which is 7. Therefore, the
equation of the unknown line can be written as y = 7x +
b, where b represents the y-intercept. In order to find the
value of b, substitute the point (1, 7) into y = 7x + b.

471
Chapter 11

y = 7x + b

Substitute (1, 7)

Geometry

7 = 7 (1) + b
7=7+b
b=0
Therefore, the equation of the line is y = 7x. Note that
a faster way to solve this problem is the trial-and-error
method. Of the four answer possibilities, choices A
and B represent lines with slopes of 7, but only choice
A contains the point (1, 7).

Peterson’s SAT® Prep Guide 2017

SUMMING IT UP
• Lines and line segments are the basic building blocks for most geometry problems.
• If a geometry problem provides a figure, mine it for clues. If a geometry problem doesn’t provide a figure, sketch one.
• If a geometry problem deals with a quadrilateral or circle, look to form triangles by drawing lines through the figure.
• Geometry diagrams on the SAT® are not always drawn to scale. If the diagram is not drawn to scale, you may want to redraw it.
•

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Peterson’s SAT® Online Course:

• Angles
• Geometry Strategy
• Hard Geometry
• Quadrilaterals
• Radian Measures
• Right Triangles
• Triangle Properties
• Trigonometry

472

• Word Problems
To purchase and access the course, go to www.petersons.com/sat.

Chapter 11
Geometry

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Chapter 12:
Functions and
Intermediate Algebra
OVERVIEW
Functions
Exercises: Functions
Integer and Rational Exponents
Exercises: Integer and Rational Exponents
Solving Complex Equations
Exercises: Solving Complex Equations
Linear, Quadratic, and Exponential Functions
Exercises: Linear, Quadratic, and Exponential
Functions

473

Summing It Up

Chapter 12
Functions
and
Intermediate
Algebra

FUNCTIONS
Definitions and Notation
Let D and R be any two sets of numbers. A function is a rule that assigns to each element of D one and only one element of R.
The set D is called the domain of the function, and the set R is called the range. A function can be specified by listing all of the
elements in the first set next to the corresponding elements in the second set or by giving a rule or a formula by which elements
from the first set can be associated with elements from the second set.
As an example, let the set D = {1, 2, 3, 4} and set R = {5, 6, 7, 8}. The diagram below indicates a particular function, f, by showing
how each element of D is associated with an element of R.

5
6
7
8

1
2
3
4
D

R

Peterson’s SAT® Prep Guide 2017

This diagram shows that the domain value of 1 is associated with the range value of 5. Similarly, 2 is associated with 6, 3 is associated with 7, and 4 is associated with 8. The function f can also be described in words by saying that f is the function that assigns
to each domain value x the range value x + 4.
Typically, the letter x is used to represent the elements of the domain and the letter y is used to represent the elements of the
range. This enables us to write the equation y = x + 4 to express the rule of association for the function above.
Note that as soon as a domain value x is selected, a range value y is determined by this rule. For this reason, x is referred to as the
independent variable, and y is called the dependent variable.
Often, the rule of association for a function is written in function notation. In this notation, the symbol f(x), which is read “f of x,”
is used instead of y to represent the range value. Therefore, the rule for our function can be written f(x) = x + 4. If you were asked
to determine which range value was associated with the domain value of, say, 3, you would compute f(x) = f(3) = 3 + 4 = 7. Note
that, in this notation, the letter f is typically used to stand for “function,” although any other letter could also be used. Therefore,
this rule could also be written as g(x) = x + 4.
Example:
Using function notation, write the rule for a function that associates, to each number in the domain, a range value
that is 7 less than 5 times the domain value.
Solution:
f(x) = 5x − 7

474
Chapter 12
Functions
and
Intermediate
Algebra

Example:
Use the function from the problem above to determine the range value that is associated with a domain value
of −12.
Solution:
f(−12) = 5(−12) − 7 = −60 − 7 = −67
Example:
If f(x) = 8x + 9, determine the value of f(5), f(q), f(p2), and f(r + 3).
Solution:
f(5) = 8(5) + 9 = 40 + 9 = 49
In the same way, to determine the value of f(q), simply substitute q for the value of x in the rule for f(x). Therefore,
f(q) = 8q + 9.
Similarly, f(p2) = 8(p2) + 9 = 8p2 + 9.
Similarly, f(r + 3) = 8(r + 3) + 9 = 8r + 24 + 9 = 8r + 33.

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Families of Functions
A family of functions is a group of functions with similar characteristics. The parent function is the most basic function in a family.
A linear function is a function whose graph is a straight line. f(x) = x is the parent function for all linear functions. A quadratic
function is a function whose graph is a parabola. The graph shown, f(x) = x2, is the parent function for all quadratic functions.

y
5
4
3
2
1
–5

–4

–3

–2

–1
–1

1

2

3

4

5

x

–2
–3
–4
–5

Example:
Compare f(x) = 2x2 to f(x) = x2.
Solution:

475
y

Chapter 12

5

Functions
and
Intermediate
Algebra

4
3
2
1
–5

–4

–3

–2

–1
–1

1

2

3

4

5

x

–2
–3
–4
–5

The graphs f(x) = 2x2 and f(x) = x2 both open up, have (0, 0) as their vertices, and have the same axis of symmetry
x = 0. The graph f(x) = 2x2 is narrower than the graph of f(x) = x2.

Peterson’s SAT® Prep Guide 2017

Example:
How does the graph f(x) = x2 + 2 compare to the parent function f(x) = x2?
Solution:

y
5
4
3
2
1
–5

–4

–3

–2

–1
–1

1

2

3

4

5

x

–2
–3
–4
–5

The graph f(x) = x2 + 2 is two units higher than f(x) = x2. The function f(x) = x2 has been shifted up 2 units. The shifting,
or translating, of a function is called a transformation. To move a function up, add a value b to the function:
f(x) = x2 + b. To move a function down, subtract a value b from the function: f(x) = x2 – b.

476

The function f(x) = (x + 2)2 looks like the following:

y

Chapter 12
5

Functions
and
Intermediate
Algebra

4
3
2
1
–5

–4

–3

–2

–1
–1

1

2

3

4

5

x

–2
–3
–4
–5

Its graph has been shifted to the left 2 units. To shift a function to the left, add a value b to inside the function:
f(x) = (x + b)2. To shift a function to the right, subtract a value b to inside the function:
f(x) = (x – b)2.

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The function –f(x) is f(x) flipped upside down (across the x-axis), while f (–x) is the mirror of f (x) (flipped across the
y-axis):
y

–f (x)

5

–5

–4

–3

–2

y

f (–x)

5

4

4

3

3

2

2

1

1

–1
–1

1

2

3

4

5

x

–5

–4

–3

–2

–1
–1

–2

–2

–3

–3

–4

–4

–5

–5

1

2

3

4

5

x

Example:
Describe how the function f(x) = –2x2 + 3 compares with the graph f(x) = x2.
Solution:
Both graphs have the same axis of symmetry but the graph f(x) = –2x2 + 3 opens downward and is narrower than the
graph of f(x) = x2. The vertex of f(x) = –2x2 + 3 is 3 units higher than the vertex of f(x) = x2.

Perform Operations on Functions and Compositions

477
Chapter 12
Functions
and
Intermediate
Algebra

You can add, subtract, multiply, and divide any two functions.
Let f(x) = 4x2 and g(x) = x2 + 9x + 6
f(x) + g(x) = 4x2 + (x2 + 9x + 6) = 5x2 + 9x + 6
f(x) – g(x) = 4x2 – (x2 + 9x + 6) = 3x2 – 9x – 6
f(x) · g(x) = 4x2 (x2 + 9x + 6) = 4x4 + 36x3 + 24x2
f ( x)
4x 2
= 2
g( x) x + 9x + 6
The composition of functions involves applying two functions in succession. The composition of a function g(x) with a function
f(x) is written as g(f(x)), read as “g of f of x”.

Peterson’s SAT® Prep Guide 2017

Let f(x) = 6x3 + 2 and g(x) = 2x2 – 8.
Example:
Find f  (g(3)) and g(f(3)).
Solution:
To evaluate f(g(3)), first find g(3).
g(3) = 2(3)2 – 8 = 10
Then f  (g(3)) = f  (10) = 6(10)3 + 2 = 6,002.
To evaluate g(f  (3)), first find f  (3).
f  (3) = 6(3)3 + 2 = 164
Then g(f(164)) = g(164) = 2(164)2 – 8 = 53,784.
Generally, f  (g(x)) ≠ g(f  (x)).

478
Chapter 12
Functions
and
Intermediate
Algebra

www.petersons.com

EXERCISES: FUNCTIONS
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

2.

3.

4.

What is the effect on the graph of the function
f(x) = 2x2 + 5 when it is changed to f(x) = 2x2 – 3?
A.

The graph gets narrower.

B.

The graph opens down.

C.

The graph gets wider.

D.

The graph shifts down the y-axis.

SHOW YOUR WORK HERE

Which function opens down?
A.

y = –7x2 – 3

B.

y = 0.5x2 + 2

C.

y = 10x2 – 1

D.

y = 7x2 + 9

3
If h(x) = 3x2 + 5, and j ( x ) =
2 , what is the value of
x − 4)
(
h(5) − j(5)?
A.

0

B.

77

C.

83

D.

87

The profit from selling x number of sweatshirts can be
described by the function p(x) = 22x – 230. What is the
profit if 200 sweatshirts are sold?
A.

$4,170

B.

$4,280

C.

$4,400

D.

$4,630

479
Chapter 12
Functions
and
Intermediate
Algebra

Peterson’s SAT® Prep Guide 2017

5.

6.

7.

480
Chapter 12
Functions
and
Intermediate
Algebra

8.

The number of blue blankets in stock can be modeled by
the equation f(x) = 2x2 – 8x + 20 and the number of red
blankets in stock can be modeled by the equation h(x) =
3x2 + 4x + 30. Which function models the total number
of blankets in stock?
A.

–x2 – 4x + 50

B.

5x2 – 12x + 50

C.

5x2 – 4x + 50

D.

–x2 + 4x + 20

A company sells designer keychains. Its monthly revenue
is modeled by R(x) = 27x, and its costs are modeled by
C(x) = 7x + 1,200, where x is the number of keychains
sold. Find R(x) – C(x).
A.

–34x + 1,200

B.

20x + 1,200

C.

20x – 1,200

D.

34x – 1,200

You have a $25 gift certificate to a clothing store. The store
is having a 30%-off sale. You decide to purchase an outfit
that costs $100 before the sale. What is the final cost of
the outfit if the gift certificate is applied before the 30%
discount?
A.

$22.50

B.

$30

C.

$45

D.

$52.50

You earn $17.50 per hour at your job. You are given a raise
of 2% after 5 months. In addition, you receive $1 per hour
for being a model employee. Find your new hourly wage
if the $1 raise is applied before the 2% raise.
A.

$18.13

B.

$18.15

C.

$18.85

D.

$18.87

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SHOW YOUR WORK HERE

ANSWER KEY AND EXPLANATIONS
1. D

3. B

5. C

7. D

2. A

4. A

6. C

8. D

1.

The correct answer is D. The vertex of the graph
changes from (0, 5) to (0, –3). The original graph is
translated down.

2.

The correct answer is A. The graph opens down since
the coefficient in front of the x2-term is less than 0.

3.

The correct answer is B.

( )

h (5) − j (5) = 3 5 2 + 5 −

7.

The correct answer is D. Let x be the regular price. The
function for the gift certificate is f(x) = x – 25. The
function for the 30% discount is g(x) = x – 0.30x = 0.7x.
The composition g(f(x)) represents the sale price when
the $25 gift certificate is applied before the 30%
discount.
g (f ( x )) = g ( x − 25)

= 0.7 ( x − 25)

= 0.7 (100 − 25)

3

(5 − 4 )

2

= 0.7 (75)

= 3 (25) + 5 − 3 = 77

4.

The correct answer is A.
p(200) = 22(200) – 230 = 4,170. The profit is $4,170.

5.

The correct answer is C.
f ( x ) + h ( x ) = 2 x 2 − 8 x + 20 + 3 x 2 + 4 x + 30
= 5 x 2 − 4 x + 50

6.

= $52.50

The correct answer is C.
R ( x ) − C ( x ) = 27 x − (7 x + 1, 200)
= 27 x − 7 x − 1, 200
= 20 x − 1, 200

So, you pay $52.50.
8.

The correct answer is D. Let x be your hourly wage.
The function for the 2% raise is f(x) = x + . 02x = 1.02x.
The function for the $1 raise is g(x) = x + 1. The
composition f(g(x)) represents the hourly wage when
the $1 raise is applied before the 2% raise.
f ( g ( x )) = f ( x + 1)

481

= 1.02 ( x + 1)

Chapter 12

= 1.02 (18.50)

Functions
and
Intermediate
Algebra

= 1.02 (17.5 + 1)
= 18.87
So, your new hourly wage is $18.87.

Peterson’s SAT® Prep Guide 2017

INTEGER AND RATIONAL EXPONENTS
Chapter 10: “Basic Algebra” contains the definitions and the rules for positive integer exponents. The following section extends
these definitions to integer and rational exponents.

Integer Exponents
Negative exponents are defined in the following way:
For any positive integer n, x − n =
 2
Similarly,  
3

−4

1
x

n

. For example, 4 −2 =

1
4

2

=

1
.
16

4

3 4 81
 3
=  = 4 = .
 2
16
2

All of the properties of exponents discussed in Chapter 11 apply to expressions with negative exponents as well. Thus, the
1
y −5
−5 −( −11)
−3
= y −5+11.= y 6
expression x −7 • x 4 is equal to x = 3 , and −11 = y
x
y
Examples:
Determine the value of the following expressions:
a. 8−2 • 84
b. x5 • x−5

482
Chapter 12
Functions
and
Intermediate
Algebra

c.

y4
y −9

−2 9
d. x y

x 7y 3

Solutions:
a. 8−2 • 84 = 82 = 64
b. x5 • x−5 = x0 = x
c.

y4
y −9

= y 4 y 9 = y 13

−2 9
9 −3
y6
d. x y = y
=
9
x 7y 3
x 7 −( −2) x

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Examples:
a. If f(x) = 5−x, what is the value of f(−2)?
b. If f(x) = 5−x, what is the value of f(2)?
Solutions:
a. f(−2) = 5−(−2) = 52 = 25
b. f(2) = 5–2 =

1
1
=
2
5 25

Rational Exponents
The definition of exponents can also be extended to include rational numbers:
For a rational number, x to the power of

1
1
is defined as the nth root of x. In other words, x n is equal to n x .
n

1

Therefore, 5 can be written as 5 2.
1

Similarly, 8 3 represents

3

8 and is thus equal to 2.
1

m

Next, x n is defined to mean ( x n ) m , for x > 0 and n > 0. Therefore, when you are given a number with a rational exponent, the
numerator represents the power to which the number is to be raised, and the denominator represents the root to be taken. The
5

expression 16 4 tells you to take the fourth root of 16 and then raise the result to the fifth power. This expression can be evaluated

5
in the following way: 16 4

=

(4 16 )

5

= (2) = 32

483

5

In summary, all of the properties of exponents apply to expressions with rational exponents.
Examples:

Chapter 12
Functions
and
Intermediate
Algebra

Determine the value of the following expressions:
1

a. 27 3
b. 64

−

2
3

5

c. 2 x 6 i 3 x

−

1
3

Solutions:
1

a. 27 3 = 3 27 = 3
b. 64

−

2
3

2

2

2
1
 13  1 
 1
=  =3
=
  =

 64 
4
16
 64 

5

c. 2 x 6 i 3 x

−

1
3

5

= 6x 6

−

1
3

1

= 6x 2

Peterson’s SAT® Prep Guide 2017

Examples:
Simplify the following expressions:
a.

1
6 2

(25a )
1

b3

b.

b

−

2
3

 − 1 3
c.  c 8 d 4 


d.

4x 3 i

16

8x 5

Solutions:

a.

(

25a 6

)

1
2

1

b.

b3
b

484
Chapter 12

−

2
3

1

1 2

4x 3 i

Functions
and
Intermediate
Algebra

1

= b 3b 3 = b 3

 − 1 3
c.  c 8 d 4 



d.

( )

= 25 2 a 6

16

1
2

+

 −
= c

1

= 25a 3 = 5a 3

2
3

= b1 = b

1  16  3  16
8 d 4 





3

1

5

3

5

8x 5 = 4 2 x 2 i 8 2 x 2
1

1

= 4 2 i 82 x 2x 2

( )

3

=2 2 2 x 2
= 4 2x 4

www.petersons.com



+

5
2

= c − 2d 12 =

d 12
c2

EXERCISES: INTEGER AND RATIONAL EXPONENTS
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

2.

3.

4.

Simplify the expression: 8 x i

4 x3

SHOW YOUR WORK HERE

3

A.

32 x 2

B.

16 2 x 4

C.

4x4

D.

4 2x2

Simplify the expression: 25 x 4 y 6 i 3 x 3 y 5
7

11

A.

3x 2 y 2

B.

5 3x 2 y 2

C.

15x12y11

D.

75x7 y11

7

11

Which of the following is equal to (xy−3)2?
A.

x
y

B.

x
y6

C.

x2
y6

D.

x2
y

Simplify the expression:
A.

a7
b6c

B.

a7 c
b2

C.

a3
b2c

D.

a3c 5
b2

485
Chapter 12
Functions
and
Intermediate
Algebra

a5b −2c −3
a −2b 4 c −2

Peterson’s SAT® Prep Guide 2017

5.

A.
B.

C.
D.

6.

b3
a 6c
b3
a 6c 7
a 8c 12
b 28
b 28
a 8c 12

(

Simplify the expression: 49 x 4 y 8 z −6
A.

49x2 y4z6

B.

7x3y4z3

C.
D.

7.

1
2

7x 2z 3
7x 2y 4
z3

Simplify the expression:
8x x

486

B.

16 x 6

C.

8x 4 x 4

Chapter 12

D.

32 x 2 x

8.

)

y4

A.

Functions
and
Intermediate
Algebra

SHOW YOUR WORK HERE

Simplify the expression: (a−2b7c−3)−4

4

256 x 2 i 16 x 4

3

−125 x 6 i

4

4

Simplify the expression:
A.

−40x6y9

B.

−20x2y

C.

−9 x 6 y

D.

20 x 3 y 3 x

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3

64 y 3

ANSWER KEY AND EXPLANATIONS

1.

1. D

3. C

5. C

7. A

2. B

4. A

6. D

8. B

5.

The correct answer is D.
8x i

4 x 3 = (8 x )

1
2

1
4x 3 2

( )

1 
3

=  2 2x 2  2x 2 

 


= 4 2x
2.

(a
6.

1
25 x 4 y 6 2

)(

1
3x 3y 5 2

)

(

= 5x 2 y 3 i

)

1
2

b 28

1

4

8

= 49 2 x 2 y 2 z

7 11
2y 2

=

7.

−6
2

The correct answer is C.

7x 2y 4
z3

The correct answer is A.
4

1

4

1

1

256 x 2 i 16 x 4 = 256 4 x 2 i 16 4 x 1
1

= x 2 y −6 =

x

= 4 x 2 i 2x

2

= 8x x

y6
8.

4.

a 8c 12

= a 8b −28c 12 =

49 x 4 y 8 z −6

1 3 5
32 x 2y 2

1
3
5
2+
3+
32 x 2y 2

= 5 3x

−3 2

b c

= 7 x 2 y 4 z −3

=5i

( xy )

)

−2 7 −3 −4

The correct answer is D.

2

The correct answer is B.

(

3.

The correct answer is C.

The correct answer is B.

The correct answer is A.
5 −2 −3

a b

c

a −2b 4 c −2

3

=a

5 −( −2) −2 − 4 −3 −( −2)

b

= a 7b −6c −1 =

c

a7
b 6c

−125 x 6 i

3

6
3
64 y 3 = −5 x 3  4 y 3 



= −20 x 2 y

487
Chapter 12
Functions
and
Intermediate
Algebra

Peterson’s SAT® Prep Guide 2017

SOLVING COMPLEX EQUATIONS
Chapter 10: “Basic Algebra” describes how to solve linear and quadratic equations. The following section discusses how to solve
some of the more complex equations and inequalities that appear on the SAT®exam.

Equations Involving Rational Expressions
A rational expression is a fraction that contains variables in the numerator and/or the denominator. The quickest way to solve
equations containing rational expressions is to determine the least common denominator (LCD) of all of the fractions in the
equation and then eliminate the fractions by multiplying each term in the equation by this LCD. The four steps in solving such
an equation are:
1.

Find the LCD of all of the rational expressions in the equation.

2.

Multiply every term on both sides of the equation by this LCD.

3.

Solve the resulting equation using the methods previously explained.

4.

Check the solution to make certain that it actually solves the equation.

Note that step 4, checking the solution, is crucial because sometimes the process produces a solution that does not actually solve
the equation. Such extraneous solutions need to be eliminated.

Example:
Solve for x:

488

7x 3
+ = 10
5 8

Solution:
The LCD of the two fractions in the equation is 40, so every term must be multiplied by 40.

Chapter 12
 3
 7x 
40   + 40   = 40 (10)
 5
 8

Functions
and
Intermediate
Algebra

Perform the multiplications.

8 (7 x ) + 5 (3) = 400

Subtract 15 from both sides.
56 x + 15 = 400
Divide both sides by 56.
56 x = 385
385 55
7
x=
=
=6
56
8
8

Check that the answer is correct by substituting 55 into the original equation.
8

www.petersons.com

Example:
Solve for x:

5
3
36
−
=
x − 4 x + 4 x 2 − 16

Solution:
Begin by finding the LCD of the three fractions. Note that since x2 − 16 = (x − 4) (x + 4), the LCD is (x − 4)(x + 4). Each
term must be multiplied by this.

( x − 4) ( x + 4) 

5 
 3   36 
 − ( x − 4 ) ( x + 4 ) 
 =
 ( x − 4) ( x + 4)
x−4
x + 4   x 2 − 16 

( x + 4)(5) − ( x − 4)(3) = 36

Distribute.

5 x + 20 − 3 x + 12 = 36 Combin
ne like terms.
2 x + 32 = 36 Subtract 32 from both sides.
2 x = 4 Divide by 2.
x =2
To check the solution, substitute 2 into the equation:
5
3
36
−
=
x − 4 x + 4 x 2 − 16
5
3
36
−
=
2 − 4 2 + 4 22 − 16
5 3
36
− − =−
2 6
12
−3 = −3
Therefore, the solution is x = 2.

Chapter 12

Example:
Solve for x:

489
Functions
and
Intermediate
Algebra

1 1 1
− =
5 6 x

Solution:
The LCD is 30x. Multiply all terms by the LCD.

(30 x ) 

1
 1 1
 − (30 x )   = (30 x )
6
5
x
6 x − 5 x = 30
x = 30

If you check the value x = 30 in the original equation, you will find that it works.

Peterson’s SAT® Prep Guide 2017

Radical Equations
Equations that have variables in their radicands are called radical equations. For example, x = 16 is a radical equation. To solve
this radical equation, square both sides:
x = 16

( x)

2

= 16 2

x = 256

Check x = 256 in the original equation.
x = 16
256 = 16 ?
16 = 16
The solution works.
Extraneous solutions may occur when you raise both sides of a radical equation to an even power. For example, if you square
both sides of the equation x = 5 you get x2 = 25. This new equation has two solutions, –5 and 5, but only 5 is the solution to the
original equation.
The steps to solve a radical equation are:

490
Chapter 12
Functions
and
Intermediate
Algebra

1.

Isolate the radical on one side of the equation.

2.

Raise both sides of the equation to the same power to eliminate the radical.

3.

Solve the resulting equation.

4.

Check the solution.

Examples:
Solve each equation.
2x + 3 = 7

a.
b.

3

4x − 5 = 3

c.

4

2x + 8 − 2 = 2

d. x + 2 = 11x + 12

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Solutions:
a.

2x + 3 = 7

(

)

2

2x + 3 = 7

2x + 3 = 7
2 (23) + 3 = 7?

2

49 = 7?
7=7

2 x + 3 = 49
2 x = 46
x = 23 Check the solution.

b.

3

(

4x − 5 = 3

)

3

3

4x − 5 = 3

3

3

3

4x − 5 = 3

4 (8) − 5 = 3?

4 x − 5 = 27
4 x = 32
x = 8 Check the solution.
c.

4

(

3

4

4

2x + 8 = 4

2x + 8

)

4

= (4)

4
4

d.

The solution is 8.

2x + 8 − 2 = 2

2 (124 ) + 8 − 2 = 2?
4

2 x + 8 = 256
2 x = 248
x = 124 Check the
e solution.

27 = 3?
3=3

2x + 8 − 2 = 2
4

The solution is 23.

256 − 2 = 2?
4 − 2 = 2?
2=2

The sollution is 124.

x + 2 = 11x + 12

( x + 2 )2 = (

11x + 12

491

)

2

x + 4 x + 4 = 11x + 12
2

Chapter 12

x2 − 7x − 8 = 0

( x − 8) ( x + 1) = 0

Functions
and
Intermediate
Algebra

x = 8 , x = −1

Check the solution x = 8.
x + 2 = 11x + 12
8 + 2 = 11(8) + 12 ?
10 = 100 ?
10 = 10
Check the solution x = –1.

The
e solution is 8.

x + 2 = 11x + 12
−1+ 2 = 11( −1) + 12 ?
1 = 1?
1= 1

The so
olution is −1.

The solutions are x = 8 and x = –1.

Peterson’s SAT® Prep Guide 2017

EXERCISES: SOLVING COMPLEX EQUATIONS
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

2.

3.

492
Chapter 12
Functions
and
Intermediate
Algebra

4.

5.

Solve for x:
A.

x = −4

B.

x = −2

C.

x=2

D.

x=4

Solve for y:

2
5
2
−
=
x − 2 x + 2 x2 − 4

4
3
12
−
=
y − 2 y + 2 y2 − 4

A.

y = −4

B.

y = −2

C.

y=2

D.

There are no solutions.

Solve for a:

3
5
7
+
=
a − 7 a2 − 13a + 42 a − 6

A.

a = −18

B.

a = −9

C.

a=9

D.

a = 18

Solve for b:
A.

b>4

B.

b>8

C.

b > 12

D.

b > 18

b b
b
+ +
>2
8 12 24

Solve the equation: 3 8 x = 4
A.

x=8

B.

x=6

C.

x=4

D.

x=2

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SHOW YOUR WORK HERE

6.

7.

8.

9.

10.

Solve the equation:
A.

x=1

B.

x=9

C.

x = 17

D.

x = 209

4

3x − 2 + 1= 6

SHOW YOUR WORK HERE

Solve the equation: −5 3 9 x + 10 = −5
A.

x=3

B.

x=1

C.

x = −1

D.

x = −3

Solve the equation: x = 16 x + 225
A.

x = 45

B.

x = 25

C.

x = −9

D.

x = −25

Solve for t:
A.

t ≤ 84

B.

t ≤ 42

C.

t ≥ 42

D.

t ≥ 84

t 2t 3t − 14
+ ≥
4 9
6

Solve the equation:
A.

x=

11
7

B.

x=

7
3

C.

x=2

D.

x=3

493
Chapter 12

16 x − 32 = 2 x + 10

Functions
and
Intermediate
Algebra

Peterson’s SAT® Prep Guide 2017

ANSWER KEY AND EXPLANATIONS

1.

1. D

3. C

5. A

7. A

9. A

2. D

4. B

6. D

8. B

10. D

The correct answer is D. The LCD of the fractions in the equation

2
5
2
−
=
is (x − 2)(x + 2).
x − 2 x + 2 x2 − 4

Multiply all terms by the LCD.

( x − 2) ( x + 2)

2
5
2
− ( x − 2) ( x + 2)
=
( x − 2) ( x + 2)
x −2
x + 2 x2 − 4
2 ( x + 2) − 5 ( x − 2) = 2
2 x + 4 − 5 x + 10 = 2
−3 x + 14 = 2
−3 x = −12
x=4

Remember that you should check the answer to make certain that it solves the equation.
2.

The correct answer is D. The LCD of the fractions in the equation is

( y − 2) ( y + 2)

494

4
3
12
is y 2 − 4.
−
=
y −2 y +2 y 2 −4

4
3
12
− ( y − 2) ( y + 2)
= ( y − 2) ( y + 2) 2
y −2
y +2
y −4
4 ( y + 2) − 3 ( y − 2) = 12

4 y + 8 − 3 y + 6 = 12
y + 14 = 12
y = −2

Chapter 12
Functions
and
Intermediate
Algebra

Note that when you substitute y = −2 into the equation, two of the denominators become 0 and are, therefore,
undefined. This means that y = −2 is an extraneous solution, and the equation actually has no solutions.
3.

The correct answer is C. Note that a2 − 13a + 42 = (a − 7)(a − 6), so the LCD of the fractions in the equation is
(a − 7)(a − 6).
Now, multiply by the LCD.

( a − 7) ( a − 6 )

3
5
7
+ ( a − 7) ( a − 6 )
=
( a − 7) ( a − 6 )
a−7
( a − 7) ( a − 6 ) a − 6
3a − 18 + 5 = 7 (a − 7)
3a − 13 = 7a − 49
4 a = 36
a=9

This solution checks.

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4.

The correct answer is B. In the inequality

8.

The correct answer is B.

b b
b
+ +
> 2 , multiply by the LCD of 24.
8 12 24

x = 16 x + 225
x 2 = 16 x + 225
x − 16 x − 225 = 0
2

(24)

( x − 25) ( x + 9) = 0

b
b
b
+ (24 ) + (24 ) > 2 (24 )
8
12
24
3b + 2b + b > 48
6b > 48

x = 25, − 9
Note when you substitute x = –9 into the equation,
you get –9 = 9. This means that –9 is an extraneous
solution. The only solution is 25.

b>8
5.

The correct answer is A.
9.
3

(

3

8x = 4

8x

)

3

= 43

(36)

8 x = 64
x =8
6.

3x − 2 + 1= 6
4

7.

3x − 2 = 5
3 x − 2 = 625
3 x = 627
x = 209

The correct answer is A.
−5 3 9 x + 10 = −5
−5 3 9 x = −15
3

9x = 3
9 x = 27
x =3

t
2t
3t − 14
+ (36)
≥ (36)
9
6
4
9t + 8t ≥ 6 (3t − 14 )
17t ≥ 18t − 84
−t ≥ −84
t ≤ 84

The correct answer is D.
4

t 2t 3t − 14
The correct answer is A. To solve + ≥
,
4 9
6
multiply by the LCD of 36.

10.

The correct answer is D.
16 x − 32 = 2 x + 10
16 x − 32 = 2 x + 10
14 x = 42
x =3

495
Chapter 12
Functions
and
Intermediate
Algebra

Peterson’s SAT® Prep Guide 2017

LINEAR, QUADRATIC, AND EXPONENTIAL FUNCTIONS
A linear function is a function of the form f(x) = mx + b, where m and b are real numbers. A quadratic function is a function of the
form g(x) = ax2 + bx + c, where a ≠ 0 and a, b, and c are real numbers. An exponential function is a function of the form f (x) = abx
where base b is a positive integer greater than 1. These functions are important, because they can be used to model many realworld occurrences.

Applications of Linear Functions
In order to manufacture a new car model, a carmaker must initially spend $750,000 to purchase the equipment needed to start
the production process. After this, it costs $7,500 to manufacture each car. In this case, the cost function that associates the cost
of manufacturing cars to the number of cars manufactured is C(x) = 7,500x + 750,000, where x represents the number of cars
manufactured, and C(x) represents the cost of x cars. For example, the cost of making 7 cars is C(7) = 7,500(7) + 750,000 = 52,500
+ 750,000 = $802,500.
The above cost function is a linear function with b = 750,000 and m = 7,500. What is the domain of this function? Note that even
though nothing has been specifically said about the domain, the only values that make sense as domain values are the nonnegative integers, 0, 1, 2, 3, 4, 5, … . In such a situation, assume that the domain only contains the values that make sense.

Example:
Using the cost function for the carmaker discussed in the example above, how much would it cost to make 24 cars?
Solution:

496

To solve this, you need to determine the value of C(24).
C(24) = 7,500(24) + 750,000 = 180,000 + 750,000 = $930,000

Chapter 12
Functions
and
Intermediate
Algebra

Example:
Using the same cost function, determine how many cars could be made for $990,000.
Solution:
In this problem, you are told that the value of C(x) is $990,000, and you need to find the value of x. To do this, solve
the equation:
990 , 000 = 75, 000 ( x ) + 750 , 000 Subtract 750,000 from both sidess.
240 , 000 = 7, 500 x
32 = x
Therefore, for $990,000, 32 cars can be manufactured.

www.petersons.com

Divide by 7,500.

Example:
In the town of Kenmore, a taxi ride costs $2.50 plus an extra $0.50 per mile. Write a function that represents the cost
of taking a taxi ride, using x to represent the number of miles traveled.
Solution:
C(x) = $2.50 + 0.50x
If a ride costs $0.50 a mile, then the cost for x miles will be $0.50x. Add to this the initial fee of $2.50 a ride.
Example:
You purchased shorts for $8 per pair, plus a shirt for $6. Write a function that represents the cost of your purchases,
where x represents the number of pairs of shorts you purchase. How much did you spend if you purchased 5 pairs of
shorts?
Solution:
f(x) = 8x +6
Let x = 5, so f(5) = 8(5) + 6 = 46.
Example:
A bus pass has a starting value of $60. Each ride costs $2.50. Write a function that represents the remaining balance
on the bus pass, where x represents the number of rides taken. How much money is left on the pass after 12 rides?

497
Chapter 12
Functions
and
Intermediate
Algebra

Solution:
f(x) = 60 – 2.5x
Let x = 12, so f(12) = 60 – 2.5(12) = 30.
There is $30 remaining on the pass after 12 rides.

The Graph of a Linear Function
Typically, when a function is graphed, the independent variable is graphed along the x-axis, and the dependent variable is
graphed along the y-axis.

Peterson’s SAT® Prep Guide 2017

The taxi ride function from the previous problem is a linear function. In order to graph this function, you must first determine
the domain. Note that the domain, once again, must consist of non-negative numbers. Next, determine a few values that satisfy
the rule for the function. For example, when x = 0, C(0) = $2.50 + 0.50(0) = $2.50. A few additional simple computations will lead
to the following table of values.

x

C(x)

0

$2.50

1

$3.00

2

$3.50

3

$4.00

If these points are plotted on a graph, you will see that they all lie on the same line. The entire graph of the taxi ride cost function
is shown here.

y

Price

6
5
4
3
2
1
1

498
Chapter 12
Functions
and
Intermediate
Algebra

2

3

4

5

6

x

Miles

In general, the graph of any linear function is either a straight line or (depending on the domain) a portion of a straight line. The
value of m represents the slope of the line, and the value of b is the y-intercept.

Applications of Quadratic Functions
Quadratic functions can also be used to model certain real-world happenings. To understand these functions better, suppose a
coffee manufacturer has a revenue function given by R(x) = 40,000x − 2,000x2, where x represents the amount of coffee produced
in tons per week. Let’s consider some of the values for this function.
If x = 0, R(x) = 40,000(0) − 2,000(0)2 = 0 represents the obvious fact that if no coffee is produced there is no revenue.
That R(1) = 40,000 − 2,000 = 38,000 tells that the revenue from 1 ton of coffee is $38,000.
Similar computations show that R(10) = $200,000 and R(11) = $198,000.
Note that the revenue is smaller if 11 tons of coffee are produced than if 10 tons are produced. There are a number of possible
reasons for this. Perhaps, for example, at the 11-ton level, more is produced than can be sold, and the coffee company must pay
to store the overage.

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The function h(t) = –16t2 + h0 models the height of an object dropped from an initial height h0 (in feet) after t seconds.
Example:
An object is dropped from the roof of a building that is 80 ft. tall. How long will it take the object to hit the ground?
Round your answer to the nearest hundredth of a second.
Solution:
h(t) = –16t2 + h0
Here h0 = 80, so h(t) = –16t2 + 80.
Substitute 0 in for h(t) and solve the equation for t.
0 = −16t 2 + 80
16t 2 = 80
t2 =5
t ≈ ±2.24
Example:
Suppose you drop a ball from a window that is 36 ft. above the ground and it lands on a porch that is 4 ft. above the
ground. How long does it take for the ball to land on the porch? Round your answer to the nearest hundredth of a
second.
Solution:

499
Chapter 12

h(t) = –16t2 + h0

Functions
and
Intermediate
Algebra

2

Here h0 = 36, so h(t) = –16t + 36.
Substitute 4 for h(t), because the ball will hit the porch at 4 feet above 0, or ground level. Solve the equation for t.
4 = −16t 2 + 36
16t 2 = 32
t2 =2
t ≈ ±1.41

It will take approximately 1.41 seconds for the object to land on the porch.

Peterson’s SAT® Prep Guide 2017

The Graph of a Quadratic Function
As you just saw, the graph of a linear function is always a straight line. To determine what the graph of a quadratic function looks
like, consider the graph of the quadratic function R(x) = 40,000x − 2,000x2. Negative numbers must be excluded from the domain.
A few computations lead to the table here.

x

R(x)

0

0

3

102,000

5

150,000

9

198,000

10

200,000

11

198,000

15

150,000

17

102,000

20

0

The graph of R(x) is shown here.

200,000

Revenue

500
Chapter 12
Functions
and
Intermediate
Algebra

150,000
100,000
50,000

2

4 6

8 10 12 14 16 18 20
Tons

The graph shown above is called a parabola. This parabola is said to “open down.” The highest point on the parabola, (10, 200,000),
is called the extreme point.
Recall that the general form of a quadratic function is g(x) = ax2 + bx + c. In general, the graph of any quadratic function will be a
parabola. If a > 0, the parabola will “open up,” and if a < 0, the parabola will “open down.” If the parabola opens up, its extreme point
is the minimum value of the function, and if the parabola opens down, its extreme point is the maximum value of the function.
 −b
The coordinates of the extreme point of a parabola are  ,
 2a

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 −b  
f   .
 2a  

Example:
Sketch the graph of the function f(x) = x2 − x − 2.
Solution:
Since the function is quadratic, the graph will be a parabola. Note that the value of a, the number in front of the
−b − (1) 1
= , and the
=
x2-term is 1, so the parabola opens up. The x-coordinate of the minimum point is x =
2a 2 (1) 2
y-coordinate of this point is
2

1
 1  1
f   =  − −2
 2  2
2
1 1
= − −2
4 2
1
= −2
4
In order to sketch a parabola, it is helpful to determine a few points on either side of the extreme point.

x

f(x)

–2

4

–1

0

0

–2

1

–2

2

0

3

4

501
Chapter 12

The graph is shown here.

Functions
and
Intermediate
Algebra

y
4
3
2
1
x
–4 –3 –2 –1
–1

1

2

3

4

–2
–3
–4

Peterson’s SAT® Prep Guide 2017

Example:
What is the relationship between the graph of the function h(x) = ax2 + bx and the graph of the function
j(x) = ax2 + bx + 7?
Solution:
If (x, y) is a point on the graph of h(x), then (x, y + 7) will be a point on the graph of j(x). Therefore, the two graphs have
exactly the same size and shape. The graph of j(x) can be obtained by taking the graph of h(x) and “lifting” each point 7
units, that is, increasing the y-coordinate of each point by 7.

Applications of Exponential Functions
If a > 0 and b > 1, then the function f (x) = abx is an exponential growth function and b is called the growth factor. If a > 0 and
0 < b < 1, then the function f (x) = abx is an exponential decay function and b is called the decay factor.
An exponential growth function can model the number of students in a high school. Suppose the number of students in 2011
is given by f(x) = 1,250(1.13)x, where x is the number of years since 2011. Let’s consider some of the values for this function. If
x = 0, then f(x) = 1,250(1.13)0 = 1,250 represents the student population in 2011. If x = 1, then f(x) = 1,250(1.13)1 =1,413 represents
the student population after 1 year. Similar computations show that f(x) = 1,596 when x = 2 and f(x) = 2,303 when x = 5.

502

An exponential decay function can model the value of an automobile. Suppose the value of the automobile is given by
f(x) = 35,000(0.85)x, where x is the number of years since the automobile was purchased. Let’s consider some of the values for this
function. If x = 0, then f(x) = 35,000(0.85)0 = 35,000 represents the value of the automobile at the time of purchase. If x = 2, then
f(x) = 35,000(0.85)2 = 25,287.50 represents the value of the automobile after 2 years. Similar computations show that
f(x) = 15,529.69 when x = 5 and f(x) = 3,057.40 when x = 15.

Chapter 12
Functions
and
Intermediate
Algebra

The Graph of an Exponential Function
To determine what the graph of an exponential function looks like, consider the graph of the exponential function
f(x) = 52,50(0.9)x, where f(x) is the value of a jet ski in dollars after x years.
The initial value of the jet ski is $5,250 and the percent of decrease per year is 10%. Some of the computations along with the
graph of f(x) is a curve that falls from left to right and gets less and less steep as x increases. The x-axis is a horizontal asymptote.

www.petersons.com

x

f(x)

0

5,250

1

4,725

5

3,100

8

2,260

12

1,482.8

15

1,080

25

339.21

Value $
5,000
4,500
4,000
3,500
3,000
2,500
2,000
1,500
1,000
500
2

4

6

8

10 12 14 16

18

Years

Comparing Linear and Exponential Growth Functions
Suppose you can choose how you will get your allowance. The first option is to get $5 a week every week. The second option is to
get $0.50 for the first week, $1 for the second week, $2 for the third, and so on, by doubling the amount each week. Which option
will pay you more?
Option 1: y = 5x, where x is the number of weeks you were paid your allowance and y is the total amount of money you
have been paid so far.
Option 2: y = (0.5)(2)x –1, where x is the number of weeks you were paid your allowance and y is the total amount of
money you have been paid so far.
As you can see from the tables, Option 1 pays more until the 8th week. After that, Option 2 will always pay more since you are
doubling a larger number.
Option 1: y = 5x

x (week)

1

2

3

4

5

6

7

8

9

10

y (total)

5

10

15

20

25

30

35

40

45

50

503
Chapter 12
Functions
and
Intermediate
Algebra

Option 2: y = (0.5)(2)x –1

x (week)

1

2

3

4

5

6

7

8

9

10

y (total)

0.5

1

2

4

8

16

32

64

128

256

Option 1 is a linear function that is increasing at a constant rate, and Option 2 is an exponential function that is
increasing rapidly as x gets bigger. You would chose Option 2 to be paid the most.

Peterson’s SAT® Prep Guide 2017

Value $
90

80

70

60

Option 2

50

Option 1
40

30

20

10

1

504
Chapter 12
Functions
and
Intermediate
Algebra

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2

3

4

5

6

7

8

9

10

11

12

Weeks

EXERCISES: LINEAR, QUADRATIC, AND EXPONENTIAL FUNCTIONS
DIRECTIONS: Work out each problem. Circle the letter of your choice.
1.

2.

3.

4.

A taxi company charges $2.25 for the first mile and $0.45
for each mile thereafter. If x stands for the number of miles
a passenger travels, which of the following functions
relates the length of the trip to its cost?
A.

f(x) = 2.25 + 0.45(x + 1)

B.

f(x) = 2.25 + 0.45(1 − x)

C.

f(x) = 2.25 + 0.45(x − 1)

D.

f(x) = 2.25 + 0.45x

SHOW YOUR WORK HERE

Amp Corporation manufactures home video systems. The
cost function for the Amp45 system is C(x) = 3x2 − 2x + 7,
where x represents the number of systems manufactured,
and the cost is in dollars. How much does it cost to manufacture 18 of these systems?
A.

$943

B.

$972

C.

$974

D.

$2,903

505

Which of the following best describes the graph of the
function f(x) = 5x − 7?

Chapter 12

A.

A horizontal line

B.

A line that increases from left to right

Functions
and
Intermediate
Algebra

C.

A line that decreases from left to right

D.

A parabola that opens up

Which of the following best describes the graph of the
function g(x) = −5x2 + 7x − 23?
A.

A horizontal line

B.

A line that increases from left to right

C.

A line that decreases from left to right

D.

A parabola that opens down

Peterson’s SAT® Prep Guide 2017

5.

6.

7.

At the Four Seasons bowling alley, it costs $1.50 to rent
shoes and $2.50 for each game played. Which of the following functions relates the number of games played, x,
to the cost in dollars?
A.

f(x) = 4.00x

B.

f(x) = 2.50 + 1.50x

C.

f(x) = 1.50x − 2.50

D.

f(x) = 2.50x + 1.50

In order to raise money for a charity, Deb makes cakes
to sell at a school bake sale. Her profit function is P(x) =
$4.50x − $15. How many cakes must she sell in order to
earn $66 for the charity?
A.

15

B.

16

C.

18

D.

20

For which domain value[s] of the function f(x) = x2 + 3x
− 10 does f(x) = 0?
A.

2

506

B.

−2

C.

−5

Chapter 12

D.

2 and −5

Functions
and
Intermediate
Algebra

8.

At which point does the graph of the function
g(x) = −12x + 6 cross the y-axis?
A.

(0, 6)

B.

(6, 0)

C.

(0, 2)

D.

 1
 0 , 
2

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SHOW YOUR WORK HERE

9.

10.

11.

12.

The functions f(x) = 16x − 7 and g(x) = 16x + 5 are graphed
on the same coordinate plane. Which of the following
best describes the appearance of the graph?
A.

Two intersecting lines, one with y-intercept of −7,
the other with y-intercept of 5

B.

Two parallel lines, one with y-intercept of −7, and
the other with y-intercept of 5

C.

Two parallel lines, one with y-intercept of 7, and
the other with y-intercept of −5

D.

Two intersecting lines, one with y-intercept of 7,
the other with y-intercept of −5

SHOW YOUR WORK HERE

What is the x-coordinate of the extreme point of the
function f(x) = 7x2 − 28x + 16?
1
2

A.

x=

B.

x=−

C.

x=2

D.

x = −2

1
2

You purchase a new bike for $850 and the value of the
bike decreases by 20% per year. Approximately, when will
the value of the bike will be less than $500?

507

A.

After 3 years

Chapter 12

B.

After 2.5 years

C.

After 2 years

D.

After 1.5 years

Functions
and
Intermediate
Algebra

What is the asymptote of the graph of f(x) = (0.75)x + 2?
A.

f(x) = –2

B.

f(x) = 0

C.

f(x) = 0.75

D.

f(x) = 2

Peterson’s SAT® Prep Guide 2017

13.

14.

A ball is dropped from a height of 20 ft. above the ground.
Given h(t) = –16t2 +h0, how long will it be before the ball
hits the ground?
A.

About 2.0 seconds

B.

About 1.56 seconds

C.

About 1.12 seconds

D.

About 0.89 seconds

The value of a car can be modeled by the equation
f(x) = 38,000(0.89)x, when x is the number of years since
the car was purchased and f(x) is the value of the car.
What is the approximate value of the car after 6 years?
A.

About $19,000

B.

About $22,000

C.

About $28,000

D.

About $30,000

508
Chapter 12
Functions
and
Intermediate
Algebra

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SHOW YOUR WORK HERE

ANSWER KEY AND EXPLANATIONS

1.

2.

1. C

4. D

7. D

10. C

13. C

2. A

5. D

8. A

11. B

14. A

3. B

6. C

9. B

12. D

The correct answer is C. The cost for the first mile on
a trip of x miles is $2.25. After this, there are still x − 1
miles to go, and the cost for each of these miles is
$0.45. Therefore, the total cost of the trip, in dollars, is
2.25 + 0.45(x − 1).

4.

The correct answer is B. The function f(x) = 5x − 7 is a
linear function, which means that its graph is a straight
line. The slope of the line is the coefficient of x, that is, 5.
A line with a positive slope increases from left to right.

Thus, f(x) = x2 + 3x − 10 = 0 at x = −5 and x = 2.
8.

The correct answer is A. For a linear function, the
y-intercept is equal to the value of the constant.
Therefore, g(x) = −12x + 6 crosses the y-axis at (0, 6).

9.

The correct answer is B. The two functions are both
linear, so their graphs are straight lines. Note that the
coefficient of the x-term in each case is 16. Since this
coefficient represents the slope of the function, both
functions are lines with slopes of 16. Lines with the
same slopes are parallel. The constant term of a linear
function is the y-intercept, so one line has a y-intercept
of −7, and the other has a y-intercept of 5.

The correct answer is D. The function g(x) = −5x2 + 7x
− 23 is a quadratic function, so its graph will be a
parabola. Because the coefficient of the x2-term is
negative, the parabola opens down.

5.

The correct answer is D. The cost of x games, at $2.50
a game, is $2.50x. Adding the $1.50 cost of shoe rental
to this leads to the function f(x) = 2.50x + 1.50.

6.

The correct answer is C. You are asked to find the
value of x for which 4.50x – 15 = 66.

4.50 x − 15 = 66
4.50 x − 15 = 66 Add 15 to both sides.
4.50 x = 81 Divide by 4.50 (or 4.5).
x = 18
Therefore, Deb must sell 18 cakes to make $66.

The correct answer is D. In order to answer this
question, you need to solve the quadratic equation
x2 + 3x − 10 = 0.
x 2 + 3 x − 10 = 0 Factor the left-hand side.
( x + 5)( x − 2) = 0 Set each factor equal to 0.
x +5= 0 x −2 = 0
x = −5
x =2

The correct answer is A. You need to determine the
value of C(18).
C(18 ) = 3(18 ) 2 − 2(18 ) + 7
= 3(324 ) − 36 + 7
= 972 − 36 + 7
= 943

3.

7.

10.

Chapter 12
Functions
and
Intermediate
Algebra

The correct answer is C. The extreme point of a
−b
quadratic function is given by the formula x =
,
2a
2
where a is the coefficient of the x -term and b is the
coefficient of the x-term. In this case, then, the extreme
point is
x =

11.

509

−b
−( −28 )
28
=
=
= 2.
2a
2(7)
14

The correct answer is B. The model giving the bike’s
value f(x) in dollars after x years is: f (x) = 850(0.8)x .
Substitute 500 in for f(x) and solve the equation.
The value of the bike will be less than $500 after
approximately 2.5 years.

Peterson’s SAT® Prep Guide 2017

12.

The correct answer is D. The graph of f(x) = abx−h + k is
the translation of the graph of f(x) = abx.
So f(x) = (0.75)x + 2 is translated up 2 units. The graph’s
asymptote is the line f(x) = 2.

13.

The correct answer is C. The model giving the height
h of the ball (in feet) after t seconds is h(t) = −16t2 + 20 .
Substitute 0 in for h(t) and solve the equation. The ball
will hit the ground about 1.12 seconds after it is
dropped.

510
Chapter 12
Functions
and
Intermediate
Algebra

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14.

The correct answer is A. Substitute 6 for x in the
equation f(x) = 38,000(0.89)x, and solve the equation.
The car will be worth approximately $19,000 after 6
years.

SUMMING IT UP
• A function is a rule that assigns exactly one output to each input. In other words, each member of the domain corresponds
to exactly one member of the range.

• In function notation, the rule of association for a function is written as f(x) to represent the range value.
• For any positive integer n, x − n =

1
xn

1

• For a rational number, x to the power of 1 is defined as the nth root of x. In other words, x n is equal to n x .
n

• A linear function is a function of the form f(x) = mx + b, where m and b are real numbers.
ºº In general, the graph of any linear function is either a straight line or depending on the domain) a portion of a straight line.

• A quadratic function is a function of the form g(x) = ax2 + bx + c, where a ≠ 0 and a, b, and c are real numbers.
ºº In general, the graph of any quadratic function will be a parabola.

• An exponential function is a function of the form f(x) = abx where base b is a positive integer greater than 1.
ºº In general, the graph of any exponential function will curve sharply as it rapidly increases or decreases.

• The four steps in solving an equation involving rational expressions are:
1. Find the LCD of all of the rational expressions in the equation.
2. Multiply every term on both sides of the equation by this LCD.
3. Solve the resulting equation using the methods previously explained.
4. Check the solution to make certain that it actually solves the equation.

• The four steps to solve a radical equation are:
1. Isolate the radical on one side of the equation.

511

2. Raise both sides of the equation to the same power to eliminate the radical.
3. Solve the resulting equation.

Chapter 12

4. Check the solution.
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intermediate algebra in Peterson’s SAT® Online Course:

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• Functions
• Hard Algebra
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• Linear Equations
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• Quadratic Systems
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Peterson’s SAT® Prep Guide 2017

Chapter 13:
Data Analysis, Statistics,
and Probability
OVERVIEW
Averages
Weighted Average
Exercises: Averages
Probability
Exercises: Probability
Data Interpretation
Exercises: Data Interpretation
Statistics
Exercises: Statistics

513

Summing It Up

Chapter 13

AVERAGES

Data Analysis,
Statistics, and
Probability

Arithmetic Mean, Median, Mode, and Range
Typically, when asked to find the average of a group of n numbers, students add up all the numbers in the group and then
divide by n. This type of average, which is more precisely called the arithmetic mean, is just one of a number of different types of
averages, each of which is computed in a different way and conveys a different type of information. There are three additional
ways to describe a set of data. Two of these are also types of averages, known as the median and the mode of the data. A third
value, range, is used to describe data. These will also occasionally appear on the SAT® exam.
In order to avoid ambiguity, when a question involves the computation of the arithmetic mean, it will use the word average
followed by the words arithmetic mean in parentheses. When an average question involves the median, the mode, or the range,
the words median, mode, and range will be specifically used. The arithmetic mean is by far the most common type of average
used on the exam.

Peterson’s SAT® Prep Guide 2017

The Arithmetic Mean
To find the average (arithmetic mean) of a group of n numbers, simply add the numbers and divide by n.
Example:
Find the average (arithmetic mean) of 32, 50, and 47.
Solution:
32
50
+47
129

43
3 129

)

Another type of arithmetic mean problem gives you the arithmetic mean and asks for the missing term.
Example:
The average (arithmetic mean) of three numbers is 43. If two of the numbers are 32 and 50, find the third number.
Solution:

514

Using the definition of arithmetic mean, write the equation:
32 + 50 + x
= 43
3
32 + 50 + x = 129
82 + x = 129

Chapter 13
Data Analysis,
Statistics, and
Probability

x = 47

Median and Mode
In order to find the median of a group of numbers, list the numbers in numerical order from smallest to largest. The median is
the number in the middle. For example, the median of the numbers 3, 3, 5, 9, and 10 is 5. Note that, typically, the median and the
arithmetic mean are not the same. In this problem, for example, the arithmetic mean is 30 ÷ 5 = 6.
If there is an even number of numbers, the median is equal to the arithmetic mean of the two numbers in the middle. For example,
to find the median of 3, 3, 5, 7, 9, and 10, note that the two middle numbers are 5 and 7. The median, then, is

5+7
=6.
2

The mode of a group of numbers is simply the number that occurs most frequently. Therefore, the mode of the group of numbers
3, 3, 5, 7, 9, and 10 is 3. If all of the numbers in a group only appear once, then there is no mode.
Example:
What is the product of the arithmetic mean, the median, and the mode of the following group of eight numbers?
2, 7, 8, 9, 9, 9, 10, and 10

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Solution:
The sum of the eight numbers is 64, so the arithmetic mean is 64 ÷ 8 = 8.
The median is the arithmetic mean of the two numbers in the middle. Since these numbers are both 9, the median is
9+9
=9 .
2
The mode is the number that occurs most often, which is also 9. The product of these three “averages” is
8 × 9 × 9 = 648.

Range
The range, or the spread, of a data set is the difference between the greatest and least data values. To find the range of a set of
data, first write the values in ascending order to make sure that you have found the least and greatest values.
Example:
Celia kept track of the average price of a gallon of gas over the last 10 years. Her data is shown in the table below.
What is the range in the average price of gas?
Solution:

Year

Average price/gallon
in U.S. dollars

2006

2.00

2007

2.08

2008

2.44

2009

3.40

2010

2.85

2011

2.90

2012

3.50

2013

4.20

2014

3.80

2015

3.25

515
Chapter 13
Data Analysis,
Statistics, and
Probability

Write the data in order from least to greatest:
2.00 2.08 2.44 2.85 2.90 3.25 3.40 3.50 3.80 4.20
Subtract the least value from the greatest value: 4.20 – 2.00 = 2.20.
$2.20 is the range or spread of the data.

Peterson’s SAT® Prep Guide 2017

WEIGHTED AVERAGE
If asked to find the arithmetic mean of a group of numbers in which some of the numbers appear more than once, simplify the
computation by using the weighted average formula. For example, suppose the question asks for the average (arithmetic mean)
age of a group of 10 friends. If four of the friends are 17, and six of the friends are 19, determine the average in the usual way:
Average age =

17 + 17 + 17 + 17 + 19 + 19 + 19 + 19 + 19 + 19 182
=
= 18.2
10
10

However, the computation can be done more quickly by taking the weights of the ages into account. The age 17 is weighted four
times, and the age 19 is weighted six times. The average can then be computed as follows:
Average age =

4(17) + 6(19 ) 182
=
= 18.2
10
10

Example:
Andrea has four grades of 90 and two grades of 80 during the spring semester of calculus. What is her average (arithmetic mean) in the course for this semester?
Solution:
90
90
90
90
80
+80

516

)

6 520
2
3

or

90 i 4 = 360

+ 80 i 2 = 160

)

6 520
= 86

2
3

Chapter 13

= 86

Data Analysis,
Statistics, and
Probability

Be sure not to average 90 and 80, since there are four grades of 90 and only two grades of 80.

Average Rate
The average rate for a trip is the total distance covered, divided by the total time spent. Recall that distance can be
determined by multiplying the rate by the time, that is, d = rt.
Example:
In driving from New York to Boston, Mr. Portney drove for 3 hours at 40 miles per hour and 1 hour at 48 miles per
hour. What was his average rate for this portion of the trip?

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Solution:

Total distance
Total time
3 ( 40 ) + 1( 48 )
Average rate =
3 +1
168
Average rate =
= 42 miles per hour
4
Average rate =

Since more of the trip was driven at 40 mph than at 48 mph, the average should be closer to 40 than to 48, which it is.
This will help you to check your answer or to pick out the correct choice in a multiple-choice question.

EXERCISES: AVERAGES
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

2.

Dan had an average (arithmetic mean) of 72 on his first
four math tests. After taking the next test, his average
(arithmetic mean) dropped to 70. Which of the following
is his most recent test grade?
A.

60

B.

62

C.

66

D.

68

What is the average (arithmetic mean) of 0.64 , 0.85,
9
and
?
10
A.

21
25

B.

2.55

C.

85%
4
5

D.

SHOW YOUR WORK HERE

517
Chapter 13
Data Analysis,
Statistics, and
Probability

Peterson’s SAT® Prep Guide 2017

3.

The costs of five different airlines’ tickets from Dallas to
Boston are shown in table below.

Airline

Ticket Cost

A

$356

B

$298

C

$312

D

$304

E

$283

A sixth airline also offers flights from Dallas to
Boston. The median price of the tickets from the six
airlines, including those shown in the table, is $308.
The range of the ticket prices is $77. What is the cost
of the sixth airline’s ticket?

518
Chapter 13

4.

A.

$385

B.

$360

C.

$279

D.

$231

A high school theater department is using sale information from previous years’ sales for the annual musical
to determine how many seats will be needed. The sale
information is shown in the table below.

Data Analysis,
Statistics, and
Probability

Year

Seats

2013

185

2014

203

2015

205

2016

197

Which correctly shows the median and range of the
data?
A.

Median = 200; Range = 20

B.

Median = 204; Range = 12

C.

Median = 200; Range = 12

D.

Median = 204; Range = 20

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SHOW YOUR WORK HERE

5.

6.

7.

What is the average (arithmetic mean) of the first 15
positive integers?
A.

7

B.

7.5

C.

8

D.

8.5

SHOW YOUR WORK HERE

A man travels a distance of 20 miles at 60 miles per hour
and then returns over the same route at 40 miles per
hour. What is his average rate for the round trip in miles
per hour?
A.

50

B.

48

C.

47

D.

46

Max is selling his car. He looks at the selling prices of the
same type of car at five local car dealerships to determine
a fair price for his car. The selling prices are listed below.
$7,505; $7,630; $7,995; $7,029; $7,135; $7,995
What is the approximate average (arithmetic mean)
selling price for the type of car Max is trying to sell?

8.

A.

$7,995

B.

$7,548

C.

$7,512

D.

$7,505

519
Chapter 13
Data Analysis,
Statistics, and
Probability

Susan has an average (arithmetic mean) of 86 in three
examinations. What grade must she receive on her next
test to raise her average (arithmetic mean) to 88?
A.

90

B.

94

C.

96

D.

100

Peterson’s SAT® Prep Guide 2017

9.

10.

The heights of the 5 high jumpers on Redwood High’s
track team are 5'11", 6'3", 6', 6'6", and 6'2". The average
(arithmetic mean) height of these players is
A.

6'2".

B.

6'3".

C.

6'4".

D.

6'5".

The ages of 14 U.S. Presidents at inauguration are listed
here in order of their presidencies. Which of the following correctly compares the average (arithmetic mean),
median, and mode of their ages?
54 51 60 62 43 55 56 61 52 69 64 46 54 47

11.

A.

mode < mean < median

B.

mode < median < mean

C.

median < mode < mean

D.

median < mean < mode

A basketball team has 5 top-scoring players. The average
(arithmetic mean) points per game scored by the 5 players during one season are shown in the list below.

520

15.5, 16.2, 18, 16.5, 17.4
There are 10 games during the season. What is the
average points scored per player for the season for
the 5 top scoring players?

Chapter 13
Data Analysis,
Statistics, and
Probability

12.

A.

16.72

B.

18.72

C.

134.8

D.

167.2

Amaya drives on two types of roads for her trip. She averages 53 miles per hour on city roads and 59 miles per
hour on the highway. Amaya drives her car on a trip that
has twice as many highway miles as city road miles. She
drives a total of 552 miles. What is her average speed for
the whole trip? Round your answer to the nearest whole.
A.

55 miles per hour

B.

56 miles per hour

C.

57 miles per hour

D.

58 miles per hour

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ANSWER KEY AND EXPLANATIONS

1.

1. B

3. B

5. C

7. B

9. A

11. D

2. C

4. A

6. B

8. B

10. B

12. C

The correct answer is B.

4.

4(72) + x
= 70
5
288 + x = 350

185, 197, 203, 205
The median is the middle value of the list. Since the
list does not have a distinct middle value, add the two
middle values 197 and 203, then divide by 2 to find
the median.

x = 62
2.

The correct answer is C. In order to find the average
(arithmetic mean) of these three numbers, they should
all be expressed as decimals.

197 + 203 = 400; 400 ÷ 2 = 200

0.64 = 0.8
0.85 = 0.85
9
= 0.9
10
0.8 + 0.85 + 0.9
Average =
3
2.55
=
= 0.85 (aarithmetic mean)
3

The median is 200.
To find the range, subtract the least from the greatest
value:
205 – 185 = 20
The range is 20.

This is equal to 85%.
3.

The correct answer is B. List the given ticket costs
from least to greatest:

The correct answer is A. List the given number of seats
from least to greatest:

5.

The correct answer is C. Positive integers begin with 1.

$283, $298, $304, $312, $356

1+ 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11+ 12 + 13 + 14 + 15
15

Add $77 to the least cost to determine if the unknown cost is also the greatest cost:

Since these numbers are evenly spaced, the average
(arithmetic mean) will be the middle number, 8.

521
Chapter 13
Data Analysis,
Statistics, and
Probability

283 + 77 = 360
Use the value to find the median of the tickets:
$283, $298, $304, $312, $356, $360
Find the average (mean) of $304 and $312, which is
$308.
The unknown ticket cost is $360.

Peterson’s SAT® Prep Guide 2017

6.

The correct answer is B.

Average rate =

10.

Total distance
Total time

43, 46, 47, 51, 52, 54, 54, 55, 56, 60, 61, 62, 64, 69

Total distance = 20 + 20 = 40

Since time =

The mode of their ages is 54, which is the only
repeated age.
The median of their ages is the middle number, which
is the average of 54 and 55, or 54.5.

distance
, time for first part of the trip is
rate

20
1
, or hour, while time for the second part of the
60
3
trip is

20
1
, or hour.
2
40

1 1
5
Total time = + , or hour
3 2
6
Average rate =

7.

The average or arithmetic mean is found by adding
their ages and dividing by 14.
43 + 46 + 47 + 51+ 52 + 54 + 54 + 55 + 56 + 60 + 61+ 62 + 64 + 69
14
774
=
= 55.3
14
Now order the three values: 54 < 54.5 < 55.3, which
means that mode < median < mean.

40
6
= 40 i = 48 mph
5
5
6

11.

The correct answer is B.
7, 505 + 7, 630 + 7, 995 + 7, 029 + 7,135 + 7, 995 45, 289
=
6
6
= 7, 548.167
The approximate average of the cars’ selling prices is
$7,548.

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Chapter 13

8.

3(86 ) + x
= 88
4
258 + x = 352
x = 94
9.

The correct answer is A.

)

29 ’ 22 " = 5 30 ’10 "
6 ’2"

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Average points for the season:
155 + 162 + 180 + 165 + 174 836
=
= 167.2 points per season
5
5
12.

5 ’11"
6 ’ 3"
6’ 0"
6’ 6"
6’ 2"

The correct answer is D. To find the average points
scored for the season by all 5 players, you can first find
the total number of points each player scored during
the season. Each player played 10 games, so to find
their total number of points scored, multiply the
average points per game by 10 games.
15.5 × 10 = 155
16.2 × 10 = 162
18 × 10 = 180
16.5 × 10 = 165
17.4 × 10 = 174

The correct answer is B.

Data Analysis,
Statistics, and
Probability

The correct answer is B. First, put the ages of the
presidents in order.

The correct answer is C. Amaya drove for a time on
city roads at 53 miles/hour and twice as much time on
the highway at 59 miles/hour, so her average speed
was:
53 x + 59(2 x ) 171x
=
= 57
3x
3x
The average speed for the entire trip was 57 miles per
hour.

PROBABILITY
In the study of probability, an experiment is a process that yields one of a certain number of possible outcomes. For example,
tossing a coin is an experiment with two possible outcomes: heads or tails. Throwing a die is an experiment with six possible
outcomes because there are six sides: 1, 2, 3, 4, 5, and 6.
Probability is a numerical way of measuring the likelihood that a specific outcome will happen. The probability of a specific outcome
is always a number between 0 and 1. An outcome with a probability of 0 cannot possibly happen, and an event with a probability
of 1 will definitely happen. Therefore, the nearer the probability of an event is to 0, the less likely the event is to happen, and the
nearer the probability of an event is to 1, the more likely the event is to happen.
If an experiment has n possible, equally likely outcomes, the probability of each specific outcome is defined to be

1
. In the coinn

1
, since heads is one of the two equally likely outcomes. When a die is thrown,
2
3
1
the probability of tossing an odd number is , or , since tossing an odd number is three of six equally likely outcomes.
6
2

tossing experiment, the probability of heads is

Here is a helpful formula for computing simple probabilities:
The probability of an event occurring =

The number of favorable outcomes
The number of possible outtcomes

Conditional Probability
The probability of an event occurring after another event has already occurred is called conditional probability.
The notation for conditional probability is P(B|A), which is read “the probability of B given A.
If both events A and B are independent, where the result of B is not affected by the result of A, then the conditional probability
of B is equal to the probability of B only.

523
Chapter 13

(B )
P
= P(B )
( A)

Data Analysis,
Statistics, and
Probability

If both events A and B are dependent, where the result of B is affected by the result of A, then the conditional probability of B may
be calculated using a few different methods. In probability situations involving dependent events, the way in which the question
is asked will determine which of the following equations to use.
P( A and B )
P( A and B )
 B
OR
OR P( A and B ) = P( A) × P  
 A
P ( A)
P(B )
Example:
Of 100 people who work out at a gym, there are 45 people who take yoga classes, 55 people who take weightlifting
classes, and 15 people who take both yoga and weightlifting. What is the probability that one of the randomly
chosen people who work out at the gym takes both yoga and weightlifting classes?

Peterson’s SAT® Prep Guide 2017

Solution:
This question is asking you to find the probability that a person is taking both a yoga class and a weightlifting class
(A and B), given that the person is in a yoga class (A).
P( A and B ) 0.15 1
=
=
P ( A)
0.45 3
The conditional probability that a person who takes yoga is also in a weightlifting class is

1
.
3

Independent Events
If events A and B are independent, then the probability of event B is not affected by the result of event A.
Example:
What is the probability of tossing two coins that both land heads up, if the first coin has already been tossed and
landed heads up?
Solution:
The result of the first toss does not affect the result of the second toss, so the conditional probability is equal to the
probability of B (the second coin toss).

524
Chapter 13
Data Analysis,
Statistics, and
Probability

P(B ) =

1
2

Example:
What is the probability of tossing two coins and having them both land heads up?

Solution:
Neither coin has been tossed, so the conditional probability of both coins landing heads up is determined using the
probability of both events. In this case, the probability of event A and event B both happening is found by multiplying
the probability of each event.
P( A and B ) =

1 1 1
× =
2 2 4

The probability of tossing two coins and having them both land heads up is

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1
= 0.25.
4

Dependent Events
If events A and B are dependent, then the probability of event B is affected by the result of event A.
Example:
What is the probability of choosing a black 5 and then a red 5 from the same standard deck of cards?
Solution:
There are two red 5s and two black 5s in a standard deck of 52 cards. Removing a black 5 from the deck will leave only 51 cards
in the deck, so the probability can be found by:
P( A and B ) =

2 2
1
× =
52 51 663

Two-Way Tables and Probability
Data on the exam is often represented in two-way tables like the one in the example below. Be careful to read the correct row or
column that represents the information in the problem. These tables are often used for two or three problems in a row.
Example:
Hattie is a member of the honor society. All members of the society are polled to determine how many hours they
spend studying per week and whether they prefer math or science classes. The results are shown in the table below.

Science

Math

Total

0−3 hours per week

4

2

6

4−6 hours per week

6

7

13

Total

10

9

19

525
Chapter 13
Data Analysis,
Statistics, and
Probability

What is the probability that an honor society member selected at random prefers math, given that the member
studies 4–6 hours per week?
Solution:
There are a total of 13 students who study 4–6 hours per week, and 7 of them prefer math. So the probability is

7
.
13

Peterson’s SAT® Prep Guide 2017

EXERCISES: PROBABILITY
DIRECTIONS: Work out each problem. Circle the letter of your choice.

1.

2.

A school cafeteria offers students three meal choices:
salad, grilled chicken, and pot roast. There are four side
choices: grilled vegetables, baked potato, rice, or soup.
What is the probability that a student who randomly
chooses a meal and a side chooses the grilled vegetables
after having chosen the grilled chicken?
A.

1
3

B.

1
4

C.

1
7

D.

1
12

A poll asked 100 visitors at a national park the distance
they hiked during their visit. The table shows the data
collected based on the visitors’ ages and hiking distances.

18−25
years old

26−35
years old

Total

0−2 miles

16

22

38

Chapter 13

2−4 miles

20

17

37

Data Analysis,
Statistics, and
Probability

4+ miles

11

14

25

Total

47

53

100

526

What is the probability that a visitor hiked greater
than 4 miles, given that the visitor was 18–25 years
old?
A.

11
14

B.

11
25

C.

11
36

D.

11
47

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3.

A game uses letter tiles that are selected randomly out
of a bag. The letters that are remaining in the bag after
each player has taken four turns are shown below.

SHOW YOUR WORK HERE

S, U, G, R, A, O, E, T, V, E, A, H, S, E
What is the probability of the player selecting two
vowel tiles, if the player selects two tiles from the
letters that remain?

4.

A.

3
13

B.

1
4

C.

6
13

D.

1
2

In a movie theater, there are 32 people who bought
popcorn, 25 people who bought a drink, and 12 people
who bought popcorn and a drink. What is the probability
that a randomly selected person who bought popcorn
also bought a drink?
A.

3
8

B.

12
25

C.

25
32

D.

12
13

527
Chapter 13
Data Analysis,
Statistics, and
Probability

Peterson’s SAT® Prep Guide 2017

5.

The two-way frequency table below shows the results
of a poll regarding video game play. The poll asked 150
randomly selected people the amount of time they spend
playing video games each week and the type of game
they most like to play. The table shows relative frequencies of each category.

1−3
hours

3−5
hours

5+
hours

Total

Roleplaying

12

15

16

43

Platform

24

19

18

61

Action

33

35

28

96

Total

69

69

62

200

What is the probability that a person plays 3–5 hours
of games per week, given that they prefer platform
games?

528
Chapter 13
Data Analysis,
Statistics, and
Probability

6.

A.

19
35

B.

19
42

C.

19
61

D.

19
69

Which question below is the probability of two dependent events?
A.

Two coins are flipped. What is the probability of
the second coin landing heads up, if the first coin
landed tails up?

B.

Two number cubes are rolled. What is the
probability of rolling two even numbers?

C.

A card is selected from a deck of 52 cards. What is
the probability of selecting 3 hearts in a row, if
each card is replaced after being selected?

D.

A bag contains 5 red marbles and 6 blue marbles.
What is the probability of selecting two marbles
of different colors, without replacement of the
marbles?

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7.

A summer camp held a fishing tournament. The participants selected the type of bait they would use. The table below
shows the first type of fish caught by each participant and the bait they used.

Perch

Bass

Trout

Catfish

Total

Live Bait

10

4

4

6

24

Artificial Bait

5

5

2

0

12

Total

15

9

6

6

36

What is the probability that a participant will catch a
bass, given that artificial bait was used?

8.

A.

1
4

B.

5
9

C.

5
12

D.

5
36

At a school dance, the DJ offers five kinds of music: Electronic, Rap, Pop, Swing, and Rock. If the kind of music
played is selected at random, what is the probability that
the next two songs the DJ will play will be an Electronic
song and then a Pop song?
A.

1
25

B.

1
10

C.

1
5

D.

2
5

SHOW YOUR WORK HERE

529
Chapter 13
Data Analysis,
Statistics, and
Probability

Peterson’s SAT® Prep Guide 2017

9.

During a science experiment, two studies of chemical
reactions are conducted. Each experiment requires a
5
hypothesis. Hypothesis A is correct of the times the
7
3
experiment is conducted. Hypothesis B is correct
of
4
the times the experiment is conducted. If the two hypotheses do not affect each other, what is the probability that
both hypotheses A and B are correct?

10.

530

A.

15
28

B.

5
7

C.

3
4

D.

8
11

Gustav finds that during daily marching band practice he
correctly completes all steps 7 of the time. During daily
8
symphonic band practice he correctly plays each piece
11
of music
of the time. The probability that he completes
12
all steps in marching band and plays each piece of music
3
correctly in symphonic band during the same day is .
5

Chapter 13

What is the probability that he correctly completes all

Data Analysis,
Statistics, and
Probability

steps during marching band, given that he has attended
symphonic band?
A.

24
35

B.

36
55

C.

11
20

D.

21
40

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ANSWER KEY AND EXPLANATIONS

1.

1. B

3. A

5. C

7. C

9. A

2. D

4. B

6. D

8. A

10. B

The correct answer is B. The probability of choosing
grilled vegetables is independent of choosing grilled
chicken. There are four sides to choose from, and
grilled vegetables is one of those four choices, or

2.

The correct answer is D. There are a total of 47 visitors
who are 18–25 years old. In that age group, 11 of those

The correct answer is A. Seven of the 14 tiles are
vowels, so the probability of selecting one vowel is
7 1
= . For the second tile, there are only 13 tiles to
14 2
choose from and 6 vowel tiles remaining.
P( A and B ) = P( A) × (B )
1 6
= ×
2 13
6
=
26
3
=
13

4.

The correct answer is B. P(A and B) is the probability
of someone having both popcorn and a drink, and P(B)
is the probability of someone having a drink. So the
probability of selecting a person with both popcorn
and a drink is P( A and B ) = 12 .
25
P(B )

5.

The correct answer is D. Situations that state there is
no replacement have an event that is dependent on
the other. In this case, the second chosen marble is
dependent on the marble that was chosen first
because there will be one less marble in the bag and it
must be a different color.

7.

The correct answer is C. There are 12 total fish caught
using artificial bait. The number of bass caught out of
5
those 12 is 5, so the probability is
.
12

8.

The correct answer is A. The probability of each type
of song being played are independent of each other.
The probability of an Electronic song being played is
1
and the probability of a Pop song being played
5
is 1 . The probability that an Electronic song and then
5
1 1 1
a Pop song will be played is found by × = .
5 5 25

1
.
4

visitors walked more than 4 miles, so the probability is
11
.
47
3.

6.

The correct answer is C. There are 61 total people
who play platform games. Of those 61 people, 19 play
19
for 3–5 hours, so the probability is
.
61

9.

The correct answer is A. The probability of each
hypothesis is independent of the other. The probability
of both events occurring can be found by
5 3 15 .
P( A and B ) = P( A) × P(B ) = × =
7 4 28

10.

531
Chapter 13
Data Analysis,
Statistics, and
Probability

The correct answer is B. The probability of correctly
completing the marching steps, given he has attended
symphonic band, can be found by
3
P( A and B ) 5 3 12 36
.
=
= × =
11 5 11 55
P(B )
12

Peterson’s SAT® Prep Guide 2017

DATA INTERPRETATION
Working with Data in Tables
Some SAT® exam questions ask you to solve mathematical problems based on data contained in tables. All such problems are
based on problem-solving techniques that have already been reviewed. The trick when working with tables is to make certain
that you select the correct data needed to solve the problem. Take your time reading each table so that you understand exactly
what information the table contains. Carefully select data from the correct row and column. As you will see, things are even trickier
when a problem involves more than one table.
In order to illustrate problem solving with tables, consider the two tables below. The three questions that follow are based on
the data within these tables.
Paul, Mark, and Bob are computer salespeople. In addition to their regular salaries, they each receive a commission for each
computer they sell. The number of computers that each salesperson sold during a particular week, as well as their commission
amounts, is shown in the tables below.

Number of Computers Sold
Monday

Tuesday

Wednesday

Thursday

Friday

Paul

9

3

12

6

4

Mark

6

3

9

1

5

Bob

8

4

5

7

8

Commission per Sale

532
Chapter 13
Data Analysis,
Statistics, and
Probability

Paul

$15

Mark

$20

Bob

$25

Example:
What is the total amount of the commissions that Bob earned over the entire week?
Solution:
This problem concerns only Bob, so ignore the information for Mark and Paul. Over the course of the week, Bob sold
8 + 4 + 5 + 7 + 8 = 32 computers. The second table tells us that Bob earns $25 per sale, so the total amount of his
commission would be $25 × 32 = $800.

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Example:
What is the total amount of commission money earned by Paul, Mark, and Bob on Thursday?
Solution:
To solve this problem, focus only on what happened on Thursday. Ignore the data for the other four days. Be careful
not to add the number of computers sold by the three people, since they each earn a different commission per sale.

• On Thursday, Paul sold 6 computers and earned a $15 commission for each computer sold, so Paul earned $15 × 6 = $90.
• Mark sold 1 computer, so, based on his $20 commission, he earned $20.
• Bob sold 7 computers and earned a $25 commission per machine, so he made $25 × 7 = $175.
• Overall, the amount of commission on Thursday is $90 + $20 + $175 = $285.
Example:
On what day did Paul and Mark earn the same amount in commission?
Solution:
You can save yourself a lot of time if you look at the tables before you start to compute. Note that Mark’s commission
is larger than Paul’s, and so the only way they could have earned the same amount is if Paul sold more computers
than Mark. The only days that Paul sold more computers than Mark were Monday, Wednesday, and Thursday, so those
are the only days that need to be considered. By observation, you can see the following:

• On Thursday, Paul made much more in commission than Mark, so eliminate Thursday.

533

• On Monday, Paul earned $15 × 9 = $135 and Mark earned $20 × 6 = $120. This means that the answer must be Wednesday.
• To be certain, note that on Wednesday Paul earned $15 × 12 = $180, and Mark earned $20 × 9 = $180 also.

Chapter 13
Data Analysis,
Statistics, and
Probability

Correlation and Scatterplots
If two variables have a relationship such that when one variable changes, the other changes in a predictable way, the two variables
are correlated. For example, there is a correlation between the number of hours an employee works each week and the amount of
money the employee earns—the more hours the employee works, the more money the employee earns. Note that in this case,
as the first variable increases, the second variable increases as well. These two variables are positively correlated.
Sometimes, when one variable increases, a second variable decreases. For example, the more that a store charges for a particular
item, the fewer of that item will be sold. In this case, these two variables are negatively correlated.
Sometimes, two variables are not correlated, that is, a change in one variable does not affect the other variable in any way. For
example, the number of cans of soda that a person drinks each day is not correlated with the amount of money the person earns.

Peterson’s SAT® Prep Guide 2017

One way to determine whether two variables are correlated or not is to sketch a scatterplot. A scatterplot is a graph on which the
x-axis represents the value of one variable and the y-axis represents the value of the other variable. Several values of one variable
and the corresponding values of the other variable are measured and plotted on the graph:

• If the points appear to form a straight line, or are close to forming a straight line, then it is likely that the variables are
correlated.

• If the line has a positive slope (rises up left to right), the variables are positively correlated.
• If the line has a negative slope (goes down left to right), the variables are negatively correlated.
• If the points on the scatterplot seem to be located more or less at random, then it is likely that the variables are not
correlated.

Positive Correlation

534

y

y

y

x

Negative Correlation x

x

It is rare that the points on a scatterplot will all lie exactly on the same line. However, if there is a strong correlation, it is likely
that there will be a line that could be drawn on the scatterplot that comes close to all of the points. Statisticians call the line that
comes the closest to all of the points on a scatterplot the “line of best fit.” Without performing any computations, it is possible to
visualize the location of the line of best fit, as the diagrams below show:

Chapter 13
Data Analysis,
Statistics, and
Probability

No Correlation

y
y

x

www.petersons.com

x

Example:
Which of the following slopes is the closest to the slope of the line of best fit for the scatterplot shown here?
y

x
A.

3

B.

1

C.

0

D . −1
Solution:
Begin by sketching in the line of best fit in its approximate location.

535

y

Chapter 13
Data Analysis,
Statistics, and
Probability

x
This line has a negative slope, since it decreases from left to right. In addition, the slope appears to be about −1, because
if you move one unit horizontally from a point on the line, you need to move one unit vertically downward to return to
the line. The correct answer is choice D.

Peterson’s SAT® Prep Guide 2017

Equation of Line of Best Fit
You may see questions that ask you to create a line of best fit and give the equation for this line. Again, there is some approximation involved in determining a line of best fit. Look at the following scatterplot. Imagine a line that would come closest to all
the data points and include an equal number of data points on either side of the line.

y

x
Notice the line of best fit below has 8 points above and 8 points below the line.
y

536
Chapter 13
Data Analysis,
Statistics, and
Probability

x

Note that the line of best fit does not have to cross any of the data points. In this case, the line of best fit goes through the middle
of the data and does not include any of the actual data points.
To determine the equation of the line of best fit, choose two points that lay on the line of best fit. You may have to approximate
points if the line does not have points that are exactly on an intersection. The line appears to have points at (6, 8) and (12, 3).
Use these two points to determine the equation of the line of best fit. First, calculate the slope, m, using the slope formula
y 2 − y1
. Plug x- and y-values into the equation: 3 − 8 = −5 The slope-intercept formula is y = mx + b.
x 2 − x1
6
12 − 6
So far, we have y =

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−5
x +b
6

Plug in one point into x and y in the equation.
−5
(6) + b
6
8 = −5 + b
b = 13

(6, 8): 8 =

The equation of the line of best fit is: y =

−5
x + 13.
6

Making Predictions
Assume the same previous scatterplot has real-world data. The following scatterplot now shows the average number of books
borrowed on a weekly basis for years 2000–2009 at a local library.

Library Books Borrowed
10
9

Books Borrowed (100s)

8
7
6
5
4
3

537

2
1

Chapter 13

0
1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

Year

Data Analysis,
Statistics, and
Probability

Example:
If the trend continued, about how many books were borrowed in 2010?
Solution:
Use the slope of the line to make predictions about data points that are not shown. According to the slope, the
average number of books borrowed weekly goes down approximately 0.6 × 100 = 60 books every year. Multiply
by 100 because, according to the title of the vertical axis, the numbers are in the 100s. The expected value for the
number of books borrowed in 2009 was 400. The slope says we should expect that number to decrease by 60 every
year, so you can predict that there were 340 books borrowed in 2010.

Peterson’s SAT® Prep Guide 2017

EXERCISES: DATA INTERPRETATION
DIRECTIONS: Work out each problem. Circle the letter of your choice.

Questions 1–3 refer to the following information.
The scatterplot below shows average monthly feed and labor costs in dollars to raise different numbers of heads of beef cattle.

Average Monthly Cost

Feed and Labor Costs
1,000
900
800
700
600
500
400
300
200
100
0
0

50

100

150

200

250

Number of Heads of Cattle

538
Chapter 13

1.

Data Analysis,
Statistics, and
Probability

2.

Which of the following is closest to the slope of a line of
best fit for this data?
A.

200

B.

1
2

C.

10
9

D.

2

Based on the scatterplot, what is the equation of the line
of best fit?
A.

y = 200x + 400

B.

y = 2x + 320

C.

y = 2x + 380
1
y = x + 380
2

D.

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SHOW YOUR WORK HERE

3.

If the trend continued, what would be the average monthly
feed and labor costs to raise 500 heads of beef cattle?
A.

$1,380.00

B.

$1,000.00

C.

$880.00

D.

$600.00

SHOW YOUR WORK HERE

Questions 4–6 refer to the following information.

Geographic Area

Attended
College

Did Not Attend
College

No Response

Total

Northeast

72,404

68,350

29,542

170,296

Northwest

88,960

125,487

48,960

263,407

Southeast

115,488

96,541

65,880

277,909

Southwest

79,880

65,874

13,840

159,594

Total

356,732

356,252

158,222

871,206

A survey was conducted in different geographic areas of a large state, covering the entire state population, about whether people
over the age of 30 attended college. The table above displays a summary of the survey results.

4.

According to the table, for which group did the highest
percentage of people report that they had attended
college?
A.

Northeast

B.

Northwest

C.

Southeast

D.

Southwest

SHOW YOUR WORK HERE

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Chapter 13
Data Analysis,
Statistics, and
Probability

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5.

6.

Of the people living in the northeast who reported that
they did not attend college, 1,000 people were selected at
random to do a follow-up survey where they were asked if
they were interested in attending adult education classes.
There were 665 people who said they were interested in
attending adult education classes. Using the data from
both the initial survey and the follow-up survey, which of
the following is most likely to be an accurate statement?
A.

About 48,149 people living in the northeast who
did not attend college would be interested in
adult education classes.

B.

About 45,453 people living in the northeast who
did not attend college would be interested in
adult education classes.

C.

About 19,645 people living in the northeast who
did not attend college would be interested in
adult education classes.

D.

Most people in the state are not interested in
taking adult education classes.

What is the relative frequency of the number of people
who attended college statewide, according to the survey?
A.

0.18

540

B.

0.41

C.

0.43

Chapter 13

D.

0.5

Data Analysis,
Statistics, and
Probability

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Questions 7–8 refer to the following information.
In this election, Haynes and Cogswell were the only ones who received votes.

7.

8.

9.

District A

District B

District C

District D

Cogswell

150

350

300

200

Haynes

100

200

450

150

What percent of the votes cast in District C did Haynes
receive?
A.

40%

B.

45%

C.

60%

D.

65%

SHOW YOUR WORK HERE

What is the ratio of the total number of votes received
by Cogswell to the total number of votes received by
Haynes?
A.

10:9

B.

9:10

C.

5:4

D.

4:5

541

The scatterplot below depicts the relationship between
two variables, x and y. Which of the following best describes the line that best fits the data in the scatterplot?

Chapter 13
Data Analysis,
Statistics, and
Probability

y

x
A.

The slope of the line of best fit is −3.

B.

The slope of the line of best fit is −1.

C.

The slope of the line of best fit is 1.

D.

The slope of the line of best fit is 3.

Peterson’s SAT® Prep Guide 2017

10.

Below is a scatterplot depicting the relationship between
the variable a and the variable b. Which of the following
best describes the relationship depicted in the graph?
b

a
A.

There is a strong positive correlation between the
two variables.

B.

There is a strong negative correlation between the
two variables.

C.

The two variables are not correlated.

D.

As the value of a increases, the value of b
decreases.

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SHOW YOUR WORK HERE

ANSWER KEY AND EXPLANATIONS

1.

1. D

3. A

5. B

7. C

9. D

2. C

4. D

6. B

8. A

10. C

The correct answer is D. Use two points that would be on the line of best fit, which would look something like this:

Feed and Labor Costs
1,000

Average Monthly Cost

900
800
700
600
500
400
300
200
100
0
0

50

100

150

200

250

Number of Heads of Cattle

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Chapter 13

Choose two points on this line: (60, 500) and (160, 700) look like they lay on, or close to, the line.
Use these coordinates in the slope formula:

y 2 − y1 700 − 500 200
=
=
=2.
x 2 − x1 160 − 60 100

2.

The correct answer is C. Using m = 2 as the slope
(from Question 1), insert a set of xy-coordinates into
the point-slope formula y = mx+ b: 500 = (2)(60) + b;
500 – 120 = b; b = 380. This gives you the equation
y = 2x + 380.

3.

The correct answer is A. Use the equation of the line
of best fit to make predictions. Plug in 500 for the
x-value: y = 2(500) + 380 = 1,000 + 380 = 1,380.

4.

Data Analysis,
Statistics, and
Probability

The correct answer is D. The percentage of people
who attended college in the southwest region is
79 , 880
≅ 0.5 = 50% .
159 , 594
The percentages for the other areas are:
72, 404
= 0.425 ≅ 43%
170 , 296
88 , 960
Northwest =
= 0.337 ≅ 34%
263, 407
115, 488
Southeast =
= 0.415 ≅ 42%
277, 909
Northeast =

Peterson’s SAT® Prep Guide 2017

5.

The correct answer is B. Extrapolating from the
second survey, we can predict that

9.

665
= 66.5%
1, 000

of the total population of the northeast will likely be

The correct answer is D. The figure here shows the
scatterplot with the approximate line of best fit
sketched in.
y

interested in taking adult education classes. Applying
this to the total northeast population who reported
that they did not attend college: 68,350 × 0.665 =
45,453 people in this population are likely to be
interested in adult education classes.
6.

The correct answer is B. The relative frequency of
college graduates in the survey is calculated by taking
the total number of people who reported having
attended college and dividing by the total number of
people in the survey:

x

Since the line increases from left to right, the line
has a positive slope. Also note that, if you move one
unit horizontally away from a point on the line, you
need to move approximately 3 units vertically to
return to the line. Therefore, the slope of the line of
best fit is approximately 3.

356 , 732
= 0.409 ≅ 0.41 .
871, 206
7.

The correct answer is C. Haynes received 450 out of a
total of 450 + 300 = 750 votes. Therefore, he received
of the vote

8.

544

450
= 60% of the vote.
750

The correct answer is A. Cogswell received 1,000
votes in total, and Haynes received 900. The ratio of
Cogswell to Haynes, therefore, is 1,000 to 900, or 10:9.

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Statistics, and
Probability

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10.

The correct answer is C. The points on the scatterplot
appear to be randomly arranged on the graph. This is
an indication that the values of a and b are not
correlated.

STATISTICS
Comparing Data Sets Using Shape, Center, and Spread
Statistics questions on the SAT® exam require using measures of central tendency, meaning the mean (average), median, and
mode of a data set. You will need to be familiar with the shape, center, and spread of data. The shape of the data refers to the
normal distribution curve, which we examine below. The center could be the average (arithmetic mean) or the median of the data
values in the set. The spread is the range of the data, or the standard deviation of the data that describes the distance between
values in a data set.
Data may be presented in tables, bar graphs, and other methods, so it is important to be familiar with different types of data
presentation.

Standard Deviation—Normal Distribution
A standard deviation describes how far the data values in a set are from the mean or how much they “deviate” from the mean.
The graphs below are both normal distribution curves. In the graph on the left, because much of the data clusters closely around
the mean, there is a small standard deviation. In the graph on the right, because the data is more spread out, there is a large
standard deviation. If these were sets of test scores for Class A and Class B on a math exam, most of the scores in Class A would
be very close to the average score, but, in Class B, the scores would be more varied.

Class A

Class B

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Chapter 13
Data Analysis,
Statistics, and
Probability

What You Really Need to Know About Standard Deviation
You won’t have to calculate the standard deviation, but it is important to understand how to use it. If data has a normal distribution, the empirical rule states that:

• Approximately 68% of the data falls within 1 standard deviation of the mean.
• Approximately 95% of the data falls within 2 standard deviations of the mean.
• Approximately 99¾% of the data falls within 3 standard deviations of the mean.
Let’s see how this works with a table of data. The table below lists the number of students enrolled in six different sections of
algebra, A, B, C, D, E, and F, offered by a college. First calculate the mean.

Peterson’s SAT® Prep Guide 2017

Number of Students

A

B

C

D

E

F

18

23

15

19

28

11

mean =

18 + 23 + 15 + 19 + 28 + 11 114
=
= 19
6
6
99.74%
95.44%
68.26%

Standard
Deviations

13%

2.15%

4

13.59%

9

34.13% 34.13%

14

19

13.59%

24

2.15%

29

13%

34

The mean is always in the middle. Here, it is 19, and each standard deviation is a distance of 5 from the mean of 19.
The standard deviation of this data set is given as 5. In this example, the standard deviation of 5 means that approximately 68%
of the algebra sections have between 14 (19 – 5) and 24 (19 + 5) students, plus or minus 1 standard deviation.

546

Approximately 95% of the algebra sections lie within 2 standard deviations, meaning the sections have between 9 and 29
students.

Chapter 13
Data Analysis,
Statistics, and
Probability

Confidence Intervals and Measurement Errors
Measurement or sampling errors will usually occur when data cannot be collected about an entire population. If, for example, we
are trying to determine the mean salary of people living in a city with 3.5 million people, we would probably use a smaller random
sample of several hundred to several thousand people that was representative of the population. The difference between the
mean salary of the actual population and that of the sample population is called a measurement or sampling error. The sampling
error decreases as the sample size increases, because there is more data that should more accurately reflect the true population.
Here’s an example:
A packaging company is gathering data about how many oranges it can fit into a crate. It takes a sample of 36 crates out of a
total shipment of 5,540 crates. The sample mean is 102 oranges with a sampling error of 6 oranges at a 95% confidence interval.

What Does the Confidence Interval Mean?
In this case, it means that based on the sample, you can be 95% confident that the true population mean for the entire shipment
is between 102 – 6 and 102 + 6, or 96 and 108 oranges per crate.
A confidence interval tells you how close the sample mean is to the actual mean of the entire population.

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Comparing Data Sets Using Mean and Standard Deviation
You may have two sets of data with similar means and ranges but different standard deviations. For the data set with the greater
standard deviation, more of the data are farther from the mean.
Example:
A coach is deciding between 2 baseball players to recruit to his team. He is looking at player performance over the
past 10 seasons. Both players have the same mean batting average of 0.270 and the same range of batting averages
(0.080) over the past 10 seasons. However, Player A’s batting averages have a higher standard deviation than Player
B’s batting averages. What does this indicate about each player?
A.

Player A is more likely to hit better than his mean batting average.

B.

Player B is more likely to hit better than his mean batting average.

C.

Player A’s batting average is more erratic.

D.

Player B’s batting average is more erratic.

Solution:
A greater standard deviation in a data set means that the data values are more spread out, or erratic, meaning performance varied more. Essentially, Player B’s batting average is more dependable; he more frequently batted close to his
batting average than did Player A. So, if the manager is looking for reliability, he may want to choose Player B for his
team. The correct answer is choice C.

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Probability

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Making Inferences Using Sample Data
Frequently, a population is too large for every data value to be measured. Instead, a random sample of the population is used to
make inferences about the true population.
Example:
An online retailer wants to determine the average dollar value of an order that it receives on a daily basis. Based on a
random sample of 200 orders, the mean dollar value is $72.00 and the standard deviation is $5.00.
The normal distribution curve would look like this:

99.74%
95.44%
68.26%

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Chapter 13
Data Analysis,
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Probability

Standard
Deviations

13%

2.15%

13.59%

34.13% 34.13%

13.59%

2.15%

13%

$57.00 $62.00 $67.00 $72.00 $77.00 $82.00 $87.00

If the sample used is representative of the true population, what can be concluded about the true population of
shoppers?
A.

The mean of the true population is $72.00.

B.

Most shoppers spend between $62.00 and $82.00.

C.

All shoppers spend between $57.00 and $87.00.

D.

The mode of the size of an order is $72.00.

Solution:
Shoppers who spent between $62.00 and $82.00 fall within 2 standard deviations of the mean of the sample population.
Approximately 95% of the shoppers fall within 2 standard deviations of the mean. Thus, it can be concluded that most
shoppers spend between $62.00 and $82.00. Though the sample is representative, it does not indicate that the mean of
the true population is equal to the mean of the sample population, so choice A is incorrect. Approximately 99¾% of the
population lies within 3 standard deviations of the mean, which, in this case, is between $57.00 and $87.00—but this
is not all shoppers, so choice C is not fully supported. We don’t know the mode of the data, so choice D is not correct.
The correct answer is choice B.

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Comparing Data Sets Using Spread
The spread is simply the difference between the least and greatest value in a set.
Example:
Todd’s meteorology class researched weekly precipitation in a tropical region during two 6-week periods. The data in
Set A covers a period from January through mid-February, and the data in Set B is from July through mid-August. The
results appear in the tables below.

Set A: Weeks 1–6 (January–February)
0.43

1.73

1.93

0.28

0.08

1.18

0.04

0.00

Set B: Weeks 27–32 (July–August)
0.20

0.01

0.00

0.08

Which of the following is true of the above sets of data?
A.

The range of the data in Set A is greater than that of Set B.

B.

The mean of the data in Set B is greater than that of Set A.

C.

The mode of the data in Set A is equal to that of the data in Set B.

D.

The mode of the data in Set A is greater than that of the data in Set B.

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Solution:
To calculate range, write the data values in order from least to greatest for each set:
		

Set A: 0.08; 0.28; 0.43; 1.18; 1.73; 1.93

		

Set B: 0.00; 0.00; 0.01; 0.04; 0.08; 0.20

Chapter 13
Data Analysis,
Statistics, and
Probability

Find the difference between the least and greatest value for each set:
		

Set A: Range = 1.93 – 0.08 = 1.85

		

Set B: Range = 0.20 – 0.00 – 0.20

The range in Set A is greater. The correct answer is choice A.

Peterson’s SAT® Prep Guide 2017

Comparing Data Sets Using Median and Mode
Using the same data sets about tropical precipitation, let’s look at problems involving median and mode.
Example:

Set A: Weeks 3–8 (January–February)
0.43

1.73

1.93

0.28

0.08

1.18

0.04

0.00

Set B: Weeks 29–34 (July–August)
0.20

0.01

0.00

0.08

Which is true about the two sets of data above?
A.

The mode of Set A is greater than the mode of Set B.

B.

The mode of Set B is greater than the mode of Set A.

C.

The median of Set A is greater than the median of Set B.

D.

The median of Set B is greater than the median of Set A.

Solution:

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Chapter 13

In this case, Set A does not have a mode, because there is no data value that appears more than once. So we cannot
make any statements involving the mode of Set A.
Calculate the median of both sets of data. The median is the number in the middle of a data set when all the values
are written in increasing order:

Data Analysis,
Statistics, and
Probability

Set A: 0.08 0.28 0.43 1.18 1.73 1.93
Here, the two middle numbers are 0.43 and 1.18, so we take their average:
Median of Set A: 0.43 + 1.18 = 1.61 = 0.805
2
2
We perform the same set of calculations for set B:
Set B: 0.00 0.00 0.01 0.04 0.08 0.20
Median of Set B: 0.01+ 0.04 = 0.05 = 0.025
2
2
From this we can see that the median of Set A is greater than the median of Set B. The correct answer is choice C.

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Evaluating Reports and Surveys
To evaluate a report about a set of data, it is important to consider the appropriateness of the data collection method.
Example:
A local politician wants to gauge how her constituents feel about the installation of a gas pipeline that will border
her district. Which of the following would allow the politician to make a valid conclusion about the opinions of her
constituents?
A.

Sending an electronic survey via e-mail

B.

Surveying people at several local movie theaters

C.

Surveying people at several local supermarkets

D.

Having people call in their opinion to a local radio show

Solution:
Let’s analyze each of these options. First is the e-mail survey in choice A. This may seem like a good option, but does
everyone in the district have an e-mail address? Will a representative group respond? Next, choice B is the option to
survey people at local movie theaters. This may work, but it would exclude people who don’t go to the movies on a
regular basis. Choice C is probably best because everyone eats, and the probability is high that most, if not all, people in
the politician’s district go to the supermarket at some point. Finally, choice D is not correct, because few people will listen
to any particular radio show, and even fewer will voluntarily call to voice their opinion. The correct answer is choice C.

When considering whether the data in a report is representative of a population, it is important to consider the demographics of
the population being studied. Their habits, behaviors, and perhaps incomes will influence their decisions and even their ability
to be included in the report. The type of survey that will include the widest range of habits, behaviors, and incomes of the people
being studied is likely the most representative.

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Peterson’s SAT® Prep Guide 2017

EXERCISES: STATISTICS
DIRECTIONS: Work out each problem. Circle the letter of your choice.
1.

The tables below show the number of employees in two different groups of an organization in different age ranges.

Production
Age Range

Number of Employees

20–24

4

25–29

8

30–34

25

35–39

19

40–44

17

Design
Age Range

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Chapter 13
Data Analysis,
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Probability

Number of Employees

20–24

2

25–29

4

30–34

6

35–39

33

40–44

38

Which of the following conclusions can be made about the data above?
A.

The range of ages is greater in production than in design.

B.

The range of ages is greater in design than in production.

C.

The median age is greater in production than in design.

D.

The median age is greater in design than in production.

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A nursery wants to know if a certain fertilizer is helping its rose bushes grow more roses per bush. The nursery uses the
fertilizer on one plot, Plot A, of rose bushes but does not use fertilizer on another plot, Plot B. All the rose bushes in both
plots were planted at the same time. The charts below show the number of flowers on each bush for the two plots.

Number of Flowers Per Bush

Plot A
10
8
6
4
2
0
1

2

3

4

5

6

7

8

9

10

Rose Bushes

Plot B
12
Number of Flowers Per Bush

2.

10

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8

Chapter 13

6

Data Analysis,
Statistics, and
Probability

4
2
0
1

2

3

4

5

6

7

8

9

10

Rose Bushes

Which of the following conclusions could be logically drawn based on the data?
A.

The mean number of roses in Plot A is greater, so plot A likely benefited from the fertilizer.

B.

The mean number of roses in Plot B is greater; so plot B likely benefited from NOT having the fertilizer.

C.

The fertilizer had no effect on the rose bushes.

D.

There is not enough information to make any conclusion.

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3.

4.

SHOW YOUR WORK HERE

If two 90% confidence intervals are of different sizes, then
the smaller interval provides a
A.

more precise estimate of the population mean.

B.

less precise estimate of the population mean.

C.

larger standard deviation.

D.

smaller standard deviation.

A property development company wants to see if a new
movie complex would be a profitable venture in a suburban area. The location is 15 miles away from an existing
movie complex in a region with approximately 650,000
residents with a mean age of 48. The company conducts
a survey of 200 people coming out of a midnight movie
on a Wednesday evening to see what they think of the
idea.
What can be said about this data collection method?

554

A.

The population surveyed is likely a good representative population.

B.

The survey probably didn’t include movie theater
employees.

C.

The sample is likely skewed toward people
younger than the mean.

D.

The sample is likely skewed toward
out-of-towners.

Chapter 13
Data Analysis,
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5.

A coffee retailer kept track of number of customers during the same 6-hour period over the course of two days and recorded
them in the table below.

1–2 p.m.

2–3 p.m.

3–4 p.m.

4–5 p.m.

5–6 p.m.

6–7 p.m.

Sunday

24

22

28

35

38

35

Monday

11

18

18

22

44

30

Which of the following is a valid conclusion to be
drawn from the data in the tables?
A.

The coffee shop is busier later in the day.

B.

There are probably never more than 50 customers
in the coffee shop.

C.

The average number of customers is higher on the
weekends than during the week.

D.

The range of data on the two days is equal.

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SHOW YOUR WORK HERE

6.

7.

8.

A farmer weighs eggs produced by his chickens. He
samples 36 eggs from a population of over 2,000 eggs.
The sample mean weight is 63.8 grams, with a sampling
error of 9.2 grams at a 90% confidence interval. Which of
the following is true, based on the sample?
A.

You can be 90% confident that another sample
will have a mean for the eggs between 54.6 and 73
grams.

B.

You can be 90% confident that the true population
mean for the eggs is between 54.6 and 73 grams.

C.

You can be 90% confident that the standard
deviation of each measurement is 9.2 grams.

D.

You can be 90% confident that the true population
mean is 63.8 grams.

SHOW YOUR WORK HERE

Using the same data from question 6, if you calculate a
90% confidence interval for the mean given a sample size
of 72 eggs, what is the effect?
A.

It increases the sample mean.

B.

It decreases the sample mean.

C.

It decreases the margin of error.

D.

It increases the margin of error.

The margin of error is +/– 20 points for a 95% confidence
interval for a given set of data. The researcher in charge
of the data wants to report a lower margin of error. Which
of the following would give a true representation of the
data and a lower margin of error?
A.

Increase the confidence level to 98%.

B.

Decrease the confidence level to 90%.

C.

Increase the sample size.

D.

Decrease the sample size.

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9.

10.

556

A poll was taken by the local university to determine
which candidate is most likely to win the upcoming
congressional race. The poll was conducted by randomly
talking to people walking downtown in the largest city
of the congressional district. Which of the following best
describes the poll?
A.

The poll is a representative sample because it was
done in the largest city in the district.

B.

The poll is a representative sample because it was
done randomly.

C.

The poll is not a representative sample because it
was done only in the largest city in the district.

D.

The poll is not a representative sample because it
was done randomly.

The double occupancy room rates at 40 five-star hotels in
Los Angeles is normally distributed with a mean of $280
and a standard deviation of $48.25. Which is the range
of values, in dollars, around the mean that includes 95%
of the room rates?
A.

$280.00–$328.25

B.

$231.75–$280.00

C.

$231.75–$328.25

D.

$183.50–$376.50

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SHOW YOUR WORK HERE

ANSWER KEY AND EXPLANATIONS

1.

2.

1. D

3. D

5. A

7. C

9. C

2. A

4. C

6. B

8. B

10. D

The correct answer is D. We don’t have exact ages,
which is why we can’t conclude anything about the
range of ages and can eliminate choices A and B. There
are 73 employees in the production group, so the
median age is the 37th data value when the ages are
written in ascending order, which would be in the
30–34 age range. There are 83 employees in the design
group. The median age is the 42nd data value, which
falls in the 35–39 age range. Thus the median age for
the design group is higher.

movie theater employees, choice B, is not relevant to
the validity of the data.
5.

The correct answer is A. It is difficult to conclude that
there are never more than 50 customers in the shop, as
choice B says, because we don’t have enough data. As
for choice C, while the mean for Sunday is higher than
the mean for Monday, the sample size of 2 days is not
enough to make a conclusion about the weekends
versus the weekdays in general. The range of the two
data sets are not equal to each other, so choice D is
incorrect.

6.

The correct answer is B. The confidence interval gives
us the ability to estimate a range for the true mean of
the population, not for a sample mean.

7.

The correct answer is C. The mean will likely be
different, but there is no reason to believe it will only
increase or only decrease. But, increasing sample size
decreases the margin of error.

The correct answer is A. First focus on the mean
number of roses on the bushes in Plot A:
6 + 6 + 4 + 5 + 9 + 5 + 5 + 8 + 5 + 4 57
=
10
10
= 5.7
Then, look at those in Plot B:
3 + 3 + 5 + 2 + 5 + 8 + 11+ 0 + 5 + 1 43
=
10
10
= 4.3

8.

The correct answer is B. The data has already been
collected, so increasing or decreasing the sample size
is not possible without redoing the study, and it would
not be an accurate representation of this data.
Decreasing the confidence level would decrease the
margin of error.

9.

The correct answer is C. A representative sample
should allow for random selection within the entire
population, not just random selection from one part of
the population.

10.

The correct answer is D. Ninety-five percent of the
rates fall within 2 standard deviations of the mean,
meaning within $48.25 × 2 = $96.50 of the mean
price of $280.00 and 280.00 + 96.50 = 376.50 and
280.00 – 96.50 = 183.50.

On average, the bushes in Plot A grew more roses and
so likely benefited from the fertilizer.
3.

4.

The correct answer is D. A 90% confidence interval
means that those who conducted the survey have
proved with 90% precision that the estimate for the
mean is within a given range, so eliminate choices A
and B. If the range of a confidence interval is less, then
the standard deviation is also less, so choice D is
correct.
The correct answer is C. People who are out after
midnight are probably younger than the mean age,
since they are not likely to be getting up early after a
late-night movie. It is probably not representative of
the general population. There is no reason to believe
the sample is skewed toward out-of-towners, choice D,
and 200 people is probably not a good sample size, so
choice A is incorrect. Whether the survey included

557
Chapter 13
Data Analysis,
Statistics, and
Probability

Peterson’s SAT® Prep Guide 2017

SUMMING IT UP
• Not all problems that deal with averages will ask you to solve for the average of a set of quantities.
• The trick when working with tables is to make sure you select the correct data needed to solve the problem.
• Take your time reading each table.
• Observe trends with scatterplots, and use lines or curves to predict unknown values.
• Read data representations carefully when comparing data sets.
• Pay attention to sampling methods when you evaluate survey data.
ONLINE
PREP

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Access more practice questions, helpful lessons, valuable tips, and expert strategies for the following data analysis, statistics, and
probability topics in Peterson’s SAT® Online Course:

• Data Interpretation
• Mean, Median, and Mode
• Percent Word Problems
• Ratios
• Word Problems
To purchase and access the course, go to www.petersons.com/sat.

558
Chapter 13
Data Analysis,
Statistics, and
Probability

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PART VII:
PRACTICE TESTS
®
FOR THE SAT
Chapter 14: SAT® Practice Tests

Chapter 14:
SAT® Practice Tests
OVERVIEW
Practice Test 1
Practice Test 2
Practice Test 3
Practice Test 4
Practice Test 5

INTRODUCTION TO THE PRACTICE TESTS
On test day, you will see these important reminders on the first page of your exam booklet:

• You must use a No. 2 pencil when taking the test; you may not use a pen or mechanical pencil.
• You may not share any questions with anyone. This is considered a violation of College Board’s“Test Security and Fairness”
policies, and it could cause your scores to be canceled.

• You are not permitted to take the test booklet out of the testing room.

561
Chapter 14

In addition to filling in your name and Test Center information on your answer sheet, you’ll be asked to fill in two different codes
that will be on your test booklet. Each test booklet has a unique code, and there will be a grid-in form to fill in with your “TEST
ID” and “FORM CODE” letters and numbers.

SAT ®
Practice
Tests

The general directions for the test will look something like this:

• You may only work on one section at a time.
• If you complete a section before time is called, check your work on that section only. You are not permitted to work on
any other section.
The directions for marking your answers will likely include the following recommendations:

•
•
•
•
•

Mark your answer sheet properly—be sure to completely fill in the answer circle.
Be careful to mark only one answer for each question.
Don’t make any stray marks on the answer sheet.
If you need to erase your answer, make sure you do so completely.
Be sure to use the answer spaces that correspond to the question numbers.

You will be able to use your test booklet for scratch work, but you won’t get credit for any work done in your test booklet. When
time is called at the end of each section, you will not be permitted to transfer answers from your test booklet to your answer sheet.

Peterson’s SAT® Prep Guide 2017

Any information, ideas, or opinions presented in any of the passages you will see on the exam that have been taken from other
sources or published material do not represent the opinions of the College Board.
Scoring on the exam is as follows:

• You will receive one point for each correct answer.
• You will not lose points for wrong answers, so you should attempt to answer every question even if you aren’t completely
sure of the correct answer.
Your testing supervisor will announce when to open the test booklet, so be sure to wait until you’re told to do so. For the purposes
of this practice test, be sure you have a timer to set for 65 minutes for the Section 1: Reading Test.
The answer sheets for each test section, including lined paper for the essay, appear on the next pages.
Following the Answer Key and Explanations for thes Practice Tests, you will find details on how to score your work.

Good
Luck!

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Practice
Tests

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Practice Test 1—Answer Sheet
Section 1: Reading Test
1.

12.

23.

33.

43.

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52.

11.

22.

Section 2: Writing and Language Test
1.

10.

19.

28.

37.

2.

11.

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29.

38.

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44.

9.

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27.

36.

Section 3: Math Test—No Calculator
1.

4.

7.

10.

13.

2.

5.

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11.

14.

3.

6.

9.

12.

15.

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Section 4: Math Test—Calculator
1.

7.

13.

19.

25.

2.

8.

14.

20.

26.

3.

9.

15.

21.

27.

4.

10.

16.

22.

28.

5.

11.

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29.

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/
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Practice Test 1—Answer Sheet
Section 5: Essay
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Practice Test 1—Answer Sheet

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Practice Test 1
SECTION 1: READING TEST
65 Minutes—52 Questions
TURN TO SECTION 1 OF YOUR ANSWER SHEET TO ANSWER THE QUESTIONS IN THIS SECTION.
DIRECTIONS: Each passage (or pair of passages) in this section is followed by a number of multiple-choice questions. After
reading each passage, select the best answer to each question based on what is stated or implied in the passage or passages
and in any supplementary material, such as a table, graph, chart, or photograph.

Questions 1–10 are based on the following passage.
PASSAGE I
The following passage has been taken from the “The Bride Comes
to Yellow Sky,” by Stephen Crane. It is a western short story that was
first published in 1898. The protagonist is Jack Potter, who returns
to the town of Yellow Sky with his bride.
		 The great Pullman was whirling onward with such
dignity of motion that a glance from the window seemed
simply to prove that the plains of Texas were pouring
Line eastward. Vast flats of green grass, dull-hued spaces
5 of mesquite and cactus, little groups of frame houses,
woods of light and tender trees, all were sweeping into
the east, sweeping over the horizon, a precipice.
		 A newly married pair had boarded this coach at San
Antonio. The man’s face was reddened from many days
10 in the wind and sun, and a direct result of his new black
clothes was that his brick-colored hands were constantly
performing in a most conscious fashion. From time to
time he looked down respectfully at his attire. He sat
with a hand on each knee, like a man waiting in a bar15 ber’s shop. The glances he devoted to other passengers
were furtive and shy.
		 The bride was not pretty, nor was she very young.
She wore a dress of blue cashmere, with small reservations of velvet here and there, and with steel buttons
20 abounding. She continually twisted her head to regard
her puff sleeves, very stiff, straight, and high. They
embarrassed her. It was quite apparent that she had
cooked, and that she expected to cook, dutifully. The
blushes caused by the careless scrutiny of some pas25 sengers as she had entered the car were strange to see
upon this plain, under-class countenance, which was
drawn in placid, almost emotionless lines.

		 “Great! And then after a while we’ll go forward
to the diner, and get a big layout. Finest meal in the
world. Charge a dollar.”
		

“Oh, do they?” cried the bride. “Charge a dollar?

35 Why, that’s too much—for us—ain’t it Jack?”

		 “Not this trip, anyhow,” he answered bravely.
“We’re going to go the whole thing.”
		 Later he explained to her about the trains. “You
see, it’s a thousand miles from one end of Texas to
40 the other; and this train runs right across it, and
never stops but for four times.” He had the pride of
an owner. He pointed out to her the dazzling fittings
of the coach, and in truth her eyes opened wider as
she contemplated the sea-green figured velvet, the
45 shining brass, silver, and glass, the wood that gleamed
as darkly brilliant as the surface of a pool of oil. At
one end a bronze figure sturdily held a support for a
separated chamber, and at convenient places on the
ceiling were frescoes in olive and silver.
50 		

To the minds of the pair, their surroundings
reflected the glory of their marriage that morning in
San Antonio. This was the environment of their new
estate; and the man’s face in particular beamed with
an elation that made him appear ridiculous to the
55 negro porter. This individual at times surveyed them
from afar with an amused and superior grin. On other
occasions he bullied them with skill in ways that did
not make it exactly plain to them that they were being
bullied. He subtly used all the manners of the most
60 unconquerable kind of snobbery. He oppressed them,
but of this oppression they had small knowledge,
and they speedily forgot that infrequently a number
of travelers covered them with stares of derisive
enjoyment. Historically, there was supposed to be
65 something infinitely humorous in their situation.

		 They were evidently very happy. “Ever been in a
parlor-car before?” he asked, smiling with delight.
30 		

“No,” she answered; “I never was. It’s fine, ain’t it?”

CONTINUE
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Practice Test 1
1

The passage might best be described as
A.

2

3

4

5

  570

6

Which choice provides the best evidence for the answer
to the previous question?

an analysis of a man’s acceptance of his social
status.

A.

Lines 4–7 (“Vast flats . . . precipice.”)

B.

an account of a couple’s anticipation of married life.

B.

Lines 12–16 (“From time . . . furtive and shy.)

C.

a description of train travel in nineteenth-century
Texas.

C.

Lines 30–33 (“‘No,’ she . . . a dollar’.”)

D.

Lines 50–55 (“To the . . . negro porter”)

D.

a criticism of class consciousness in the nineteenth
century.

7

It can be inferred from the passage that Jack

In lines 23–27 (“The blushes . . . emotionless lines.”), why
does the narrator note that the bride’s blushing seemed
so out of place on her face?

A.

wants to impress his new bride.

A.

B.

is likely a farmhand or rancher.

To emphasize the degree to which other passengers are staring

C.

is used to being treated as an inferior.

B.

To express the bride’s extreme happiness with her
marriage

D.

wants to change his station in life.

C.

To communicate that the bride is lacking in
self-confidence

D.

To underscore that the bride is strikingly
unattractive

Which choice provides the best evidence for the answer
to the previous question?
A.

Lines 12–15 (“From time . . . barber’s shop.”)

B.

Lines 17–22 (“The bride . . . embarrassed her.”)

C.

Lines 31–37 (“‘Great! . . . whole thing.”)

A.

project.

D.

Lines 50–55 (“To the . . . negro porter.”)

B.

interests.

C.

property.

D.

standing.

8

As used in line 2, “dignity” most nearly means
A.

splendor.

B.

respectability.

C.

superiority.

D.

gracefulness.

9

As used in line 53, “estate” most nearly means

In lines 55–64 (“This individual . . . derisive enjoyment.”),
the narrator maintains that the
A.

porter mocks the couple, making Jack and his bride
believe they are being catered to.

Jack and his bride might best be described as

B.

couple barely notices the contemptuous way they
are treated by the porter.

A.

firm and resolute in their decisions.

C.

B.

nervous and fearful about their trip.

porter is openly hostile to the couple, making the
trip painful for them.

C.

awkward and self-conscious in the setting.

D.

D.

amazed and bewildered by the landscape.

couple has become used to the rude behavior of
the porter.

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Practice Test 1
10 What is the main rhetorical effect of lines 10–13 (“and a . . .
his attire”)?
A.

To illustrate how nervous and awkward the
groom is

B.

To show how much the groom is used to using his
hands

C.

To convey how unaccustomed the groom is to
wearing dress clothes

D.

To show how happy the groom is about being
married

Questions 11–21 are based on the following passage.
The following passage has been taken from American Ornithology
by Alexander Wilson, a Scottish-American naturalist. Dubbed “the
Father of American Ornithology,” Wilson is regarded as the greatest
American ornithologist after Audubon. His nine-volume American
Ornithology was published between 1808 and 1814.

Line
5

10

15

20

25

30

		 About the twenty-fifth of April the Hummingbird
usually arrives in Pennsylvania; and about the tenth of
May begins to build its nest. This is generally fixed on
the upper side of a horizontal branch, not among the
twigs, but on the body of the branch itself. Yet I have
known instances where it was attached by the side to
an old moss-grown trunk; and others where it was fastened on a strong rank stalk, or weed, in the garden; but
these cases are rare. In the woods it very often chooses
a white oak sapling to build on; and in the orchard, or
garden, selects a pear tree for that purpose. The branch
is seldom more than ten feet from the ground. The nest is
about an inch in diameter, and as much in depth. A very
complete one is now lying before me, and the materials
of which it is composed are as follows: —The outward
coat is formed of small pieces of bluish grey lichen that
vegetates on old trees and fences, thickly glued on with
the saliva of the bird, giving firmness and consistency
to the whole, as well as keeping out moisture. Within
this are thick matted layers of the fine wings of certain
flying seeds, closely laid together; and lastly, the downy
substance from the great mullein, and from the stalks of
the common fern, lines the whole. The base of the nest
is continued round the stem of the branch, to which it
closely adheres; and, when viewed from below, appears
a mere mossy knot or accidental protuberance. The
eggs are two, pure white, and of equal thickness at both
ends. . . . On a person’s approaching their nest, the little
proprietors dart around with a humming sound, passing
frequently within a few inches of one’s head; and should
the young be newly hatched, the female will resume her

place on the nest even while you stand within a yard or
two of the spot. The precise period of incubation I am
unable to give; but the young are in the habit, a short
35 time before they leave the nest, of thrusting their bills
into the mouths of their parents, and sucking what they
have brought them. I never could perceive that they
carried them any animal food; tho, from circumstances
that will presently be mentioned, I think it highly
40 probable they do. As I have found their nest with eggs
so late as the twelfth of July, I do not doubt but that they
frequently, and perhaps usually, raise two broods in the
same season.
45

50

55

60

65

70

75

		 The hummingbird is extremely fond of tubular
flowers, and I have often stopt, with pleasure, to observe
his maneuvers among the blossoms of the trumpet
flower. When arrived before a thicket of these that are full
blown, he poises, or suspends himself on wing, for the
space of two or three seconds, so steadily, that his wings
become invisible, or only like a mist; and you can plainly
distinguish the pupil of his eye looking round with great
quickness and circumspection; the glossy golden green
of his back, and the fire of his throat, dazzling in the sun,
form altogether a most interesting appearance. The
position into which his body is usually thrown while in
the act of thrusting his slender tubular tongue into the
flower, to extract its sweets, is exhibited in the figure
on the plate. When he alights, which is frequently, he
always prefers the small dead twigs of a tree, or bush,
where he dresses and arranges his plumage with great
dexterity. His only note is a single chirp, not louder than
that of a small cricket or grasshopper, generally uttered
while passing from flower to flower, or when engaged
in fight with his fellows; for when two males meet at
the same bush, or flower, a battle instantly takes place;
and the combatants ascend in the air, chirping, darting
and circling around each other, till the eye is no longer
able to follow them. The conqueror, however, generally
returns to the place, to reap the fruits of his victory. I
have seen him attack, and for a few moments tease the
King-bird; and have also seen him, in his turn, assaulted
by a humble-bee, which he soon put to flight. He is one
of those few birds that are universally beloved; and
amidst the sweet dewy serenity of a summer’s morning,
his appearance among the arbours of honeysuckles, and
beds of flowers, is truly interesting.

CONTINUE
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Practice Test 1
11 The author is mostly concerned with
A.

describing the characteristics of the hummingbird.

B.

explaining how hummingbirds build their nests.

C.

convincing the reader that hummingbirds are
interesting.

D.

interpreting the meaning of certain hummingbird
behaviors.

16 Which choice provides the best evidence for the answer
to the previous question?
A.

Lines 28–33 (“On a person’s . . . the spot.”)

B.

Lines 33–37 (“The precise . . . brought them.”)

C.

Lines 37–40 (“I never . . . they do.”)

D.

Lines 61–64 (“His only . . . with his fellows;”)

17 As it is used in line 60, “dresses” most nearly means
12 Based on lines 1–3 (“About the twenty-fifth of April . . . to
build its nest.”), it can generally be assumed that hummingbirds

oils.

B.

dons.

A.

take two weeks to build their nests.

C.

shuffles.

B.

migrate elsewhere for the winter.

D.

neatens.

C.

cannot be found in places farther north.

D.

are mostly solitary animals.

13 As used in line 3, “fixed” most nearly means
A.

adjusted.

B.

intended.

C.

aligned.

D.

secured.

14 In lines 13–26 (“A very complete one . . . accidental
protuberance.”), the author is mainly concerned with
how the hummingbird

18 In lines 49–50, the author notes that the hummingbird’s
“wings become invisible, or only like a mist” to
A.

show how transparent the wings are.

B.

emphasize how fast the wings are moving.

C.

point out that the sun reflects off the wings.

D.

reiterate that the hummingbird is beautiful.

19 In lines 68–69 (“The conqueror . . . fruits of his victory.”),
“fruits of his victory” refers to
A.

the dead twigs of a tree.

B.

Lines 58–61 (“When he alights, . . . dexterity.”)

A.

uses nearby plants in the nest.

C.

Lines 63–65 (“passing from . . . or flower”)

B.

builds compact, complicated nests.

D.

the other combatant.

C.

builds the nest over a period of time.

D.

constructs a nest that is waterproof.

15 It can be inferred from the passage that hummingbirds

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A.

A.

stay in the nest for years at a time.

B.

builds compact, complicated nests.

C.

builds the nest over a period of time.

D.

feed only on the sweet nectar of flowers.

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20 Which choice provides the best evidence for the answer
to the previous question?
A.

Lines 1–3 (“About the twenty-fifth . . . its nest.”)

B.

Lines 58–61 (“When he alights, . . . dexterity.”)

C.

Lines 63–65 (“passing from, . . . or flower.”)

D.

Lines 66–68 (“and the combatants . . . follow them.”)

Practice Test 1
21 The author most likely references the kingbird in lines
69–71 (“I have seen . . . King-bird.”) to
A.

highlight the many dangers that confront
hummingbirds.

B.

describe how the hummingbird stays close to its
nest.

C.

emphasize that the hummingbird is an aggressive
bird.

D.

introduce the idea that the hummingbird does not
only fight for self-defense.

Questions 22–31 are based on the following passage and
supplementary material.
The passage is excerpted from information provided by the
National Oceanic and Atmospheric Administration (NOAA) at
http://www.noaa.gov.
		 Over half a mile taller . . . than Mt. Everest, Mauna
Kea in Hawai’i is more than 6 miles tall, from its base
on the ocean floor to its summit two miles above the
Line surface of the Pacific Ocean. This island mountain is only
5 one of many features found on the ocean floor. Besides
being the base for islands, the ocean floor also includes
continental shelves and slopes, canyons, oceanic ridges,
trenches, fracture zones, abyssal hills, abyssal plains,
volcanoes, and seamounts. Not just rock and mud, these
10 locations are the sites of exotic ecosystems that have
rarely been seen or even explored.
Plate Tectonics and the Ocean Floor
		 The shape of the ocean floor, its bathymetry, is
largely a result of a process called plate tectonics. The
outer rocky layer of the Earth includes about a dozen
15 large sections called tectonic plates that are arranged
like a spherical jigsaw puzzle floating on top of the
Earth’s hot flowing mantle. Convection currents in the
molten mantle layer cause the plates to slowly move
about the Earth a few centimeters each year. Many ocean
20 floor features are a result of the interactions that occur
at the edges of these plates.
		 The shifting plates may collide (converge), move
away (diverge), or slide past (transform) each other. As
plates converge, one plate may dive under the other,
25 causing earthquakes, forming volcanoes, or creating
deep ocean trenches such as the Mariana Trench. Where
plates are pulled away (diverge) from each other, molten
magma flows upward between the plates, forming
mid-ocean ridges, underwater volcanoes, hydrothermal

30 vents, and new ocean floor crust. The Mid-Atlantic Ridge

is an example of this type of plate boundary. . . .
Marine Life and Exploration on the Ocean Floor
		 Over the last decade, more than 1500 new species
have been discovered in the ocean by marine biologists and other ocean scientists. Many of these newly
35 discovered species live deep on the ocean floor in
unique habitats dependent on processes resulting from
plate movement, underwater volcanoes, and cold water
seeps. The discovery of deep ocean hydrothermal vent
ecosystems in 1977 forced scientists to redefine living
40 systems on our planet. Considered one of the most
important scientific discoveries of the last century,
organisms in this deep, dark ecosystem rely on chemicals
and a process called chemosynthesis as the base of their
food web and not on sunlight and photosynthesis as in
45 other previously described ecosystems. . . .
		 Hydrothermal vents form along mid-ocean ridges,
in places where the sea floor moves apart very slowly (6
to 18 cm per year) as magma wells up from below. (This
is the engine that drives Earth’s tectonic plates apart,
50 moving continents and causing volcanic eruptions and
earthquakes.) When cold ocean water seeps through
cracks in the sea floor to hot spots below, hydrothermal
vents belch a mineral-rich broth of scalding water.
Sometimes, in very hot vents, the emerging fluid turns
55 black, creating a “black smoker,” because dissolved sulfides of metals (iron, copper, and several heavy metals)
instantaneously precipitate out of solution when they
mix with the cold surrounding seawater.
		

Unlike plants that rely on sunlight, bacteria living

60 in and around the dark vents extract their energy from

hydrogen sulfide (HS) and other molecules that billow
out of the seafloor. Just like plants, the bacteria use their
energy to build sugars out of carbon dioxide and water.
Sugars then provide fuel and raw material for the rest of
65 the microbes’ activities.
Why Is Chemosynthesis Important?
		 Chemosynthetic deep-sea bacteria form the base
of a varied food web that includes shrimp, tubeworms,
clams, fish, crabs, and octopi. All of these animals must
be adapted to endure the extreme environment of the
70 vents—complete darkness; water temperatures ranging
from 2°C (in ambient seawater) to about 400°C (at the
vent openings); pressures hundreds of times that at
sea level; and high concentrations of sulfides and other
noxious chemicals.

CONTINUE
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Practice Test 1
Why is photosynthesis important?
75 Aquatic and terrestrial plants form the base of varied food webs

that may include small fish and crabs, larger fish, and eventually,
humans.

plume
of black
‘smoke’
black
smoker
chimneys
containing
sulphides
cold
seawater

100 m

hydrothermal fluids
enter ocean at
up to 350°C
or more

cold
seawater
(typically
2–4°C)

sulphides
metalliferous
deposited
in cracks
sediment
and veins
(stockwork)
oceanic
crust
high permeability

seawater
leaching of
metal ions
from rock

low permeability

seawater

high-temperature
reaction zone
heat from magma

22 Which best describes the function of the opening sentence
(“Over half a mile taller . . . the Pacific Ocean.”)?

  574

A.

It shows how unusual and extraordinary some
ocean features are.

B.

It emphasizes that Mt. Everest is not the world’s
tallest mountain.

C.

It describes the geography of island mountains in
the Pacific Ocean.

D.

It compares the geography of Mt. Everest to that of
Mauna Kea.

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23 The author uses the simile “like a spherical jigsaw puzzle”
(line 16) to illustrate that
A.

each plate plays a critical role.

B.

the earth is sphere-shaped.

C.

each plate is symmetrical.

D.

the plates fit together.

Practice Test 1
24 The movement of the earth’s tectonic plates is a
function of

28 The author devotes the first half of the passage to an
explanation of plate tectonics in order to

A.

earthquakes and volcanoes.

A.

describe the geography of the ocean floor.

B.

the movement of ocean tides.

B.

C.

new ocean floor crust.

explain the conditions that create hydrothermal
vents.

D.

the moving molten mantle layer.

C.

compare hydrothermal vents to underwater
volcanoes.

D.

argue that hydrothermal vents are a unique
ecosystem.

25 Which inference can you make about scientists’ pre-1977
understanding of deep ocean hydrothermal vent ecosystems?
A.

Scientists did not know how organisms in deep
ocean hydrothermal vent ecosystems survived.

B.

Scientists believed that colliding plates caused the
formation of deep ocean hydrothermal vents.

C.

D.

Scientists determined that some sea life can move
easily between shallow waters and deep ocean
hydrothermal vent ecosystems.
Scientists believed minerals could be mined from
deep ocean hydrothermal vent ecosystems.

26 Which choice provides the best evidence for the answer
to the previous question?

29 Which choice provides the best evidence for the answer
to the previous question?
A.

Lines 5–9 (“Besides being . . . and seamounts.”)

B.

Lines 34–38 (“Many of . . . water seeps.”)

C.

Lines 46–48 (“Hydrothermal vents . . . from below.”)

D.

Lines 48–51 (“This is the engine . . . and
earthquakes.”)

30 As used in line 74, “noxious” most nearly means
A.

annoying.

B.

harmful.

A.

Lines 32–34 (“Over the . . . ocean scientists.”)

C.

offensive.

B.

Lines 40–44 (“Considered one . . . web.”)

D.

unusual.

C.

Lines 46–48 (“Hydrothermal vents . . . from below.”)

D.

Lines 54–58 (“Sometimes, in . . . surrounding
seawater.”)

27 As it is used in line 61, “billow” most nearly means

31 Based on the diagram, what happens to the sulfides inside
the black smoker?
A.

They are carried by ocean currents.

B.

They are pushed out into the atmosphere.

A.

flow.

C.

They remain inside of the black smoker’s veins.

B.

crest.

D.

They drop back into the black smoker.

C.

wave.

D.

inflate.

CONTINUE
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Practice Test 1
Questions 32–42 are based on the following passage
and supplementary material.
The passage is an excerpt from “How the Other Half Lives,” by Jacob
Riis. It was published in 1890 and documented the squalid living
conditions in the tenements of New York City.
		 The old question, what to do with the boy, assumes
a new and serious phase in the tenements. Under the
best conditions found there, it is not easily answered.
Line In nine cases out of ten he would make an excellent
5 mechanic, if trained early to work at a trade, for he is
neither dull nor slow, but the short-sighted despotism
of the trades unions has practically closed that avenue
to him. Trade-schools, however excellent, cannot supply
the opportunity thus denied him, and at the outset the
10 boy stands condemned by his own to low and ill-paid
drudgery, held down by the hand that of all should labor
to raise him.

15

20

25

30

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		 Home, the greatest factor of all in the training of
the young, means nothing to him but a pigeon-hole
in a coop along with so many other human animals. Its
influence is scarcely of the elevating kind, if it have any.
The very games at which he takes a hand in the street
become polluting in its atmosphere. With no steady
hand to guide him, the boy takes naturally to idle ways.
Caught in the street by the truant officer, or by the
agents of the Children’s Societies, peddling, perhaps, or
begging, to help out the family resources, he runs the
risk of being sent to a reformatory, where contact with
vicious boys older than himself soon develop the latent
possibilities for evil that lie hidden in him. The city has
no Truant Home in which to keep him, and all efforts
of the children’s friends to enforce school attendance
are paralyzed by this want. The risk of the reformatory
is too great. What is done in the end is to let him take
chances—with the chances all against him. The result is
the rough young savage, familiar from the street. Rough

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as he is, if any one doubt that this child of common clay
have in him the instinct of beauty, of love for the ideal of
which his life has no embodiment, let him put the matter
35 to the test. Let him take into a tenement block a handful
of flowers from the fields and watch the brightened
faces, the sudden abandonment of play and fight that
go ever hand in hand where there is no elbow-room,
the wild entreaty for “posies,” the eager love with which
40 the little messengers of peace are shielded, once possessed; then let him change his mind. I have seen an
armful of daisies keep the peace of a block better than
a policeman and his club, seen instincts awaken under
their gentle appeal, whose very existence the soil in
45 which they grew made seem a mockery. . . .
		 Yet, as I knew, that dismal alley with its bare brick
walls, between which no sun ever rose or set, was
the world of those children. It filled their young lives.
Probably not one of them had ever been out of the sight
50 of it. They were too dirty, too ragged, and too generally
disreputable, too well hidden in their slum besides, to
come into line with the Fresh Air summer boarders.
		 With such human instincts and cravings, forever
unsatisfied, turned into a haunting curse; with appetite
55 ground to keenest edge by a hunger that is never fed,
the children of the poor grow up in joyless homes to
lives of wearisome toil that claims them at an age when
the play of their happier fellows has but just begun. Has
a yard of turf been laid and a vine been coaxed to grow
60 within their reach, they are banished and barred out
from it as from a heaven that is not for such as they. I
came upon a couple of youngsters in a Mulberry Street
yard a while ago that were chalking on the fence their
first lesson in “writin’.”
65 		

And this is what they wrote: “Keeb of te Grass.” They
had it by heart, for there was not, I verily believe, a green
sod within a quarter of a mile. Home to them is an empty
name.

Practice Test 1
32 The passage is mainly concerned with establishing that
A.

there are very few job opportunities available for
poor people.

B.

the condition of the tenements condemns children
to lives of misery.

C.

more education is needed to help elevate the
children of the poor.

D.

children who live in poverty naturally turn to crime
to support themselves.

33 As it is used in line 18, “polluting” most nearly means

6 37

As it is used in line 19, “idle” most nearly means
A.

unproductive.

B.

vain.

C.

immature.

D.

simple.

38 The author suggests taking flowers to the tenement child
to show that
A.

he has the capacity to appreciate exquisite things.

B.

he is always bored and will play with anything.

A.

dirty.

C.

there are few flowers growing in the tenements.

B.

dangerous.

D.

the policemen patrolling the tenements are brutal.

C.

corrupting.

D.

rowdy.

34 In line 10, “condemned by his own” means that the boy is
A.

denounced because of his character.

B.

criticized by his family members.

C.

held back by his own community.

D.

ridiculed by teachers in his school.

35 Which choice provides the best evidence for the answer
to the previous question?
A.

Lines 17–18 (“The very . . . atmosphere.”)

B.

Lines 20–21 (“Caught . . . Societies”)

C.

Lines 25–26 (“The city . . . keep him”)

D.

Lines 29–30 (“What is done . . . against him.”)

36 In lines 14–15, the author uses the metaphor of the pigeon
coop to establish that the tenement is
A.

as filthy as a cage in which animal are kept.

B.

completely devoid of privacy for its tenants.

C.

the only housing available for any person.

D.

merely a place to live and lacks any positive
influences.

39 Which choice provides the best evidence for the answer
to the previous question?
A.

Lines 30–31 (“The result . . . the street.”)

B.

Lines 31–35 (“Rough as . . . the test.”)

C.

Lines 41–45 (“I have . . . a mockery.”)

D.

Lines 46–48 (“Yet, as . . . those children.”)

40 The photos suggest which of the following statements
about tenements that the author did not address?
A.

Tenements are extremely unsanitary.

B.

Tenements are joyless.

C.

Tenement are dangerous.

D.

Tenement offer no job opportunities.

41 The author provides the story of the “writin’” in the last
paragraph to show that tenement children
A.

do not know how to spell.

B.

are in need of good schools.

C.

have no access to green spaces.

D.

are prone to vandalism.

CONTINUE
Peterson's SAT® Prep Guide 2017

  577

Practice Test 1
42 The author suggests in lines 25–28 that a truant home
A.

is no different from a reformatory.

B.

does not enforce school attendance.

C.

is full of vicious young boys.

D.

is necessary but unavailable.

Questions 43–52 are based on the following two passages.
Passage 1 is a letter written by Robert Schumann to Clara Wieck
in 1838.

		 You cannot think how your letter has raised and
strengthened me. . . . You are splendid, and I have much
more reason to be proud of you than of me. I have made
up my mind, though, to read all your wishes in your face.
35 Then you will think, even though you don’t say it, that
your Robert is a really good sort, that he is entirely yours,
and loves you more than words can say. You shall indeed
have cause to think so in the happy future. I still see you
as you looked in your little cap that last evening. I still
40 hear you call me du. Clara, I heard nothing of what you
said but that du. Don’t you remember?
PASSAGE 2
Napoleon Bonaparte to
Josephine Bonaparte (1796)

Passage 2 is a letter from Napoleon Bonaparte to his wife Josephine,
written in 1796.
PASSAGE 1
Robert Schumann to Clara Wieck (1838)

Line
5

10

15

20

25

30

  578

		 I have a hundred things to write to you, great
and small, if only I could do it neatly, but my writing
grows more and more indistinct, a sign, I fear, of heart
weakness. There are terrible hours when your image
forsakes me, when I wonder anxiously whether I have
ordered my life as wisely as I might, whether I had any
right to bind you to me, my angel, or can really make you
as happy as I should wish. These doubts all arise, I am
inclined to think, from your father’s attitude towards me.
It is so easy to accept other people’s estimate of oneself.
Your father’s behaviour makes me ask myself if I am really
so bad—of such humble standing—as to invite such
treatment from anyone. Accustomed to easy victory over
difficulties, to the smiles of fortune, and to affection, I
have been spoiled by having things made too easy for
me, and now I have to face refusal, insult, and calumny.
I have read of many such things in novels, but I thought
too highly of myself to imagine I could ever be the hero
of a family tragedy of the Kotzebue sort myself. If I had
ever done your father an injury, he might well hate me;
but I cannot see why he should despise me and, as you
say, hate me without any reason. But my turn will come,
and I will then show him how I love you and himself; for
I will tell you, as a secret, that I really love and respect
your father for his many great and fine qualities, as no
one but yourself can do. I have a natural inborn devotion
and reverence for him, as for all strong characters, and
it makes his antipathy for me doubly painful. Well, he
may some time declare peace, and say to us, “Take each
other, then.”

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45

50

55

60

65

		 I have not spent a day without loving you; I have not
spent a night without embracing you; I have not so much
as drunk a single cup of tea without cursing the pride
and ambition which force me to remain separated from
the moving spirit of my life. In the midst of my duties,
whether I am at the head of my army or inspecting the
camps, my beloved Josephine stands alone in my heart,
occupies my mind, fills my thoughts. If I am moving away
from you with the speed of the Rhône torrent, it is only
that I may see you again more quickly. If I rise to work in
the middle of the night, it is because this may hasten by
a matter of days the arrival of my sweet love. Yet in your
letter of the 23rd and 26th Ventôse, you call me vous.
Vous yourself! Ah! wretch, how could you have written
this letter? How cold it is! And then there are those four
days between the 23rd and the 26th; what were you
doing that you failed to write to your husband? . . . Ah,
my love, that vous, those four days make me long for
my former indifference. Woe to the person responsible!
May he, as punishment and penalty, experience what my
convictions and the evidence (which is in your friend’s
favour) would make me experience! Hell has no torments
great enough! Vous! Vous! Ah! How will things stand in
two weeks? . . . My spirit is heavy; my heart is fettered
and I am terrified by my fantasies. . . . You love me less;
but you will get over the loss. One day you will love me
no longer; at least tell me; then I shall know how I have
come to deserve this misfortune. …

43 Both passages are primarily concerned with the subject of
A.

jealousy.

B.

commitment.

C.

work.

D.

being apart.

Practice Test 1
44 As it is used in line 28, “antipathy” most nearly means

49 As it is used in line 52, “hasten” most nearly means

A.

indifference.

A.

stretch out.

B.

mistrust.

B.

speed up.

C.

rudeness.

C.

force.

D.

dislike.

D.

cause.

45 We can infer from Passage 1 that Clara’s father’s feelings
about Robert have caused Robert to

50 We can infer that Clara’s letter to Robert was different from
Josephine’s letter to Napoleon in that Clara’s letter

A.

dislike Clara’s father.

A.

made Robert doubt himself.

B.

question his and Clara’s future.

B.

made Robert question her loyalty.

C.

doubt Clara’s love for him.

C.

left Robert feeling encouraged.

D.

question Clara’s character.

D.

left Robert feeling confused.

46 Which choice provides the best evidence for the answer
to the previous question?
A.

Lines 4–9 (“There are . . . towards me.”)

B.

Lines 11–13 (“Your father’s . . . from anyone.”)

C.

Lines 22–26 (“But my . . . can do.”)

D.

Lines 35–37 (“Then you . . . can say.”)

51 In each passage, the author recollects something his beloved said. But the effect of Josephine’s words (lines 53–56)
differ from the effect of Clara’s words (lines 39–41) in that
A.

Josephine’s words have hurt Napoleon.

B.

Clara’s words have offended Robert.

C.

Josephine’s words have calmed Napoleon.

D.

Clara’s words have humbled Robert.

47 Napoleon’s letter to Josephine suggests that she
52 We can infer that both Robert and Napoleon
A.

wishes she could fight alongside her husband.

B.

has little to occupy her time.

C.

is distracting him from his responsibilities.

D.

has never regretted her marriage to him.

A.

are modest people.

B.

anger quickly.

C.

tend to be indecisive.

D.

suffer from self-doubt.

48 Which choice provides the best evidence for the answer
to the previous question?
A.

Lines 42–43 (“I have . . . you”)

B.

Lines 43–46 (“I have . . . my life.”)

C.

Lines 46–48 (“In the . . . heart”)

D.

Lines 49–51 (“If I . . . quickly.”)

STOP
If you finish before time is called, you may check your work on this section only.
Do not turn to any other section.
Peterson's SAT® Prep Guide 2017

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Practice Test 1
SECTION 2: WRITING AND LANGUAGE TEST
35 Minutes—44 Questions
TURN TO SECTION 2 OF YOUR ANSWER SHEET TO ANSWER THE QUESTIONS IN THIS SECTION.
DIRECTIONS: Each passage below is accompanied by a number of multiple-choice questions. For some questions, you
will need to consider how the passage might be revised to improve the expression of ideas. Other questions will ask you
to consider how the passage might be edited to correct errors in sentence structure, usage, or punctuation. A passage may
be accompanied by one or more graphics—such as a chart, table, or graph—that you will need to refer to in order to best
answer the question(s).
Some questions will direct you to an underlined portion of a passage—it could be one word, a portion of a sentence, or the full
sentence itself. Other questions will direct you to a particular paragraph or to certain sentences within a paragraph, or you’ll
be asked to think about the passage as a whole. Each question number refers to the corresponding number in the passage.
After reading each passage, select the answer to each question that most effectively improves the quality of writing in the
passage or that makes the passage follow the conventions of Standard Written English. Many questions include a “NO
CHANGE” option. Select that option if you think the best choice is to leave that specific portion of the passage as it is.

Questions 1–11 are based on the following passage.
Burma: has it truly changed?
A rigid red-and-white sign 1 erected on a rural road in

1

Burma reads, “[The Military] AND THE PEOPLE IN ETERNAL

A.

NO CHANGE

B.

born

It’s no wonder that Burma, also known as Myanmar, inspired

C.

floated

two of the most well-known books about totalitarianism: 2

D.

dripped

the British Imperial Police Force in the 1920s. Five years after

A.

NO CHANGE

Eric Arthur Blair began his tour of duty in the far-flung, 3

B.

Animal Farm and Nineteen Eighty-Four.

obscure Asian colony, he returned to his homeland, shed his

C.

Animal Farm and Nineteen Eighty-Four,

uniform, changed his name to George Orwell, and started a

D.

Animal Farm and Nineteen Eighty-Four!

A.

NO CHANGE

B.

darkened

C.

obvious

D.

uncertain

UNITY. ANYONE ATTEMPTING TO DIVIDE THEM IS OUR ENEMY.”

Animal Farm and Nineteen Eighty-Four? The books sprang from
the mind of a man who, as a teenager, sought adventure with

2

new career as a novelist.
3

CONTINUE
Peterson's SAT® Prep Guide 2017

  581

Practice Test 1
The contrasts Orwell saw still exist. This land of haunting

4

beauty, with its history of human rights violations, 4 had
also sparked filmmaker Ron Fricke’s imagination. He and his
crew traveled to 25 countries to shoot images for his nonverbal film Samsara (2013). Samsara means “the ever-turning
wheel of life” in Sanskrit. The sequences shot in Bagan, Burma,
are almost dreamlike in quality, especially because, as in the

A.

NO CHANGE

B.

have also sparked

C.

also sparked

D.

will also spark

A.

NO CHANGE

B.

As they seemingly float above seas of green foliage,
only music accompanies scenes of hundreds of
Buddhist temples.

C.

Only music accompanies scenes, as they seemingly
float above seas of green foliage, of hundreds of
Buddhist temples.

D.

Only music accompanies scenes, of hundreds of
Buddhist temples, as they seemingly float above a
sea of green foliage.

A.

NO CHANGE

B.

the countries major industries

C.

the countrys major industries

D.

the countries’ major industries

5

rest of the film, there is no dialogue or narration. 5 Only
music accompanies scenes of hundreds of Buddhist temples
as they seemingly float upon seas of green foliage. Watching
these scenes, it is hard to believe that Burma has seen violent
years of civil war, ethnic cleansing, and forced labor. But
such problems, as well as those of economic stagnation
and corruption, can usually be traced back to the military
regime, which took power in 1962 through a coup. The
military manages 6 the country’s major industries and has
also been accused of controlling Burma’s substantial heroin
exports.

6

The Irish rock band U2 dedicated their song “Walk On”
to Burmese academic Aung San Suu Kyi, who was under
house arrest from 1989 until 2010 because of her prodemocracy stance. 7 The members of U2 have a history
of incorporating their political views into their music. Her
National League for Democracy (NLD) won the 1990 electi