GMAT® OFFICIAL GUIDE 2018 GMAT

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The definitive guide from the makers of the GMAT® exam

Official Guide
2018
Book +

Online
The ONLY
source of real
GMAT® questions
from past
exams

This edition includes
130 never-before-seen questions

Graduate Management Admission Council®

mba.com

GMAT® OFFICIAL GUIDE 2018
Copyright © 2017 by the Graduate Management Admission Council®. All rights reserved.
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10

Letter from the President and CEO, GMAC®

1.0
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11

What Is the GMAT® Exam?
What Is the GMAT® Exam?
Why Take the GMAT® Exam?
GMAT® Exam Format
What Is the Content of the Test Like?
Integrated Reasoning Section
Quantitative Section
Verbal Section
Analytical Writing Assessment
What Computer Skills Will I Need?
What Are the Test Centers Like?
How Are Scores Calculated?
Test Development Process

2
3
3
4
5
6
6
6
7
7
7
7
8

2.0
2.0
2.1
2.2
2.3
2.4
2.5

How to Prepare
How to Prepare
How Should I Prepare to Take the Test?
What About Practice Tests?
How Should I Use the Diagnostic Test?
Where Can I Get Additional Practice?
General Test-Taking Suggestions

10
11
11
11
12
12
12

3.0
3.0
3.1
3.2
3.3
3.4
3.5
3.6

Diagnostic Test
Diagnostic Test
Quantitative Questions
Verbal Questions
Quantitative and Verbal Answer Keys
Interpretive Guide
Quantitative Answer Explanations
Verbal Answer Explanations

14
15
16
23
40
40
41
60

4.0
4.0
4.1
4.2
4.3
4.4

Math Review
Math Review
Arithmetic
Algebra
Geometry
Word Problems

100
101
102
114
122
134

5.0
5.0
5.1
5.2
5.3
5.4
5.5

Problem Solving
Problem Solving
Test-Taking Strategies
The Directions
Practice Questions
Answer Key
Answer Explanations

142
143
144
144
146
181
183

6.0
6.0
6.1
6.2
6.3
6.4
6.5

Data Sufficiency
Data Sufficiency
Test-Taking Strategies
The Directions
Practice Questions
Answer Key
Answer Explanations

264
265
266
268
270
288
289

Table of Contents

7.0
7.0
7.1
7.2
7.3
7.4
7.5
7.6

Reading Comprehension
Reading Comprehension
What Is Measured
Test-Taking Strategies
The Directions
Practice Questions
Answer Key
Answer Explanations

356
357
358
360
361
362
423
424

8.0
8.0
8.1
8.2
8.3
8.4
8.5
8.6

Critical Reasoning
Critical Reasoning
What Is Measured
Test-Taking Strategies
The Directions
Practice Questions
Answer Key
Answer Explanations

504
505
506
506
507
508
546
547

9.0
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8

Sentence Correction
Sentence Correction
Basic English Grammar Rules
Study Suggestions
What Is Measured
Test-Taking Strategies
The Directions
Practice Questions
Answer Key
Answer Explanations

672
673
673
678
678
678
679
680
709
710

10.0
10.0
10.1
10.2
10.3
10.4

Integrated Reasoning
Integrated Reasoning
What Is Measured
The Question Types
Test-Taking Strategies
The Directions

788
789
789
790
797
799

11.0
11.0
11.1
11.2
11.3
11.4
11.5
11.6

Analytical Writing Assessment
Analytical Writing Assessment
What Is Measured
Test-Taking Strategies
The Directions
GMAT® Scoring Guide: Analysis of an Argument
Sample: Analysis of an Argument
Analysis of an Argument Sample Topics

800
801
801
802
802
803
805
809

Answer Sheets
Diagnostic Answer Sheet—Quantitative
Diagnostic Answer Sheet—Verbal
Problem Solving Answer Sheet
Data Sufficiency Answer Sheet
Reading Comprehension Answer Sheet
Critical Reasoning Answer Sheet
Sentence Correction Answer Sheet

841
841
841
842
843
844
845
846

Appendix A

Dear GMAT Test-Taker,
Thank you for your interest in graduate management education. Taking the GMAT® exam lets schools
know that you’re serious about your graduate business education. By using the Official Guide to prepare
for the GMAT, you’re taking a very important step toward achieving your goals and pursuing admission
to a high-quality business school or master’s program.
This book, GMAT ® Official Guide 2018, is designed to help you prepare for and build confidence to do
your best on the GMAT exam. It’s the only guide of its kind on the market that includes real GMAT
exam questions published by the Graduate Management Admission Council (GMAC), the makers of
the exam.
GMAC was founded by the world’s leading business schools in 1953. The GMAT exam was developed
to help people who aspire to careers in management demonstrate their command of the skills needed
for success in the classroom. Schools use and trust the GMAT exam as part of their admissions process
because it’s an excellent predictor of classroom success and your ability to excel in your chosen program.
Today more than 6,500 graduate programs around the world use the GMAT exam to establish their
MBA, graduate-level management degrees and specialized programs as hallmarks of excellence. Nine out
of 10 new MBA enrollments at Top 50 US full-time MBA programs are made using a GMAT
score.
These facts make us proud and drive us to keep improving the GMAT as well as play a role in helping
you find and gain admission to the best school or program for you. We’re committed to ensuring
that no talent goes undiscovered, and that more people around the world can pursue opportunities in
graduate management education.
I applaud your commitment to educational success, and I know that this book and the other Official
GMAT preparation materials available at mba.com will give you the confidence to achieve your
personal best on the GMAT exam and launch or reinvigorate a rewarding career.
I wish you the best success on all your educational and professional endeavors in the future.
Sincerely,

Sangeet Chowfla
CEO of the Graduate Management Admission Council

GMAT® Ofﬁcial Guide 2018

1.0 What Is the GMAT® Exam?

1.0 What Is the GMAT® Exam?
The Graduate Management Admission Test® (GMAT®) exam is a standardized exam used in
admissions decisions by more than 6,500 graduate management programs worldwide. It helps you gauge,
and demonstrate to schools, your academic potential for success in graduate-level management studies.
The four‐part exam measures your Analytical Writing, Verbal, Quantitative, and Integrated Reasoning
skills—higher‐order reasoning skills that management faculty worldwide have identified as important
for incoming students to have. “Higher-order” reasoning skills involve complex judgments, and include
critical thinking, analysis, and problem solving. Unlike undergraduate grades and curricula, which vary
in their meaning across regions and institutions, your GMAT scores provide a standardized, statistically
valid and reliable measure of how you are likely to perform academically in the core curriculum of a
graduate management program. The GMAT exam’s validity, fairness, and value in admissions have been
well‐established through numerous academic studies.
The GMAT exam is delivered entirely in English and solely on a computer. It is not a test of business
knowledge, subject matter mastery, English vocabulary, or advanced computational skills. The GMAT
exam also does not measure other factors related to success in graduate management study, such as job
experience, leadership ability, motivation, and interpersonal skills. Your GMAT score is intended to be
used as one admissions criterion among other, more subjective, criteria, such as admissions essays and
interviews.

1.1 Why Take the GMAT® Exam?
Launched in 1954 by a group of nine business schools to
provide a uniform measure of the academic skills needed to
succeed in their programs, the GMAT exam is now used
by more than 6,500 graduate management programs at
approximately 2,100 institutions worldwide.
Taking the GMAT exam helps you stand out in the
admissions process and demonstrate your readiness and
commitment to pursuing graduate management education.
Schools use GMAT scores to help them select the most
qualified applicants—because they know that candidates
who take the GMAT exam are serious about earning a
graduate business degree, and it’s a proven predictor of a
student’s ability to succeed in his or her chosen program.
When you consider which programs to apply to, you can
look at a school’s use of the GMAT exam as one indicator
of quality. Schools that use the GMAT exam typically list
score ranges or average scores in their class profiles, so you
may also find these profiles helpful in gauging the academic
competitiveness of a program you are considering and how
well your performance on the exam compares with that of
the students enrolled in the program.

Myth -vs- FACT
M – If I don’t achieve a high score
on the GMAT, I won’t get into
my top choice schools.
F – There are great schools
available for candidates at any
GMAT score range.
Fewer than 50 of the more than 250,000
people taking the GMAT exam each year get
a perfect score of 800; and many more get
into top business school programs around
the world each year. Admissions Ofﬁcers
use GMAT scores as one component in their
admissions decisions, in conjunction with
undergraduate records, application essays,
interviews, letters of recommendation, and
other information when deciding whom
to accept into their programs. Visit School
Finder on mba.com to learn about schools
that are the best ﬁt for you.

No matter how you perform on the GMAT exam, you should contact the schools that interest you
to learn more and to ask how they use GMAT scores and other criteria (such as your undergraduate
grades, essays, and letters of recommendation) in their admissions processes. School admissions
3

GMAT® Ofﬁcial Guide 2018

offices, web sites, and materials published by schools are the key sources of information when you are
doing research about where you might want to go to business school.
For more information on the GMAT, test preparation materials, registration, how to use and send your
GMAT scores to schools, and applying to business school, please visit mba.com.

1.2 GMAT® Exam Format
The GMAT exam consists of four separately timed sections
(see the table on the next page). For the Analytical Writing
Assessment (AWA) essay prompt, you will have 30 minutes
to type your essay on a computer keyboard. The 30-minute
Integrated Reasoning section features 12 question prompts
in four different formats. There are two 75-minute,
multiple-choice sections: the Quantitative section, with 37
questions, and the Verbal section, with 41 questions.
The Verbal and Quantitative sections of the GMAT exam
are computer adaptive, which means that the test draws
from a large bank of questions to tailor itself to your ability
level, and you won’t get many questions that are too hard
or too easy for you. The first question will be of medium
difficulty. As you answer each question, the computer scores
your answer and uses it—as well as your responses to all
preceding questions—to select the next question.
Computer‐adaptive tests become more difficult the more
questions you answer correctly, but if you get a question
that seems easier than the last one, it does not necessarily
mean you answered the last question incorrectly. The test
has to cover a range of content, both in the type of question
asked and the subject matter presented.

Myth -vs- FACT
M – Getting an easier question
means I answered the last
one wrong.
F – You should not become
distracted by the difﬁculty level
of a question.
Most people are not skilled at estimating
question difﬁculty, so don’t worry when
taking the test or waste valuable time trying
to determine the difﬁculty of the question you
are answering.
To ensure that everyone receives the same
content, the test selects a speciﬁc number
of questions of each type. The test may call
for your next problem to be a relatively hard
data sufﬁciency question involving arithmetic
operations. But, if there are no more
relatively difﬁcult data sufﬁciency questions
involving arithmetic, you might be given an
easier question.

Because the computer uses your answers to select your next questions, you may not skip questions or go
back and change your answer to a previous question. If you don’t know the answer to a question, try to
eliminate as many choices as possible, then select the answer you think is best.
Though the individual questions are different, the mix of question types is the same for every GMAT
exam. Your score is determined by the difficulty and statistical characteristics of the questions you
answer as well as the number of questions you answer correctly. By adapting to each test‐taker, the
GMAT exam is able to accurately and efficiently gauge skill levels over a full range of abilities, from
very high to very low.
The test includes the types of questions found in this book and in the online Integrated Reasoning
component, but the format and presentation of the questions are different on the computer. When you
take the test:
• Only one question or question prompt at a time is presented on the computer screen.
• The answer choices for the multiple‐choice questions will be preceded by circles, rather than by
letters.
4

1.3 What Is the GMAT® Exam? What Is the Content of the Test Like?

• Different question types appear in random order in the multiple‐choice and Integrated Reasoning
sections.
• You must select your answer using the computer.
• You must choose an answer and conﬁrm your choice before moving on to the next question.
• You may not go back to previous screens to change answers to previous questions.

Format of the GMAT® Exam
Questions

Timing

Analytical Writing
Analysis of an Argument

30 min.

Integrated Reasoning
Multi‐Source Reasoning
Table Analysis
Graphics Interpretation
Two‐Part Analysis

30 min.

Optional break
Quantitative
Problem Solving
Data Sufﬁciency

up to 8 min.
37

Optional break
Verbal
Reading Comprehension
Critical Reasoning
Sentence Correction

75 min.

up to 8 min.
41

75 min.

Total Time:

210 min.

1.3 What Is the Content of the Test Like?
The GMAT exam measures higher‐order analytical skills encompassing several types of reasoning. The
Analytical Writing Assessment asks you to analyze the reasoning behind an argument and respond
in writing; the Integrated Reasoning section asks you to interpret and synthesize information from
multiple sources and in different formats to make reasoned conclusions; the Quantitative section asks
you to reason quantitatively using basic arithmetic, algebra, and geometry; and the Verbal section asks
you to read and comprehend written material and to reason and evaluate arguments.
Test questions may address a variety of subjects, but all of the information you need to answer the
questions will be included on the exam, with no outside knowledge of the subject matter necessary. The
GMAT exam is not a test of business knowledge, English vocabulary, or advanced computational skills.
You will need to read and write in English and have basic math and English skills to perform well on
the test, but its difficulty comes from the English analytical and critical thinking abilities.
The questions in this book are organized by question type and from easiest to most difficult, but keep in mind
that when you take the test, you may see different types of questions in any order within each section.
5

GMAT® Ofﬁcial Guide 2018

1.4 Integrated Reasoning Section
The Integrated Reasoning section highlights the relevant skills that business managers in today’s datadriven world need in order to analyze sophisticated streams of data and solve complex problems. It
measures your ability to understand and evaluate multiple sources and types of information—graphic,
numeric, and verbal—as they relate to one another; use both quantitative and verbal reasoning to solve
complex problems; and solve multiple problems in relation to one another.
Four types of questions are used in the Integrated Reasoning section:
• Multi‐Source Reasoning
• Table Analysis
• Graphics Interpretation
• Two‐Part Analysis
Integrated Reasoning questions may be quantitative, verbal, or a combination of both. You will have to
interpret graphics and sort tables to extract meaning from data, but advanced statistical knowledge and
spreadsheet manipulation skills are not necessary. You will have access to an on-screen calculator with
basic functions for the Integrated Reasoning section, but note that the calculator is not available on the
Quantitative section.
To review the Integrated Reasoning question types and test‐taking tips, see chapter 10. For practice
questions of each format, with full answer explanations, visit the Integrated Reasoning online
component using your unique access code found in the inside front cover of the book.

1.5 Quantitative Section
The GMAT Quantitative section measures your ability to reason quantitatively, solve quantitative
problems, and interpret graphic data.
Two types of multiple‐choice questions are used in the Quantitative section:
• Problem Solving
• Data Sufficiency
Both are intermingled throughout the Quantitative section, and require basic knowledge of arithmetic,
elementary algebra, and commonly known concepts of geometry.
To review the basic mathematical concepts that you will need to answer Quantitative questions, see the
math review in chapter 4. For test‐taking tips speciﬁc to the question types in the Quantitative section,
practice questions, and answer explanations, see chapters 5 and 6.

1.6 Verbal Section
The GMAT Verbal section measures your ability to read and comprehend written material and to
reason and evaluate arguments. The Verbal section includes reading sections from several different
content areas. Although you may be generally familiar with some of the material, neither the reading
passages nor the questions assume detailed knowledge of the topics discussed.
6

1.7 What Is the GMAT® Exam? Analytical Writing Assessment

Three types of multiple‐choice questions are intermingled throughout the Verbal section:
• Reading Comprehension
• Critical Reasoning
• Sentence Correction
All three require basic knowledge of the English language, but the Verbal section is not a test of
advanced vocabulary.
For test‐taking tips speciﬁc to each question type in the Verbal section, practice questions, and answer
explanations, see chapters 7 through 9.

1.7 Analytical Writing Assessment
The Analytical Writing Assessment (AWA) consists of one 30‐minute writing task: Analysis of an
Argument. The AWA measures your ability to think critically, communicate your ideas, and formulate
an appropriate and constructive critique. You will type your essay on a computer keyboard.
For test‐taking tips, sample essay responses, answer explanations, and sample Analysis of an Argument
topics, see chapter 11.

1.8 What Computer Skills Will I Need?
The GMAT exam requires only computer skills. You will type your AWA essay on the computer
keyboard using standard word‐processing keystrokes. In the Integrated Reasoning and
multiple‐choice sections, you will select your responses using either your computer mouse or the
keyboard. The Integrated Reasoning section includes basic computer navigation and functions, such as
clicking on tabs and using drop‐down menus to sort tables and select answers.

1.9 What Are the Test Centers Like?
The GMAT exam is administered under standardized conditions at test centers worldwide. Each test
center has a proctored testing room with individual computer workstations that allow you to sit for the
exam under quiet conditions and with some privacy. You will be able to take two optional 8-minute
breaks during the course of the exam. You may not take notes or scratch paper with you into the testing
room, but an erasable notepad and marker will be provided for you to use during the test.

1.10 How Are Scores Calculated?
Verbal and Quantitative sections are scored on a scale of 0 to 60, with scores below 6 or above 51
extremely rare. The Total GMAT score ranges from 200 to 800 and is based on your performance in
these two sections. Your score is determined by:
• The number of questions you answer
• The number of questions you answer correctly or incorrectly
• The level of difficulty and other statistical characteristics of each question

GMAT® Ofﬁcial Guide 2018

Your Verbal, Quantitative, and Total GMAT scores are determined by a complex mathematical
procedure that takes into account the difficulty of the questions that were presented to you and how
you answered them. When you answer the easier questions correctly, you get a chance to answer harder
questions, making it possible to earn a higher score. After you have completed all the questions on the
test, or when your time is expired, the computer will calculate your scores. Your scores on the Verbal and
Quantitative sections are combined to produce your Total score which ranges from 200 to 800 in
10 point increments.
The Analytical Writing Assessment consists of one writing task, Analysis of an Argument, and your essay
will be scored two times independently. Essays are evaluated by college and university faculty members
from a variety of disciplines, including management education, who rate the overall quality of your critical
thinking and writing. (For details on how readers are qualiﬁed, visit mba.com.) In addition, your response
may be scored by an automated scoring program designed to reﬂect the judgment of expert readers.
Your essay is scored on a scale of 0 to 6, in half point increments, with 6 being the highest score and
0 the lowest. A score of zero is given for responses that are off topic, are in a foreign language, merely
attempt to copy the topic, consist only of keystroke characters, or are blank. Your AWA score is typically
the average of two independent ratings. If the independent scores vary by more than a point, a third
reader adjudicates, but because of ongoing training and monitoring, discrepancies are rare.
Your Integrated Reasoning section is scored on a scale of 1 to 8, in one-point increments. You must
answer all parts of a single question correctly in order to receive credit. No partial credit is given. Like
your AWA score, your Integrated Reasoning score will not count toward your Total score.
Your Analytical Writing Assessment and Integrated Reasoning scores are computed and reported separately
from the other sections of the test and have no effect on your Verbal, Quantitative, or Total scores. The
schools that you have designated to receive your scores may receive a copy of your Analytical Writing
Assessment essay with your score report. Your own copy of your score report will not include your essay.
Your GMAT score includes a percentile ranking that compares your skill level with other test-takers
from the past three years. The percentile rank of your score shows the percentage of tests taken with
scores lower than your score. Every July, percentile ranking tables are updated. Visit mba.com to view
the most recent percentile rankings tables.

1.11 Test Development Process
The GMAT exam is developed by experts who use standardized procedures to ensure high‐quality,
widely-appropriate test material. All questions are subjected to independent reviews and are revised
or discarded as necessary. Multiple‐choice questions are tested during GMAT exam administrations.
Analytical Writing Assessment tasks are tested on mba.com registrants and then assessed for their
fairness and reliability. For more information on test development, visit mba.com.

1.11 What Is the GMAT® Exam? Test Development Process

To register for the GMAT exam, go to www.mba.com
9

2.0 How to Prepare

2.0 How to Prepare
2.1 How Should I Prepare to Take the Test?
The GMAT® exam is designed specifically to measure reasoning skills needed for management
education, and the test contains several question formats unique to the GMAT exam. At a minimum,
you should be familiar with the test format and the question formats before you sit for the test. Because
the GMAT exam is a timed exam, you should practice answering test questions, not only to better
understand the question formats and the skills they require, but also to help you learn to pace yourself
so you can finish each section when you sit for the exam.
Because the exam measures reasoning rather than
subject matter knowledge, you most likely will not find
it helpful to memorize facts. You do not need to study
advanced mathematical concepts, but you should be sure
your grasp of basic arithmetic, algebra, and geometry is
sound enough that you can use these skills in quantitative
problem solving. Likewise, you do not need to study
advanced vocabulary words, but you should have a firm
understanding of basic English vocabulary and grammar
for reading, writing, and reasoning.

Myth -vs- FACT
M – You need very advanced math
skills to get a high GMAT score.
F – The GMAT measures your
reasoning and critical thinking
abilities, rather than your
advanced math skills.
The GMAT exam only requires basic quantitative

This book and other study materials released by the
skills. You should review the math skills
(algebra, geometry, basic arithmetic) presented
Graduate Management Admission Council (GMAC)
in this guide (chapter 4) and the GMAT®
are the ONLY source of questions that have been retired
Ofﬁcial Guide Quantitative Review 2018. The
from the GMAT exam. All questions that appear or
difﬁculty of GMAT Quantitative questions stems
have appeared on the GMAT exam are copyrighted and
from the logic and analysis used to solve the
problems and not the underlying math skills.
owned by GMAC, which does not license them to be
reprinted elsewhere. Accessing live Integrated Reasoning,
Quantitative, or Verbal test questions in advance or sharing test content during or after you take the test
is a serious violation, which could cause your scores to be canceled and schools to be notified. In cases of
a serious violation, you may be banned from future testing and other legal remedies may be pursued.

2.2 What About Practice Tests?
The Quantitative and Verbal sections of the GMAT exam are computer adaptive, and the Integrated
Reasoning section includes questions that require you to use the computer to sort tables and navigate to
different sources of information. Our official practice materials will help you get comfortable with the
format of the test and better prepare for exam day. Two full-length GMAT practice exams are available at no
charge for those who have created an account on mba.com. The practice exams include computer‐adaptive
Quantitative and Verbal sections, plus additional practice questions, information about the test, and tutorials
to help you become familiar with how the GMAT exam will appear on the computer screen at the test center.
To maximize your studying efforts with the free practice exams, you should leverage official practice
materials as you start to prepare for the test. Take one practice test to make yourself familiar with the
exam and to get a baseline score. After you have studied using this book and other study materials, take
the second practice test to determine whether you need to shift your focus to other areas you need to
strengthen. Note that the free practice tests may include questions that are also published in this book.
As your test day approaches, consider taking more official practice tests to help measure your progress
and give you a better idea of how you might score on exam day.
11

GMAT® Ofﬁcial Guide 2018

2.3 How Should I Use the Diagnostic Test?
This book contains a Diagnostic Test to help you determine the types of Quantitative and Verbal
questions that you need to practice most. You should take the Diagnostic Test around the same time
that you take the first GMAT practice test. The Diagnostic Test will give you a rating—below average,
average, above average, or excellent—of your skills in each type of GMAT test question. These ratings will
help you identify areas to focus on as you prepare for the GMAT exam.
The Diagnostic Test does not include Integrated Reasoning or Analytical Writing Assessment questions.
Use the results of the Diagnostic Test to help you select the right chapter of this book to start with.
Next, read the introductory material carefully, and answer the practice questions in that chapter.
Remember, the questions in the chapters are organized by difficulty, from easiest to most difficult. Make
sure you follow the directions for each type of question and try to work as quickly and as efficiently as
possible. Then review the explanations for the correct answers, spending as much time as necessary to
familiarize yourself with the range of questions or problems presented.

2.4 Where Can I Get Additional Practice?
If you would like additional practice, GMAT® Official Guide 2018 Quantitative Review and GMAT®
Official Guide 2018 Verbal Review each offer 300 additional questions that are not published in this
guide. You can also find more Quantitative, Verbal, and Integrated Reasoning practice questions, fulllength, computer-adaptive practice exams, Analytical Writing Assessment practice prompts, and other
helpful study materials at mba.com.

2.5 General Test-Taking Suggestions
Speciﬁc test‐taking strategies for individual question types are presented later in this book. The
following are general suggestions to help you perform your best on the test.
1.

Use your time wisely.

Although the GMAT exam stresses accuracy more than speed, it is important to use your time wisely. On
average, you will have about 1¾ minutes for each Verbal question, about 2 minutes for each Quantitative
question, and about 2½ minutes for each Integrated Reasoning question, some of which have multiple
questions. Once you start the test, an onscreen clock will show the time you have left. You can hide this
display if you want, but it is a good idea to check the clock periodically to monitor your progress. The clock
will automatically alert you when 5 minutes remain for the section you are working on.
2.

Answer practice questions ahead of time.

After you become generally familiar with all question types, use the practice questions in this book
and the online Integrated Reasoning component to prepare for the actual test. It may be useful to time
yourself as you answer the practice questions to get an idea of how long you will have for each question
when you sit for the actual test, as well as to determine whether you are answering quickly enough to
finish the test in the allotted time.

2.0 How to Prepare General Test-Taking Suggestions

Read all test directions carefully.

The directions explain exactly what is required to
answer each question type. If you read hastily, you may
miss important instructions and impact your ability to
answer correctly. To review directions during the test,
click on the Help icon. But be aware that the time you
spend reviewing directions will count against your time
allotment for that section of the test.
4.

Read each question carefully and thoroughly.

Before you answer a question, determine exactly what is
being asked and then select the best choice. Never skim
a question or the possible answers; skimming may cause
you to miss important information or nuances.
5.

Do not spend too much time on any one question.

If you do not know the correct answer, or if the
question is too time consuming, try to eliminate choices
you know are wrong, select the best of the remaining
answer choices, and move on to the next question.
Not completing sections and randomly guessing
answers to questions at the end of each test section
can significantly lower your score. As long as you have
worked on each section, you will receive a score even if
you do not finish one or more sections in the allotted
time. You will not earn points for questions you never
get to see.
6.

Conﬁrm your answers ONLY when you are ready to
move on.

On the Quantitative and Verbal sections, once you have
selected your answer to a multiple‐choice question,
you will be asked to conﬁrm it. Once you conﬁrm your
response, you cannot go back and change it. You may
not skip questions. In the Integrated Reasoning section,
there may be several questions based on information
provided in the same question prompt. When there
is more than one response on a single screen, you can
change your response to any of the questions on the
screen before moving on to the next screen. However,
you may not navigate back to a previous screen to
change any responses.
7.

Myth -vs- FACT
M – It is more important to respond
correctly to the test questions
than it is to ﬁnish the test.
F – There is a signiﬁcant penalty
for not completing the GMAT
exam.
Pacing is important. If you are stumped by a
question, give it your best guess and move
on. If you guess incorrectly, the computer
program will likely give you an easier
question, which you are likely to answer
correctly, and the computer will rapidly
return to giving you questions matched to
your ability. If you don’t ﬁnish the test, your
score will be reduced. Failing to answer ﬁve
verbal questions, for example, could reduce
your score from the 91st percentile to the
77th percentile.

Myth -vs- FACT
M – The ﬁrst 10 questions are
critical and you should invest
the most time on those.
F – All questions count.
The computer-adaptive testing algorithm
uses each answered question to obtain an
initial estimate. However, as you continue to
answer questions, the algorithm self-corrects
by computing an updated estimate on the
basis of all the questions you have answered,
and then administers items that are closely
matched to this new estimate of your ability.
Your ﬁnal score is based on all your responses
and considers the difﬁculty of all the
questions you answered. Taking additional
time on the ﬁrst 10 questions will not game
the system and can hurt your ability to ﬁnish
the test.

Plan your essay answer before you begin to write.

The best way to approach the Analysis of an Argument
section is to read the directions carefully, take a few minutes to think about the question, and plan a
response before you begin writing. Take time to organize your ideas and develop them fully, but leave
time to reread your response and make any revisions that you think would improve it.

3.0 Diagnostic Test

3.0 Diagnostic Test
Like the practice sections later in the book, the Diagnostic Test uses questions from real GMAT®
exams. The purpose of the Diagnostic Test is to help you determine how skilled you are in answering
each of the ﬁve types of questions on the GMAT exam: problem solving, data sufﬁciency, reading
comprehension, critical reasoning, and sentence correction.
Scores on the Diagnostic Test are designed to help you answer the question, “If all the questions on the
GMAT exam were like the questions in this section, how well would I do?” Your scores are classiﬁed as
being excellent, above average, average, or below average, relative to the scores of other test‐takers. You
can use this information to focus your test‐preparation activities.

Instructions
1. Take your time answering these questions. The Diagnostic Test is not timed.
2. If you are stumped by a question, you should guess and move on, just like you should do on the
real GMAT exam.
3. You can take one segment at a time, if you want. It is better to ﬁnish an entire section
(Quantitative or Verbal) in one sitting, but this is not a requirement.
4. You can go back and change your answers in the Diagnostic Test.
5. After you take the test, check your answers using the answer key that follows the test. The number
of correct answers is your raw score.
6. Convert your raw score, using the table provided.
Note: The Diagnostic Test is designed to give you guidance on how to prepare for the GMAT exam;
however, a strong score on one type of question does not guarantee that you will perform as well on the
real GMAT exam. The statistical reliability of scores on the Diagnostic Test ranges from 0.75 to 0.89,
and the subscale classiﬁcation is about 85%–90% accurate, meaning that your scores on the Diagnostic
Test are a good, but not perfect, measure of how you are likely to perform on the real test. Use the tests
on the free online software to obtain a good estimate of your expected GMAT Verbal, Quantitative, and
Total scores.
You should not compare the number of questions you got right in each section. Instead, you should
compare how your responses are rated in each section.

GMAT® Official Guide 2018

3.1 Quantitative Questions
Problem Solving
Solve the problem and indicate the best of the answer choices given.
Numbers: All numbers used are real numbers.
Figures: All ﬁgures accompanying problem solving questions are intended to provide information
useful in solving the problems. Figures are drawn as accurately as possible. Exceptions will
be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also
straight. The positions of points, angles, regions, etc., exist in the order shown, and angle
measures are greater than zero. All ﬁgures lie in a plane unless otherwise indicated.
1.

Last month a certain music club offered a discount
to preferred customers. After the ﬁrst compact disc
purchased, preferred customers paid $3.99 for each
additional compact disc purchased. If a preferred
customer purchased a total of 6 compact discs and
paid $15.95 for the ﬁrst compact disc, then the dollar
amount that the customer paid for the 6 compact
discs is equivalent to which of the following?
(A)

5(4.00) + 15.90

(B)

5(4.00) + 15.95

(C)

5(4.00) + 16.00

(D)

5(4.00 − 0.01) + 15.90

(E)

5(4.00 − 0.05) + 15.95

The average (arithmetic mean) of the integers from
200 to 400, inclusive, is how much greater than the
average of the integers from 50 to 100, inclusive?
(A)

150

(B)

175

(C)

200

(D)

225

(E)

300

The sequence a1, a2 , a3 , , an , is such that
a +a
an = n −1 n −2 for all n ≥ 3. If a3 = 4 and
2
a5 = 20 what is the value of a6 ?

Among a group of 2,500 people, 35 percent invest
in municipal bonds, 18 percent invest in oil stocks,
and 7 percent invest in both municipal bonds and oil
stocks. If 1 person is to be randomly selected from the
2,500 people, what is the probability that the person
selected will be one who invests in municipal bonds but
NOT in oil stocks?
(A)

9
50

(B)

7
25

(C)

7
20

(D)

21
50

(E)

27
50

A closed cylindrical tank contains 36π cubic feet of
water and is ﬁlled to half its capacity. When the tank
is placed upright on its circular base on level ground,
the height of the water in the tank is 4 feet. When the
tank is placed on its side on level ground, what is the
height, in feet, of the surface of the water above the
ground?
(A)

(B)

(C)

(A)

(D)

(B)

(E)

(C)

(D)

(E)

3.1 Diagnostic Test Quantitative Questions

A marketing ﬁrm determined that, of 200 households
surveyed, 80 used neither Brand A nor Brand B soap,
60 used only Brand A soap, and for every household
that used both brands of soap, 3 used only Brand B
soap. How many of the 200 households surveyed used
both brands of soap?
(A)

(B)

(C)

(D)

(E)

A researcher computed the mean, the median, and the
standard deviation for a set of performance scores.
If 5 were to be added to each score, which of these
three statistics would change?
(A)

The mean only

(B)

The median only

(C)

The standard deviation only

(D)

The mean and the median

(E)

The mean and the standard deviation

y°

A certain club has 10 members, including Harry. One
of the 10 members is to be chosen at random to be
the president, one of the remaining 9 members is to
be chosen at random to be the secretary, and one of
the remaining 8 members is to be chosen at random
to be the treasurer. What is the probability that Harry
will be either the member chosen to be the secretary
or the member chosen to be the treasurer?
(A)

1
720

(B)

1
80

1
10
1
9
1
5

x°

v°

10.

z°

w°

In the ﬁgure shown, what is the value of
v+x+y+z+w?
(A) 45
(B) 90
(C)

180

(D)

270

(E)

360

2
5
of its revenue in December and its revenue in January
1
was of its revenue in November, then the store’s
4
revenue in December was how many times the average
(arithmetic mean) of its revenues in November and
January?
If a certain toy store’s revenue in November was

(A)

1
4

(B)

1
2

(C)

2
3

(D)

(E)

GMAT® Official Guide 2018

11.

12.

13.

Of the three‐digit integers greater than 700, how many
have two digits that are equal to each other and the
remaining digit different from the other two?

15.

109

(B)

108

(B)

(C)

107

(C)

(D)

106

(D)

(E)

105

(E)

16.

Positive integer y is 50 percent of 50 percent of
positive integer x, and y percent of x equals 100. What
is the value of x ?
(A)

(B)

100

(C)

200

(D)

1,000

(E)

2,000

17.

(E)

Of the 84 parents who attended a meeting at a school,
35 volunteered to supervise children during the school
picnic and 11 volunteered both to supervise children
during the picnic and to bring refreshments to the
picnic. If the number of parents who volunteered
to bring refreshments was 1.5 times the number
of parents who neither volunteered to supervise
children during the picnic nor volunteered to bring
refreshments, how many of the parents volunteered to
bring refreshments?
(A)

(B)

(C)

(D)

(E)

(A)

(B)

(C)

2 − 2x

(D)

4x − 2

(E)

6x – 1

If n =

(B)
(C)
(D)

If 3 − 2x = 2x + 1, then 4x2 =

(A)

s
If s and t are positive integers such that = 64.12,
t
which of the following could be the remainder when s
is divided by t ?

(B) 4

(A)

(A) 2

14.

The product of all the prime numbers less than 20 is
closest to which of the following powers of 10 ?

(E)
18.

19.

16
, what is the value of n ?
81
1
9
1
4
4
9
2
3
9
2

If n is the product of the integers from 1 to 8,
inclusive, how many different prime factors greater
than 1 does n have?
(A)

Four

(B)

Five

(C)

Six

(D)

Seven

(E)

Eight

If k is an integer and 2 < k < 7, for how many different
values of k is there a triangle with sides of lengths
2, 7, and k ?
(A)

One

(B)

Two

(C)

Three

(D)

Four

(E)

Five

3.1 Diagnostic Test Quantitative Questions

20.

A right circular cone is inscribed in a hemisphere so
that the base of the cone coincides with the base of
the hemisphere. What is the ratio of the height of the
cone to the radius of the hemisphere?
(A)

II.

(B)

1:1

III.

(C)

1
:1
2

(A)

II only

(B)

I and II only

(C)

I and III only

(D)

II and III only

(E)

I, II, and III

(E)

22.

If the positive integer x is a multiple of 4 and the
positive integer y is a multiple of 6, then xy must be a
multiple of which of the following?
I. 8

3 :1

2 :1

(D)

21.

23.

2:1

John deposited $10,000 to open a new savings
account that earned 4 percent annual interest,
compounded quarterly. If there were no other
transactions in the account, what was the amount of
money in John’s account 6 months after the account
was opened?
(A)

$10,100

(B)

$10,101

(C)

$10,200

(D)

$10,201

(E)

$10,400

A container in the shape of a right circular cylinder is
1
full of water. If the volume of water in the container
2
is 36 cubic inches and the height of the container
is 9 inches, what is the diameter of the base of the
cylinder, in inches?
(A)

16
9π

(B)

4
π

(C)

12
π

(D)

2
π

(E)

24.

Aaron will jog from home at x miles per hour and then
walk back home by the same route at y miles per hour.
How many miles from home can Aaron jog so that he
spends a total of t hours jogging and walking?
(A)
(B)
(C)
(D)
(E)

xt
y
x +t
xy
xyt
x+y
x + y +t
xy
y +t t
−
x
y

2
π

GMAT® Official Guide 2018

Data Sufficiency
Each data sufficiency problem consists of a question and two statements, labeled (1) and (2), which
contain certain data. Using these data and your knowledge of mathematics and everyday facts (such
as the number of days in July or the meaning of the word counterclockwise), decide whether the data
given are sufficient for answering the question and then indicate one of the following answer choices:
A
B
C
D
E

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

Note: In data sufficiency problems that ask for the value of a quantity, the data given in the statements
are sufficient only when it is possible to determine exactly one numerical value for the quantity.
Example:

yº

P
xº
zº

In ΔPQR, what is the value of x ?
(1)

PQ = PR

(2)

y = 40

Explanation: According to statement (1) PQ = PR; therefore, ΔPQR is isosceles and y = z. Since
x + y + z = 180, it follows that x + 2y = 180. Since statement (1) does not give a value for y, you cannot
answer the question using statement (1) alone. According to statement (2), y = 40; therefore, x + z = 140.
Since statement (2) does not give a value for z, you cannot answer the question using statement (2)
alone. Using both statements together, since x + 2y = 180 and the value of y is given, you can find the
value of x. Therefore, BOTH statements (1) and (2) TOGETHER are sufficient to answer the question, but
NEITHER statement ALONE is sufficient.
Numbers: All numbers used are real numbers.
Figures:
• F igures conform to the information given in the question, but will not necessarily conform to the
additional information given in statements (1) and (2).
• Lines shown as straight are straight, and lines that appear jagged are also straight.
• The positions of points, angles, regions, etc., exist in the order shown, and angle measures are
greater than zero.
• All figures lie in a plane unless otherwise indicated.

3.1 Diagnostic Test Quantitative Questions

25.

If the units digit of integer n is greater than 2, what is
the units digit of n ?
(1)

The units digit of n is the same as the units digit
of n2.

(2)

The units digit of n is the same as the units digit
of n3.

31.

32.
26.

27.

28.

29.

(1)

The variance for the set of measurements is 4.

(2)

For each measurement, the difference between
the mean and that measurement is 2.

What is the value of the integer p ?

Is the range of the integers 6, 3, y, 4, 5, and x greater
than 9 ?

(1)

Each of the integers 2, 3, and 5 is a factor of p.

(1)

y > 3x

(2)

Each of the integers 2, 5, and 7 is a factor of p.

(2)

y>x>3

If the length of Wanda’s telephone call was rounded
up to the nearest whole minute by her telephone
company, then Wanda was charged for how many
minutes for her telephone call?
(1)

The total charge for Wanda’s telephone call was
$6.50.

(2)

Wanda was charged $0.50 more for the ﬁrst
minute of the telephone call than for each minute
after the ﬁrst.

33.

34.

What is the perimeter of isosceles triangle MNP ?
(1)

MN = 16

(2)

NP = 20

In a survey of retailers, what percent had purchased
computers for business purposes?
(1)

(2)

30.

Is the standard deviation of the set of measurements
x1, x2, x3, x4, . . ., x20 less than 3 ?

85 percent of the retailers surveyed who owned
their own store had purchased computers for
business purposes.
40 percent of the retailers surveyed owned their
own store.

The only gift certiﬁcates that a certain store sold
yesterday were worth either $100 each or $10 each. If
the store sold a total of 20 gift certiﬁcates yesterday,
how many gift certiﬁcates worth $10 each did the
store sell yesterday?
(1)

(2)

35.

The gift certiﬁcates sold by the store yesterday
were worth a total of between $1,650 and
$1,800.

36.

37.

5x +2
< 1?
25

(1)

5x < 1

(2)

x<0

Of the companies surveyed about the skills they
required in prospective employees, 20 percent
required both computer skills and writing skills. What
percent of the companies surveyed required neither
computer skills nor writing skills?
(1)

Of those companies surveyed that required
computer skills, half required writing skills.

(2)

45 percent of the companies surveyed required
writing skills but not computer skills.

What is the value of w + q ?
(1)

3w = 3 – 3q

(2)

5w + 5q = 5

If X and Y are points in a plane and X lies inside the
circle C with center O and radius 2, does Y lie inside
circle C ?
(1)

The length of line segment XY is 3.

(2)

The length of line segment OY is 1.5.

Is x > y ?
(1)
(2)

x=y+2
x
= y −1
2

Yesterday the store sold more than 15 gift
certiﬁcates worth $100 each.

GMAT® Official Guide 2018

38.

39.

40.

41.

42.

43.

If Paula drove the distance from her home to her
college at an average speed that was greater than
70 kilometers per hour, did it take her less than
3 hours to drive this distance?
(1)

The distance that Paula drove from her home to
her college was greater than 200 kilometers.

(2)

The distance that Paula drove from her home to
her college was less than 205 kilometers.

44.

(1)
(2)
45.

In the xy‐plane, if line k has negative slope and passes
through the point (−5,r), is the x‐intercept of line
k positive?
(1)

The slope of line k is −5.

(2)

r>0

46.

If $5,000 invested for one year at p percent simple
annual interest yields $500, what amount must be
invested at k percent simple annual interest for one
year to yield the same number of dollars?
(1)

k = 0.8p

(2)

k=8

x+y
> 0 , is x < 0 ?
z

If m and n are positive integers, is

47.

( m ) is an integer.
( n ) is an integer.

( m)

an integer?

Of the 66 people in a certain auditorium, at most
6 people have their birthdays in any one given month.
Does at least one person in the auditorium have a
birthday in January?
(1)

More of the people in the auditorium have their
birthday in February than in March.

(2)

Five of the people in the auditorium have their
birthday in March.

Last year the average (arithmetic mean) salary of the
10 employees of Company X was $42,800. What is
the average salary of the same 10 employees this
year?
(1)

For 8 of the 10 employees, this year’s salary is
15 percent greater than last year’s salary.

(2)

For 2 of the 10 employees, this year’s salary is
the same as last year’s salary.

(1)

x 7.
For k values 3, 4, and 5, the sum of the smaller
two sides is not larger than the third side; thus,
6 is the only possible value of k that satisﬁes the
conditions.
The correct answer is A.

3.5 Diagnostic Test Quantitative Answer Explanations

20.

1:1

(C)

1
:1
2

(E)

Since John’s account is compounded quarterly, he
1
receives of his annual interest, or 1%, every
4
3 months. This is added to the amount already in
the account to accrue interest for the next quarter.
After 6 months, this process will have occurred
twice, so the amount in John’s account will then be
($10,000)(1.01)(1.01) = $10,000(1.01)2 = $10,201.

3 :1

(B)

(D)

Arithmetic Operations on rational numbers

The correct answer is D.

2 :1
2:1
22.

Geometry Volume

As the diagram below shows, the height of the
cone will be the radius of the hemisphere, so the
ratio is 1:1.

The correct answer is B.
21.

A container in the shape of a right circular cylinder is
1 full of water. If the volume of water in the container
2
is 36 cubic inches and the height of the container
is 9 inches, what is the diameter of the base of the
cylinder, in inches?
(A)

16
9π

(B)

4
π

(C)

12
π

(D)

2
π

(E)

2
π

Geometry Volume

For a right cylinder, volume = π(radius)2(height).
Since the volume of water is 36 cubic inches and
since this represents 1 the container, the water
2
is occupying 1 the container’s height, or
2
9  1  = 4.5 inches. Let r be the radius of the
 2
cylinder.

(A)

$10,100

(B)

$10,101

(C)

$10,200

(D)

$10,201

36 = π r 2 ( 4.5)

(E)

$10,400

8 = πr 2

divide both sides by 4.5

8 = r2
π

divide both sides by π

8 =r
π

take the square root of both sides

2 2
=r
π

simplify the 8 to get the radius

GMAT® Official Guide 2018

Then, since the diameter is twice the length of the
radius, the diameter equals
 2 2
2
2
=4
=4 2.

 π 
π
π

23.

24.

The correct answer is E.

(A)

xt
y

If the positive integer x is a multiple of 4 and the
positive integer y is a multiple of 6, then xy must be a
multiple of which of the following?

(B)

x +t
xy

(C)

xyt
x+y

(D)

x + y +t
xy

(E)

y +t t
−
x
y

I. 8
II.

III.

(A)

II only

(B)

I and II only

(C)

I and III only

(D)

II and III only

(E)

I, II, and III

Arithmetic Properties of numbers

The product xy must be a multiple of 4(6) = 24
and any of its factors. Test each alternative.
I.

24 = 3
8 is a factor of 24
8
MUST be a multiple of 8

24 = 2
12 is a factor of 24
II.
12
MUST be a multiple of 12
III.

24 = 1 1
18 is NOT a factor of 24
18
3
NEED NOT be a multiple of 18

The correct answer is B.

Algebra Simplifying algebraic expressions

Let j be the number of hours Aaron spends
jogging; then let t – j be the total number of
hours he spends walking. It can be stated that
Aaron jogs a distance of xj miles and walks a
distance of y(t – j) miles. Because Aaron travels
the same route, the miles jogged must equal the
miles walked, and they can be set equal.
xj = y (t − j ) set number of miles equal to
each other
xj = yt − jy

distribute the y

xj + jy = yt add jy to both sides to get all
terms with j to one side
j ( x + y ) = yt
j=

yt
.
x+ y

factor out the j
divide both sides by x + y

So, the number of hours Aaron spends jogging is
yt
j=
.
x+ y
The number of miles he can jog is xj or, by
 yt 
xyt
=
substitution of this value of j, x 
.

 x+ y x+ y
The correct answer is C.

3.5 Diagnostic Test Quantitative Answer Explanations

Data Sufficiency
The following section on data sufficiency is intended to familiarize you with the most efficient and
effective approaches to the kinds of problems common to data sufficiency. The particular questions in
this chapter are generally representative of the kinds of data sufficiency questions you will encounter
on the GMAT exam. Remember that it is the problem solving strategy that is important, not the specific
details of a particular question.

25.

If the units digit of integer n is greater than 2, what is
the units digit of n ?

26.

What is the value of the integer p ?
(1)

Each of the integers 2, 3, and 5 is a factor of p.

(1)

The units digit of n is the same as the units digit
of n2.

(2)

Each of the integers 2, 5, and 7 is a factor of p.

(2)

The units digit of n is the same as the units digit
of n3.

Arithmetic Properties of numbers

(1) These are factors of p, but it is not clear that
they are the only factors of p; NOT sufﬁcient.

Arithmetic Arithmetic operations

(2) These are factors of p, but it is not clear that
they are the only factors of p; NOT sufﬁcient.

If the units digit of n is greater than 2, then it can
only be the digits 3, 4, 5, 6, 7, 8, or 9.
(1) To solve this problem, it is necessary to
ﬁnd a digit that is the same as the units digit of
its square. For example, both 43 squared (1,849)
and 303 squared (91,809) have a units digit of 9,
which is different from the units digit of 43 and
303. However, 25 squared (625) and 385 squared
(148,225) both have a units digit of 5, and 16 and
226 both have a units digit of 6 and their squares
(256 and 51,076, respectively) do, too. However,
there is no further information to choose between
5 or 6; NOT sufﬁcient.

Taken together, (1) and (2) overlap, but again
there is no clear indication that these are the only
factors of p.
The correct answer is E;
both statements together are still not sufﬁcient.
27.

If the length of Wanda’s telephone call was rounded
up to the nearest whole minute by her telephone
company, then Wanda was charged for how many
minutes for her telephone call?

(2) Once again, 5 and 6 are the only numbers
which, when cubed, will both have a 5 or 6
respectively in their units digits. However, the
information given does not distinguish between
them; NOT sufﬁcient.

(1)

The total charge for Wanda’s telephone call was
$6.50.

(2)

Wanda was charged $0.50 more for the ﬁrst
minute of the telephone call than for each minute
after the ﬁrst.

Since (1) and (2) together yield the same
information but with no direction as to which
to choose, there is not enough information to
determine the answer.

Arithmetic Arithmetic operations

The correct answer is E;
both statements together are still not sufﬁcient.

(1) This does not give any information as to the
call’s cost per minute; NOT sufﬁcient.
(2) From this, it can be determined only that
the call was longer than one minute and that the
charge for the ﬁrst minute was $0.50 more than
the charge for each succeeding minute; NOT
sufﬁcient.

GMAT® Official Guide 2018

Taking (1) and (2) together, the number of
minutes cannot be determined as long as the
cost of each minute after the ﬁrst is unknown.
For example, if the cost of each minute after the
ﬁrst minute were $0.40, then the cost of the ﬁrst
minute would be $0.90. Then the total cost of the
other minutes would be $6.50 – $0.90 = $5.60,
and $5.60 ÷ $0.40 would yield 14. In this case,
the time of the call would be 1 + 14 = 15 minutes.
If, however, the cost of each minute after the
ﬁrst minute were $0.15, then the cost of the ﬁrst
minute would be $0.65. Then $6.50 – $0.65 would
be $5.85, and this in turn, when divided by $0.15,
would yield 39 minutes, for a total call length of
40 minutes. More information on the cost of each
minute after the ﬁrst minute is still needed.

29.

MN = 16

(2)

NP = 20

Since it is unclear whether MN or NP is one of
the equal sides, it is not possible to determine the
length of the third side or the perimeter of the
triangle. The perimeter could be either
((2)(16)) + 20 = 52 or ((2)(20)) + 16 = 56.
The correct answer is E;
both statements together are still not sufﬁcient.

40 percent of the retailers surveyed owned their
own store.

Using the information from both (1) and (2),
the percent of surveyed owner‐retailers who had
purchased computers can be deduced, and the
percent of nonowner‐retailers can also be deduced.
However, the information that would permit a
determination of either the percent of nonowner‐
retailers who had purchased computers or the overall
percent of all retailers (both owners and nonowners)
who had purchased computers is still not provided.

The perimeter of a triangle is the sum of all three
sides. In the case of an isosceles triangle, two of
the sides are equal. To determine the perimeter
of this triangle, it is necessary to know both the
length of an equal side and the length of the base
of the triangle.

(2) Only gives the length of one side; NOT
sufﬁcient.

(2)

(2) While this permits the percent of owners
and nonowners in the survey to be deduced, the
overall percent of retailers who had purchased
computers cannot be determined; NOT sufﬁcient.

Geometry Triangles

(1) Only gives the length of one side; NOT
sufﬁcient.

85 percent of the retailers surveyed who owned
their own store had purchased computers for
business purposes.

(1) With only this, it cannot be known
what percent of the retailers not owning their
own store had purchased computers, and so it
cannot be known how many retailers purchased
computers overall; NOT sufﬁcient.

What is the perimeter of isosceles triangle MNP ?
(1)

(1)

Arithmetic Percents

The correct answer is E;
both statements together are still not sufﬁcient.
28.

In a survey of retailers, what percent had purchased
computers for business purposes?

The correct answer is E;
both statements together are still not sufﬁcient.
30.

The gift certiﬁcates sold by the store yesterday
were worth a total of between $1,650 and
$1,800.

(2)

Yesterday the store sold more than 15 gift
certiﬁcates worth $100 each.

Algebra Applied problems; Simultaneous
equations; Inequalities

Let x represent the number of $100 certiﬁcates
sold, and let y represent the number of $10
certiﬁcates sold. Then the given information can
be expressed as x + y = 20 or thus y = 20 – x. The
52

3.5 Diagnostic Test Quantitative Answer Explanations

value of the $100 certiﬁcates sold is 100x, and the
value of the $10 certiﬁcates sold is 10y.

(2) For each measurement, the difference
between the mean and that measurement is 2.
Therefore, the square of each difference is 4, and
the sum of all the squares is 4 × 20 = 80. The
standard deviation is 80 = 4 = 2, which is less
20
than 3; SUFFICIENT.

(1) From this, it is known that 100x + 10y >1,650.
Since y = 20 − x, this value can be substituted for y,
and the inequality can be solved for x:
100 x + 10 y > 1, 650
100 x + 10(20 − x ) > 1, 650
substitute for y
100x + 200 − 10x > 1,650
distribute
90x + 200 > 1,650
simplify
90x > 1,450 subtract 200 from
both sides
x > 16.1
Thus, more than 16 of the $100 certiﬁcates were
sold. If 17 $100 certiﬁcates were sold, then it
must be that 3 $10 certiﬁcates were also sold for
a total of $1,730, which satisﬁes the condition of
being between $1,650 and $1,800. If, however,
18 $100 certiﬁcates were sold, then it must
be that 2 $10 certiﬁcates were sold, and this
totals $1,820, which is more than $1,800 and
fails to satisfy the condition. Therefore, 3 of the
$10 certiﬁcates were sold; SUFFICIENT.
(2) From this it can be known only that the
number of $10 certiﬁcates sold was 4 or fewer;
NOT sufﬁcient.
The correct answer is A;
statement 1 alone is sufﬁcient.
31.

Is the standard deviation of the set of measurements
x1, x2, x3, x4, …, x20 less than 3 ?
(1)

The variance for the set of measurements is 4.

(2)

For each measurement, the difference between
the mean and that measurement is 2.

Arithmetic Statistics

In determining the standard deviation, the
difference between each measurement and
the mean is squared, and then the squared
differences are added and divided by the number
of measurements. The quotient is the variance
and the positive square root of the variance is the
standard deviation.
(1) If the variance is 4, then the standard
deviation = 4 = 2, which is less than 3;
SUFFICIENT.

The correct answer is D;
each statement alone is sufﬁcient.
32.

Is the range of the integers 6, 3, y, 4, 5, and x greater
than 9 ?
(1)

y > 3x

(2)

y>x>3

Arithmetic Statistics

The range of a set of integers is equal to the
difference between the largest integer and the
smallest integer. The range of the set of integers
3, 4, 5, and 6 is 3, which is derived from 6 – 3.
(1) Although it is known that y > 3x, the value
of x is unknown. If, for example, x = 1, then the
value of y would be greater than 3. However, if
x = 2, then the value of y would be greater than
6, and, since 6 would no longer be the largest
integer, the range would be affected. Because the
actual values of x and y are unknown, the value of
the range is also unknown; NOT sufﬁcient.
(2) If x > 3 and y > x, then x could be 4 and
y could be 5. Then the range of the 6 integers
would still be 6 – 3 or 3. However, if x were 4 and
y were 15, then the range of the 6 integers would
be 15 – 3, or 12. There is no means to establish
the values of x and y, beyond the fact that they
both are greater than 3; NOT sufﬁcient.
Taking (1) and (2) together, it is known that
x > 3 and that y > 3x. Since the smallest integer
that x could be is thus 4, then y > 3(4) or y > 12.
Therefore, the integer y must be 13 or larger.
When y is equal to 13, the range of the 6 integers
is 13 – 3 = 10, which is larger than 9. As y
increases in value, the value of the range will also
increase.
The correct answer is C;
both statements together are sufﬁcient.

GMAT® Official Guide 2018

33.

It is given that 20 percent of the surveyed
companies fell into category 1. It is necessary
to determine what percent of the surveyed
companies fell into category 4.

5x +2
< 1?
25

(1)

5x < 1

(2)

x <0

(1) This helps identify the percentage in
1
category 2. Since the companies that required
2
computer skills also required writing skills (i.e.,
those in category 1), then the other 1 of the
2
companies that required computer skills did not
require writing skills (thus category 2 = category
1). However, this information only establishes
that 20 percent required computer skills, but not
writing skills; NOT sufficient.

Algebra Inequalities

Note that x r +s = ( x r )( x s )
5x + 2 < 1
(1) If 5x < 1, then
since
25
5x + 2 = 5x ⋅ 5 2 = 5x ; SUFFICIENT.
25
25
(2) If x < 0 , then
x+2<2

x+2

5 <1
25

add 2 to both sides

because a < b implies 5a < 5b

(2) While this establishes category 3, that
is, that 45 percent required writing skills but
not computer skills, no further information is
available; NOT sufficient.

divide both sides by 52 = 25;

SUFFICIENT.

Taking (1) and (2) together, the first three
categories add up to 85 percent (20 + 20 + 45).
Therefore, category 4 would be equal to
100 – 85 = 15 percent of the surveyed companies
required neither computer skills nor writing skills.

The correct answer is D;
each statement alone is sufficient.
34.

Of those companies surveyed that required
computer skills, half required writing skills.

(2)

45 percent of the companies surveyed required
writing skills but not computer skills.

Arithmetic Percents

The surveyed companies could be placed into one
of the following four categories:
1. Requiring computer skills and requiring
writing skills
2. Requiring computer skills but not requiring
writing skills
3. Not requiring computer skills but requiring
writing skills
4. Not requiring either computer skills or writing
skills

The correct answer is C;
both statements together are sufficient.
35.

What is the value of w + q ?
(1)

3w = 3 − 3q

(2)

5w + 5q = 5

Algebra First- and second-degree equations

(1) If 3q is added to both sides of this equation,
it can be rewritten as 3w + 3q = 3. When each
term is then divided by 3, it yields w + q = 1;
SUFFICIENT.
(2) When each term in this equation is divided
by 5, it becomes w + q = 1; SUFFICIENT.
The correct answer is D;
each statement alone is sufficient.

3.5 Diagnostic Test Quantitative Answer Explanations

36.

If X and Y are points in a plane and X lies inside the
circle C with center O and radius 2, does Y lie inside
circle C ?
(1)

The length of line segment XY is 3.

(2)

The length of line segment OY is 1.5.

38.

Geometry Circles

(1) The maximum distance between two points
that lie on a circle is equal to the diameter, or
2 times the radius. Since the radius of circle C
is 2, the diameter in this case is 4. It cannot be
assumed, however, that X and Y are points on the
diameter; X can lie anywhere within the circle.
When the distance between X and Y is 3, it is still
possible either that Y is within the circle or that Y
is outside the circle; NOT sufﬁcient.

(2)

x
= y −1
2

Algebra First- and second-degree equations

(1) Since 2 has to be added to y in order to
make it equal to x, it can be reasoned that x > y;
SUFFICIENT.

(2)

The distance that Paula drove from her home to
her college was less than 205 kilometers.

(1) If D is the distance Paula drove then
D > 200 and D > 200 = 2 6 so t > 2 6 and
7
7
70 70
t may or may not be less than 3; NOT sufficient.
(2)

If D is the distance Paula drove then
D < 205 and D < 205 = 2 13 so t < 2 13 < 3;
14
14
70 70
SUFFICIENT.

Is x > y ?
x = y +2

The distance that Paula drove from her home to
her college was greater than 200 kilometers.

A distance problem uses the formula distance =
rate × time. To ﬁnd the time, the formula would
be rearranged as time = distance . To solve this
rate
problem, it is necessary to know the rate (given
here as 70 kilometers per hour) and the distance.

The correct answer is B;
statement 2 alone is sufﬁcient.

(1)

Arithmetic Distance problem

(2) If the length of the line segment OY is 1.5
and the circle has a radius of 2, then the distance
from the center O to point Y is less than the
radius, and point Y must therefore lie within the
circle; SUFFICIENT.

37.

If Paula drove the distance from her home to her
college at an average speed that was greater than
70 kilometers per hour, did it take her less than
3 hours to drive this distance?

The correct answer is B;
statement 2 alone is sufﬁcient.
39.

In the xy‐plane, if line k has negative slope and passes
through the point (–5,r), is the x‐intercept of line k
positive?
(1)

The slope of line k is –5.

(2)

r>0

(2) Multiplying both sides of this equation by
2 results in x = 2(y – 1) or x = 2 y – 2. If y were 0,
then x would be – 2, and y would be greater than
x. If y were a negative number like –2, then
x = 2(–2) – 2 = 6, and again y would be greater
than x. However, if y were a positive number such
as 4, then x = 2(4) –2 = 6, and x > y. Since there is
no other information concerning the value of y, it
cannot be determined if x > y; NOT sufﬁcient.
The correct answer is A;
statement 1 alone is sufﬁcient.

GMAT® Official Guide 2018

(2) If k = 8, then the rate is 8 percent, and
the same formula and procedure as above are
employed again; SUFFICIENT.

Geometry Coordinate geometry

The x‐intercept is the x‐coordinate of the point
in which the line k crosses the x‐axis and would
have the coordinates (x,0).
(1) Knowing the slope of the line does not
help in determining the x‐intercept, since from
point (–5,r) the line k extends in both directions.
Without knowing the value of r, the x‐intercept
could be –5 if r were 0, or it could be other
numbers, both positive and negative, depending
on the value of r; NOT sufﬁcient.
(2) Knowing that r > 0 suggests that the
x‐intercept is not –5; the point (–5,r), where r
is a positive number, does lie in quadrant II. It
could, however, be any point with an x‐coordinate
of –5 in that quadrant and line k could have any
negative slope, and so the line k would vary with
the value of r. Therefore, the x‐intercept of line k
cannot be determined; NOT sufﬁcient.
Using (1) and (2) together does not help in
the determination of the x‐intercept, since the
point (–5,r) could have any positive y‐coordinate
and thus line k could cross the x‐axis at many
different places.
The correct answer is E;
both statements together are still not sufﬁcient.
40.

If $5,000 invested for one year at p percent simple
annual interest yields $500, what amount must be
invested at k percent simple annual interest for one
year to yield the same number of dollars?
(1)

k = 0.8p

(2)

k=8

Arithmetic Interest problem

With simple annual interest, the formula to use is
interest = principal × rate × time. It is given
p
that $500 = $5, 000 ×
× 1 (year), so
100
p = 10 percent interest.
(1) If p is 10 percent, then k = 0.8 p is 0.08.
Using the same formula, the time is again 1 year;
the interest is the same amount; and the rate is
0.08, or 8 percent. Thus, $500 = principal × 0.08 × 1,
or principal $6,250; SUFFICIENT.
56

The correct answer is D;
each statement alone is sufﬁcient.
41.

x+y
> 0, is x < 0 ?
z

(1)

x 0, then either one of two cases holds
z
true. Either ( x + y ) > 0 and z > 0, or ( x + y ) < 0
and z < 0. In other words, in order for the term to
be greater than zero, it must be true that either 1)
both the numerator and denominator are greater
than 0 or 2) both the numerator and denominator
are less than 0.

(1) Regardless of whether ( x + y ) is positive or
negative, the positive or negative value of z must
be in agreement with the sign of ( x + y ) in order
x+ y
for
> 0. However, there is no information
z
about z here; NOT sufﬁcient.
(2) If z < 0, then ( x + y ) must be less than 0.
However, this statement gives no information
about ( x + y ); NOT sufﬁcient.
This can be solved using (1) and (2) together.
From (2), it is known that z < 0, and, going back
to the original analysis, for the term to be greater
than zero, ( x + y ) must also be less than 0.
If x + y < 0 then x < –y. But x < y from (1) so
x+x < −y + y
2x < 0

x < 0.

The correct answer is C;
both statements together are sufﬁcient.

3.5 Diagnostic Test Quantitative Answer Explanations

42.

(1) These data will produce an average of
2160 = 72°
for last April in City X. However,
30
there is no information regarding the median for
comparison; NOT sufﬁcient.

Does the integer k have at least three different positive
prime factors?
(1)

k
is an integer.
15

(2)

k
is an integer.
10

(2) The median is the middle temperature of
the data. As such, 50 percent of the daily high
temperatures will be at or above the median, and
50 percent will be at or below the median. If
60 percent of the daily high temperatures were
less than the average daily high temperature, then
the average of the daily highs must be greater
than the median; SUFFICIENT.

Arithmetic Properties of numbers

(1) The prime factors of 15 are 3 and 5. So in
this case, k has at least 2 different positive prime
factors, but it is unknown if there are more
positive prime factors; NOT sufﬁcient.
(2) The prime factors of 10 are 2 and 5, showing
that k has at least these 2 different positive prime
factors, but k might or might not have more;
NOT sufﬁcient.
Taking (1) and (2) together, since k is divisible
by both 10 and 15, it must be divisible by their
different positive prime factors of 2, 3, and 5.
Thus k has at least 3 different positive prime
factors.
The correct answer is C;
both statements together are sufﬁcient.
43.

In City X last April, was the average (arithmetic mean)
daily high temperature greater than the median daily
high temperature?
(1)

In City X last April, the sum of the 30 daily high
temperatures was 2,160°.

(2)

In City X last April, 60 percent of the daily high
temperatures were less than the average daily
high temperature.

Arithmetic Statistics

The formula for calculating the arithmetic mean,
or the average, is as follows:

The correct answer is B;
statement 2 alone is sufﬁcient.
44.

( m)

If m and n are positive integers, is
(1)
(2)

( m ) is an integer.
( n ) is an integer.

an integer?

Arithmetic Properties of numbers

( )
( )

(1) If m is an integer and n is a positive
n
integer, then m is an integer because an
integer raised to a positive integer is an integer;
SUFFICIENT.

( )

(2) The information that n is an integer
is not helpful in answering the question. For
example, if m = 2 and n = 9, 9 = 3, which is
9
an integer, but 2 = 16 2 , which is not
an integer. But if m = 4 and n = 9, then 9 = 3,

( ) (

)

( )

which is an integer, and 4
integer; NOT sufficient.

= 29 = 512 is an

The correct answer is A;
statement 1 alone is sufﬁcient.

Average = sum of v values
v

GMAT® Official Guide 2018

45.

Of the 66 people in a certain auditorium, at most 6
people have birthdays in any one given month. Does at
least one person in the auditorium have a birthday in
January?
(1)

More of the people in the auditorium have
birthdays in February than in March.

(2)

Five of the people in the auditorium have
birthdays in March.

Arithmetic Statistics

(1) Since all 10 employees did not receive the
same 15 percent increase, it cannot be assumed
that the mean this year is 15 percent higher than
last year. It remains unknown whether these
8 salaries were the top 8 salaries, the bottom
8 salaries, or somewhere in‐between. Without
this type of information from last year, the
mean for this year cannot be determined; NOT
sufﬁcient.

Algebra Sets and functions

(2) If 2 salaries remained the same as last
year, then 8 salaries changed. Without further
information about the changes, the mean for this
year cannot be determined; NOT sufﬁcient.

Because it is given that 6 is the greatest number
of individuals who can have birthdays in any
particular month, these 66 people could be evenly
distributed across 11 of the 12 months of the
year. That is to say, it could be possible for the
distribution to be 11 × 6 = 66, and thus any given
month, such as January, would not have a person
with a birthday. Assume that January has no
people with birthdays, and see if this assumption
is disproved.
(1) The information that more people have
February birthdays than March birthdays
indicates that the distribution is not even.
Therefore, March is underrepresented and must
thus have fewer than 6 birthdays. Since no month
can have more than 6 people with birthdays, and
every month but January already has as many
people with birthdays as it can have, January
has to have at least 1 person with a birthday;
SUFFICIENT.
(2) Again, March is underrepresented with only
5 birthdays, and none of the other months can
have more than 6 birthdays. Therefore, the extra
birthday (from March) must occur in January;
SUFFICIENT.
The correct answer is D;
each statement alone is sufﬁcient.
46.

Last year the average (arithmetic mean) salary of the
10 employees of Company X was $42,800. What is
the average salary of the same 10 employees this
year?
(1)

For 8 of the 10 employees, this year’s salary is
15 percent greater than last year’s salary.

(2)

For 2 of the 10 employees, this year’s salary is
the same as last year’s salary.

Even taking (1) and (2) together, it remains
impossible to tell the mean salary for this year
without additional data.
The correct answer is E;
both statements together are still not sufﬁcient.
47.

In a certain classroom, there are 80 books, of which
24 are ﬁction and 23 are written in Spanish. How many
of the ﬁction books are written in Spanish?
(1)

Of the ﬁction books, there are 6 more that are
not written in Spanish than are written in Spanish.

(2)

Of the books written in Spanish, there are 5
more nonﬁction books than ﬁction books.

Algebra Sets and functions

Let x represent the ﬁction books that are written
in Spanish. A table could be set up like the one
below, ﬁlling in the information that is known or
able to be known:
Spanish
Fiction

Non‐Spanish

Nonfiction
Total

Total
56

(1) If x represents the ﬁction books written in
Spanish, then x + 6 can now be used to represent
the ﬁction books that are not written in Spanish.
From the table above, it can be seen then that
x + x + 6 = 24, or 2x = 18. Therefore, x, or the
number of ﬁction books written in Spanish, is 9;
SUFFICIENT.

3.5 Diagnostic Test Quantitative Answer Explanations

(2) If x represents the ﬁction books written in
Spanish, then x + 5 can now be used to represent
the nonﬁction books written in Spanish. From
the table, it can be said that x + x + 5 = 23, or
2x = 18. Therefore, x, or the number of ﬁction
books written in Spanish, is 9; SUFFICIENT.
The correct answer is D;
each statement alone is sufﬁcient.
48.

If p is the perimeter of rectangle Q, what is the value of
p?
(1)

Each diagonal of rectangle Q has length 10.

(2)

The area of rectangle Q is 48.

Geometry Rectangles; Perimeter; Area

The perimeter of a rectangle is equal to 2 times
the rectangle’s length plus 2 times the rectangle’s
width, or p = 2l + 2w. The diagonals of a rectangle
are equal. In a rectangle, because a diagonal forms
a right triangle, the length of a diagonal is equal
to the square root of the length squared plus the
width squared, or d = l 2 + w 2.

Using (1) and (2) together, it is possible to solve
this problem. Since from (2) lw = 48, then
w = 48 . Substituting this into 100 = l 2 + w 2
l
from (1) the equation can be solved as follows:
100 = l 2 +  48 
 l 

substitution

100l 2 = l 4 + 2, 304

multiply both sides by l2

factor like a quadratic

l 4 − 100l 2 + 2, 304 = 0 move all terms to one
side
2

− 64 )( l 2 − 36 ) = 0
2

l = 64, l = 36

solve for l2

Since l is a length, it must be positive, so l is
48 = 6
either 8 or 6. When l = 8,w =
, and when
8
48
l = 6,w =
= 8, both of which give the same
6
perimeter.
The correct answer is C;
both statements together are sufﬁcient.

(1) If a diagonal = 10, then 10 = l 2 + w 2 , or,
by squaring both sides, 100 = l 2 + w 2 . Without
knowing the value or the relationship between
the other two sides of the right triangle, it is
impossible to solve for l or w, and thus for the
perimeter of the rectangle; NOT sufﬁcient.
(2) If the area of the rectangle is 48, then it can
be stated that lw = 48. However, without further
information, the perimeter cannot be determined.
For example, l could be 6 and w could be 8,
and the perimeter would then be 12 + 16 = 28.
However, it could also be that l is 4 and w is
12, and in that case the perimeter would be
8 + 24 = 32; NOT sufﬁcient.

GMAT® Official Guide 2018

3.6 Verbal Answer Explanations
Reading Comprehension
The following discussion is intended to familiarize you with the most efficient and effective approaches
to the kinds of problems common to reading comprehension. The particular questions in this chapter
are generally representative of the kinds of reading comprehension questions you will encounter on the
GMAT exam. Remember that it is the problem solving strategy that is important, not the specific details
of a particular question.

A The sentence does not explain a failure of
the economic signaling theory.

Questions 1–5 refer to the passage on page 23.
49.

Which of the following best describes the purpose of
the sentence in lines 10–15 ?
(A)

To show that economic signaling theory fails to
explain a ﬁnding

B Economic signaling theory is about
perceptions of quality, but this explanation is
about actual quality and its correlation with
advertising.

(B)

To introduce a distinction not accounted for by
economic signaling theory

(C)

To account for an exception to a generalization
suggested by Marquardt and McGann

C No exception is mentioned in Marquardt
and McGann’s work.

(D)

To explain why Marquardt and McGann’s research
was conducted

(E)

To offer an explanation for an observation
reported by Marquardt and McGann

D The sentence does not examine why or how
the research was undertaken.
E

The correct answer is E.

Logical structure

Marquardt and McGann found a correlation
between highly advertised products and high‐
quality products. The connection can be explained
by understanding that companies may invest
heavily in such advertising, anticipating that
recurring purchases of high‐quality products will
eventually recover these advertising costs. The
consumers will continue to buy these products
over time because of loyalty to their high quality.
The statement in bold provides this explanation
for the correlation noted by Marquardt and
McGann.

C
orrect. This statement provides an
explanation of why highly advertised
products did indeed rank high on certain
measures of product quality.

50.

The primary purpose of the passage is to
(A)

present ﬁndings that contradict one explanation
for the effects of a particular advertising practice

(B)

argue that theoretical explanations about the
effects of a particular advertising practice are of
limited value without empirical evidence

(C)

discuss how and why particular advertising
practices may affect consumers’ perceptions

(D)

contrast the research methods used in two
different studies of a particular advertising
practice

(E)

explain why a ﬁnding about consumer responses
to a particular advertising practice was
unexpected

3.6 Diagnostic Test Verbal Answer Explanations

Main idea

The primary purpose can be determined only by
evaluating the whole passage. The ﬁrst paragraph
discusses consumers’ perceptions of quality
based on frequency of advertising. The second
paragraph discusses three studies that show how
consumers base their evaluations of products on
the kinds of advertising they see. Therefore, the
purpose of the whole passage is to show how
consumers’ perceptions of products are shaped by
certain advertising practices.

51.

Kirmani’s research, as described in the passage,
suggests which of the following regarding consumers’
expectations about the quality of advertised products?
(A)

Those expectations are likely to be highest if a
manufacturer runs both black‐and‐white and color
advertisements for the same product.

(B)

Those expectations can be shaped by the
presence of color in an advertisement as well as
by the frequency with which an advertisement
appears.

(C)

Those expectations are usually high for
frequently advertised new brands but not for
frequently advertised familiar brands.

(D)

Those expectations are likely to be higher for
products whose black‐and‐white advertisements
are often repeated than for those whose color
advertisements are less often repeated.

C Correct. The passage shows how the
frequency and the kind of advertising
inﬂuence consumers’ perceptions about the
quality of the products advertised.

(E)

Those expectations are less deﬁnitively shaped
by the manufacturer’s advertisements than by
information that consumers gather from other
sources.

D The passage reports the ﬁndings of four
studies but does not mention research
methods.

Inference

A The passage shows that expensive
advertising works to a certain point, but not
after it; this method examines a continuum,
not a contradiction.
B Most of the passage is devoted to empirical
evidence.

E The passage does not indicate that any of
the ﬁndings were unexpected.
The correct answer is C.

The question’s use of the word suggests means that
the answer depends on making an inference. This
research is discussed in the second paragraph.
Kirmani found that too much advertising
tended to make the consumers believe that
manufacturers were desperate. The use of color
was also found to affect consumers’ perceptions of
brand quality. Realizing that color advertising is
more expensive than black‐and‐white, consumers
react more quickly to what they perceive to be its
overuse than they do to a repetition of black‐and‐
white advertisements.

GMAT® Official Guide 2018

A This situation is not discussed in the
research, at least as it is reported in this
passage.
B Correct. It can be inferred that consumers’
perceptions of product quality are inﬂuenced
by the use of color in an advertisement and
by the frequency of the advertisement’s
appearance.
C The research does not make a distinction
between new and familiar brands.
D The research indicates only that consumers
can tolerate black‐and‐white advertisements
for a longer time than color advertisements
before dismissing them as excessive.
E There is no discussion of what consumers
learn from other sources.
The correct answer is B.
52.

Kirmani’s third study, as described in the passage,
suggests which of the following conclusions about a
black‐and‐white advertisement?
(A)

(B)

It can be repeated more frequently than a
comparable color advertisement could before
consumers begin to suspect low manufacturer
conﬁdence in the quality of the advertised
product.
It will have the greatest impact on consumers’
perceptions of the quality of the advertised
product if it appears during periods when a color
version of the same advertisement is also being
used.

(C)

It will attract more attention from readers of the
print publication in which it appears if it is used
only a few times.

(D)

It may be perceived by some consumers as
more expensive than a comparable color
advertisement.

(E)

It is likely to be perceived by consumers as
a sign of higher manufacturer conﬁdence in
the quality of the advertised product than a
comparable color advertisement would be.

Inference

Kirmani’s third study is discussed in the ﬁnal
two sentences. Consumers suspect expensive
advertising results from a manufacturer’s lack
of conﬁdence in the quality of the product.
Consumers reach the point at which they
ﬁnd advertising excessive more quickly with
color advertising than with black‐and‐white
advertising because they understand that the
addition of color increases advertising expenses. It
is reasonable to infer that the reverse is also true
and thus that consumers will tolerate lengthier
repetitions of black‐and‐white advertising
without becoming suspicious of product quality.
Correct. Consumers find color advertising
A
excessive more quickly and thus can
be expected to find black‐and‐white
advertising excessive less quickly.
B The study does not discuss concurrent
appearances of color and black‐and‐white
advertisements for the same product.
C The sole conclusion about frequency is that
consumers can tolerate a greater frequency
of black‐and‐white advertisements than
color advertisements.
D It is stated that consumers understand that
color advertisements are more expensive.
E The research certainly does not report this
finding.
The correct answer is A.

3.6 Diagnostic Test Verbal Answer Explanations

53.

A Kirmani’s research, as reported here, does
not support this claim.

The passage suggests that Kirmani would be most
likely to agree with which of the following statements
about consumers’ perceptions of the relationship
between the frequency with which a product is
advertised and the product’s quality?
(A)

B Kirmani’s research examines how consumers
respond to the frequency of advertising; the
research does not indicate that consumers
do not notice frequency.

Consumers’ perceptions about the frequency
with which an advertisement appears are their
primary consideration when evaluating an
advertisement’s claims about product quality.

(B)

Because most consumers do not notice the
frequency of advertisement, it has little impact
on most consumers’ expectations regarding
product quality.

(C)

Consumers perceive frequency of advertisement
as a signal about product quality only when the
advertisement is for a product that is newly on
the market.

(D)

The frequency of advertisement is not always
perceived by consumers to indicate that
manufacturers are highly conﬁdent about their
products’ quality.

(E)

Consumers who try a new product that has been
frequently advertised are likely to perceive the
advertisement’s frequency as having been an
accurate indicator of the product’s quality.

C The research does not distinguish between
new and familiar products.
Correct. Excessive advertising may lead
D
consumers to believe that the manufacturer
lacks conﬁdence in the quality of the
product.
E Kirmani’s research does not speciﬁcally
address new products.
The correct answer is D.
Questions 6–11 refer to the passage on
page 25.
54.

The main purpose of the passage is to
(A)

propose an experiment

(B)

analyze a function

(C)

refute an argument

Inference

(D)

explain a contradiction

The ﬁrst sentence of the second paragraph
provides the answer to this question: at some level
of spending the manufacturer’s advertising effort
may be perceived as unreasonably high, implying low
manufacturer conﬁdence in product quality. Thus, it
is logical to assume that if a product is advertised
too frequently, consumers may believe that the
manufacturer is spending excessive amounts on
advertising because that manufacturer is not
conﬁdent of the product’s quality.

(E)

simulate a process

Main idea

Determining the main purpose comes from
considering the passage as a whole. The ﬁrst
paragraph begins by noting that the idea of the
brain as an information processor is generally
accepted by neuroscientists. The author then
presents Searle as an enemy of this position and
explains Searle’s belief that human thought is
more than information processing. The second
paragraph questions Searle’s position, and the
third asserts that the brain is an information
processor, refuting Searle’s argument.

GMAT® Official Guide 2018

A The author uses the idea of a mechanical
simulation of a stomach as a metaphor for a
computer’s simulation of thought; this is not
a proposal for an experiment.

A
Correct. Searle believes that meaning and
content cannot be reduced to algorithms.
B The author argues for the mechanical
simulation, but offers no evidence that
Searle would agree.

B The author analyzes Searle’s position, but no
function is analyzed.

C This statement reﬂects the author’s position,
but it is the opposite of Searle’s.

Correct. The author explains Searle’s
C
argument in order to refute it.

D Searle asserts that only people, not
computers, have the causal powers of the
brain.

D The author points out a weakness in Searle’s
thinking, but not a contradiction.
E The simulation of a process is included as
a metaphor, but it is not essential to the
passage.
The correct answer is C.
55.

The correct answer is A.

Which of the following is most consistent with Searle’s
reasoning as presented in the passage?

The author of the passage would be most likely to
agree with which of the following statements about the
simulation of organ functions?

(A)

Meaning and content cannot be reduced to
algorithms.

(A)

(B)

The process of digestion can be simulated
mechanically, but not on a computer.

An artiﬁcial device that achieves the functions of
the stomach could be considered a valid model
of the stomach.

(B)

Simulated thoughts and real thoughts are
essentially similar because they are composed
primarily of information.

Computer simulations of the brain are best
used to crack the brain’s codes of meaning and
content.

(C)

A computer can use “causal powers” similar
to those of the human brain when processing
information.

Computer simulations of the brain challenge
ideas that are fundamental to psychology and
neuroscience.

(D)

Computer simulations of the world can achieve
the complexity of the brain’s representations of
the world.

Because the brain and the stomach both act
as processors, they can best be simulated by
mechanical devices.

(E)

The computer’s limitations in simulating digestion
suggest equal limitations in computer‐simulated
thinking.

(C)

(D)

(E)

Evaluation

Searle’s position is stated in the ﬁrst paragraph:
because computers merely follow algorithms, they
cannot deal with important aspects of human thought
such as meaning and content. Thus, Searle believes
that meaning and content cannot be reduced to
algorithms.

E The passage does not discuss computer
simulations of the world.

56.

Application

To answer this question, think about how
the author would respond to each statement.
Anticipating the author’s response depends on
understanding the author’s point of view. In this
passage, the author is arguing against Searle’s
view of the brain and in favor of the brain as
information processor. The author believes that
the computer can be a model of the brain and
uses the example of the mechanical stomach to
support his position on simulations.

3.6 Diagnostic Test Verbal Answer Explanations

A
Correct. The ﬁrst two sentences of the third
paragraph imply that a mechanical device is
a valid model.
B The author believes a computer can simulate
the brain but does not comment on how
these simulations should be used. There is
no way to predict the author’s reaction to
this statement.
C The author would reject this statement since
neuroscience and psychology do in fact see
the brain as an information processor.
D The author agrees that both the brain and
the stomach act as processors; believes that
the computer, a nonmechanical device,
can simulate the brain; and offers a way
that a mechanical device could simulate
the stomach. The author does not suggest
that mechanical devices are the best way to
simulate both their processes.
E This statement reﬂects Searle’s viewpoint,
which the author rejects.
The correct answer is A.
57.

It can be inferred that the author of the passage
believes that Searle’s argument is ﬂawed by its
failure to
(A)

distinguish between syntactic and semantic
operations

(B)

explain adequately how people, unlike
computers, are able to understand meaning

(C)

provide concrete examples illustrating its claims
about thinking

(D)

understand how computers use algorithms to
process information

(E)

decipher the code that is transmitted from
neuron to neuron in the brain

Inference

The author’s attitude toward Searle’s argument is
apparent in the ﬁrst paragraph, which ends with
the author’s summary of what Searle is saying.
Computers understand structures, Searle argues,
but only people understand meaning. How do
people understand meaning? The author notes
that Searle is not able to answer this question
and is able only to assert that people have causal
powers of the brain.
A The author makes it clear in the first
paragraph that Searle does distinguish
between the two. In Searle’s view computers
are syntactic, interpreting structure or
arrangement, rather than semantic,
understanding meaning.
B
Correct. The first paragraph ends with the
contrast between people and computers:
People, on the other hand, understand meaning
because they have something Searle obscurely
calls the causal powers of the brain. By calling
Searle’s explanation obscure, the author
implies that Searle has not adequately
clarified how people understand meaning.
C Nothing in the passage criticizes Searle for
not providing concrete examples. Indeed, in
the second paragraph, the author anticipates
how Searle would react to one concrete
example, the computer simulation of the
stomach.
D In the first paragraph, the author says
that Searle argues that computers simply
follow algorithms; whether or not Searle
understands how they use algorithms is
irrelevant.
E Since, as the author suggests in the
first paragraph, Searle does not believe
information could be a code transmitted
from neuron to neuron, he cannot be
expected to decipher that code.
The correct answer is B.

GMAT® Official Guide 2018

58.

From the passage, it can be inferred that the author
would agree with Searle on which of the following
points?

59.

Which of the following most accurately represents
Searle’s criticism of the brain‐as‐computer metaphor,
as that criticism is described in the passage?

(A)

Computers operate by following algorithms.

(A)

The metaphor is not experimentally veriﬁable.

(B)

The human brain can never fully understand its
own functions.

(B)

The metaphor does not take into account the
unique powers of the brain.

(C)

The comparison of the brain to a machine is
overly simplistic.

(C)

(D)

The most accurate models of physical processes
are computer simulations.

The metaphor suggests that a brain’s functions
can be simulated as easily as those of a
stomach.

(D)

The metaphor suggests that a computer can
simulate the workings of the mind by using the
codes of neural transmission.

(E)

The metaphor is unhelpful because both the
brain and the computer process information.

(E)

Human thought and computer‐simulated thought
involve similar processes of representation.

Inference

An inference requires going beyond the material
explicitly stated in the passage to the author’s
ideas that underlie that material. The author and
Searle take opposite points of view on the brain
as information processor. Their area of agreement
is narrow. However, they do both agree that
computers work by following algorithms.
Correct. The ﬁrst paragraph explains that
A
Searle dismisses computers because they
simply follow algorithms; while the author
disagrees with Searle on virtually every
other point, no disagreement is voiced here.

Inference

Searle’s criticism of the brain‐as‐computer
metaphor is discussed in the ﬁrst paragraph.
Computers are merely machines; only people are
endowed with causal powers of the brain that allow
them to understand meaning and content.
A Searle does not believe in the value of the
metaphor, so its veriﬁcation is beside the
point.

B The ﬁrst paragraph shows this to be Searle’s
position, but not the author’s.

B
Correct. Searle believes that people have
something computers do not, causal powers
of the brain for understanding important
aspects of human thought.

C The ﬁrst paragraph shows this to be Searle’s
position, but not the author’s.

C Comparing the brain to a computer, the
metaphor does not make this suggestion.

D The second paragraph explains Searle’s
rejection of this position.

D In the second paragraph, the author says, but
even if a computer could simulate the workings
of the mind, making it clear that presently it
cannot; this statement does not reﬂect why
Searle rejects the metaphor.

E The ﬁnal paragraph establishes this as the
author’s position, but not Searle’s.
The correct answer is A.

E This is not the basis of Searle’s objection
since he does not accept the premise that
the brain is an information processor.
The correct answer is B.

3.6 Diagnostic Test Verbal Answer Explanations

60.

Questions 12–17 refer to the passage on
page 27.

A Lines 6–10 emphasize the victories of the
reformers.

The primary purpose of the passage is to

B The passage discusses the effects, rather than
the origins, of child labor laws.

(A)

explain why women reformers of the Progressive
Era failed to achieve their goals

(B)

discuss the origins of child labor laws in the late
nineteenth and early twentieth centuries

(C)

compare the living conditions of working‐class
and middle‐class women in the Progressive Era

(D)

discuss an oversight on the part of women
reformers of the Progressive Era

(E)

revise a traditional view of the role played by
women reformers in enacting Progressive Era
reforms

Main idea

Understanding the author’s purpose comes only
from reﬂecting on the passage as a whole. The
beginning of the passage notes the success of
middle‐class women reformers in improving
working conditions for women and children.
The middle discusses the position of working‐
class mothers, who were more concerned with
the economic survival of their families than
with labor reform and consequently tried to
circumvent the laws. The close of the passage
observes that, although middle‐class reformers
were right to point out exploitation of children,
they failed to understand the economic plight of
working‐class families, who needed the income
earned by every possible member. The purpose of
this passage is to show the failure of middle‐class
reformers to understand the economic position of
working‐class families.

C Living conditions of middle‐class and
working‐class women are not compared.
D
Correct. As is made clear, especially in
the ﬁnal sentence of the passage, women
reformers failed to understand the economic
needs of working‐class families.
E A traditional view is not compared with a
newer, revised view of the reformers.
The correct answer is D.
61.

The view mentioned in line 17 of the passage refers to
which of the following?
(A)

Some working‐class mothers’ resistance to the
enforcement of child labor laws

(B)

Reformers’ belief that child labor and industrial
home work should be abolished

(C)

Reformers’ opinions about how working‐class
families raised their children

(D)

Certain women historians’ observation that there
was a lack of consensus between women of
different classes on the issue of child labor and
industrial home work

(E)

Working‐class families’ fears about the adverse
consequences that child labor laws would have
on their ability to earn an adequate living

Inference

To ﬁnd what this appearance of view refers to, it
is necessary to look back to the beginning of the
sentence. This view, not shared by working‐class
mothers, refers to the reformers’ conviction that
child labor and industrial home work were equally
inhumane practices that should be outlawed.

GMAT® Official Guide 2018

A
This view must refer back to a point already
stated; resistance to child labor laws is not
discussed until the following sentence.

A
Correct. The author uses the recent work of
women historians to support the statement
that women of different social classes were
pitted against one another.

B
Correct. This view refers to the position
of reformers stated earlier in the same
sentence: that child labor and industrial home
work…should be outlawed.

B The women historians have recently observed;
the verb observed introduces a statement
rather than a question.

C
This view must refer back to a point already
stated; the reformers’ belief that resistance to
child labor laws was due to poor parenting
is discussed later in the passage.

C The reference to women historians has to
do with working‐class mothers; it does
not challenge women’s activism and role in
social reform.

D A number of women historians have said
that working‐class mothers did not always
share the view of middle‐class women
reformers about child labor.

D The passage supports what the women
historians say about working‐class mothers.
E The author does not deﬁne or present the
traditional view of child labor reform, nor is
any inherent contradiction pointed out.

E
This view must refer back to a point already
stated; the fears of working‐class families
are examined in the following sentence.
The correct answer is B.
62.

The author of the passage mentions the observations
of women historians (lines 15–17) most probably in
order to

63.

The passage suggests that which of the following was
a reason for the difference of opinion between working‐
class mothers and women reformers on the issue of
child labor?
(A)

(A)

provide support for an assertion made in the
preceding sentence (lines 10–12)

Reformers’ belief that industrial home work was
preferable to child labor outside the home

(B)

raise a question that is answered in the last
sentence of the passage (lines 27–32)

Reformers’ belief that child labor laws should
pertain to working conditions but not to pay

(C)

introduce an opinion that challenges a statement
made in the ﬁrst sentence of the passage

Working‐class mothers’ resentment at reformers’
attempts to interfere with their parenting

(D)

offer an alternative view to the one attributed in
the passage to working‐class mothers

Working‐class mothers’ belief that child labor was
an inhumane practice

(E)

point out a contradiction inherent in the
traditional view of child labor reform as it is
presented in the passage

Working‐class families’ need for every
employable member of their families to earn
money

Evaluation

In lines 10–12, the author asserts that child labor
laws pitted women of different classes against one
another. The view of the middle‐class women
reformers is stated, and then, to show that
working‐class mothers did not hold the same
opinion, the author turns to the recent work of
women historians to support this statement.

The correct answer is A.

3.6 Diagnostic Test Verbal Answer Explanations

Inference

The question’s use of the word suggests means
that the answer depends on making an inference.
Lines 12–23 examine the different views of
middle‐class reformers and working‐class
mothers on child labor laws. While the reformers
saw child labor as an inhumane practice that
should be outlawed, working class mothers
understood the necessity of pooling the wages of as
many family members as possible and viewed child
labor legislation as a personal economic disaster.
A Lines 12–14 show that reformers regarded
both kinds of work as equally inhumane
practices that should be outlawed.
B Pay is not speciﬁcally discussed in the
passage.
C Lines 24–27 indicate that the reformers
believed working‐class resistance to child
labor laws was a sign of poor parenting,
but nothing is said about the working‐class
response to this view.
D Lines 12–17 say that the reformers held this
position, but working class mothers did not
always share this view.
E
Correct. Lines 17–23 explain that working‐
class families needed the wages of as many
family members as possible.
The correct answer is E.

64.

The author of the passage asserts which of the
following about women reformers who tried to abolish
child labor?
(A)

They alienated working‐class mothers by
attempting to enlist them in agitating for
progressive causes.

(B)

They underestimated the prevalence of child
labor among the working classes.

(C)

They were correct in their conviction that child
labor was deplorable but shortsighted about the
impact of child labor legislation on working‐class
families.

(D)

They were aggressive in their attempts to
enforce child labor legislation, but were
unable to prevent working‐class families from
circumventing them.

(E)

They were prevented by their nearly total
disenfranchisement from making signiﬁcant
progress in child labor reform.

Supporting ideas

This question is based on information explicitly
stated in the ﬁnal sentence of the passage.
Women reformers viewed child labor as a terribly
exploitative practice but they failed to take account of
the economic needs of working‐class families.
A The passage does not say that reformers
tried to enlist working‐class mothers in
progressive causes.
B No evidence is offered to support such a
statement.
C
Correct. The ﬁnal sentence makes clear
that the reformers recognized child labor
as exploitative but did not understand the
economic needs of working‐class families.
D The reformers’ activities involved promoting
legislation; there is no evidence in the
passage that the reformers themselves
attempted to enforce these laws.
E. Lines 6–10 show that the reformers
improved working conditions for
women and children, despite their
disenfranchisement.
The correct answer is C.

GMAT® Official Guide 2018

65.

According to the passage, one of the most striking
achievements of white middle‐class women reformers
during the Progressive Era was
(A)

gaining the right to vote in school elections

(B)

mobilizing working‐class women in the ﬁght
against child labor

(C)

uniting women of different classes in grassroots
activism

(D)

improving the economic conditions of working‐
class families

(E)

improving women’s and children’s working
conditions

Supporting ideas

The question’s use of the phrase according to the
passage indicates that the answer can be found
through careful reading of the passage. This
question is based on information explicitly stated
in lines 7–10, which state that white middle‐class
women reformers won a variety of victories, notably
in the improvement of working conditions, especially
for women and children.
A Lines 6–7 show that women already had the
right to vote in school elections.
B Lines 20–24 show that working‐class
families tried to circumvent child labor laws.
C Lines 11–12 say that one product of
grassroots activism, child labor legislation,
pitted women of different classes against one
another.
D Lines 31–32 say that the reformers failed to
take account of the economic needs of working‐
class families.
E
Correct. The passage states that reformers
improved the working conditions of women
and children.
The correct answer is E.

3.6 Diagnostic Test Verbal Answer Explanations

Critical Reasoning
The following discussion is intended to familiarize you with the most efficient and effective approaches
to critical reasoning questions. The particular questions in this chapter are generally representative of
the kinds of critical reasoning questions you will encounter on the GMAT exam. Remember that it is the
problem solving strategy that is important, not the specific details of a particular question.

66.

Vasquez‐Morrell Assurance specializes in insuring manufacturers. Whenever a policyholder makes a claim, a claims adjuster
determines the amount that Vasquez‐Morrell is obligated to pay. Vasquez‐Morrell is cutting its staff of claims adjusters by
15 percent. To ensure that the company’s ability to handle claims promptly is affected as little as possible by the staff cuts,
consultants recommend that Vasquez‐Morrell lay off those adjusters who now take longest, on average, to complete work
on claims assigned to them.
Which of the following, if true, most seriously calls into question the consultants’ criterion for selecting the staff to be laid off?
(A)

If the time that Vasquez‐Morrell takes to settle claims increases significantly, it could lose business to other insurers.

(B)

Supervisors at Vasquez‐Morrell tend to assign the most complex claims to the most capable adjusters.

(C)

At Vasquez‐Morrell, no insurance payments are made until a claims adjuster has reached a final determination on the claim.

(D)

There are no positions at Vasquez‐Morrell to which staff currently employed as claims adjusters could be reassigned.

(E)

The premiums that Vasquez‐Morrell currently charges are no higher than those charged for similar coverage by
competitors.

Evaluation of a Plan
Situation An insurance company must reduce its staff of claims adjusters. To ensure continuing promptness

in handling claims, consultants advise the company to lay off those adjusters who take the longest
to complete claims.

Reasoning
What problem could there be with the criterion? The consultants’ criterion is the time an adjuster takes

to settle a claim. However, some claims are naturally more complicated and require more time. If
it is true that the company now assigns these time‐consuming cases to its most capable adjusters,
then these adjusters would be likely to be the ones who take longest to complete their cases.
Laying off the adjusters who take the longest would thus mean laying off the company’s most
capable staff, which could very well decrease its ability to handle claims promptly.

The consultants’ advice makes sense if increased time to handle claims causes the company to lose business.

Correct. This statement properly identiﬁes the problem with the consultants’ criterion.

C This statement merely describes the process of handling a claim; it does not provide any information about
the criterion for layoffs.
D The consultants make no recommendations for reassigning staff, so indicating that there are no positions
available does not call their advice into question.
E The consultants do not recommend a change in premiums; noting that they are similar to competitors’
premiums does not undermine the plan that the consultants recommend.
The correct answer is B.

GMAT® Official Guide 2018

67.

Prolonged spells of hot, dry weather at the end of the grape‐growing season typically reduce a vineyard’s yield, because the
grapes stay relatively small. In years with such weather, wine producers can make only a relatively small quantity of wine
from a given area of vineyards. Nonetheless, in regions where wine producers generally grow their own grapes, analysts
typically expect a long, hot, dry spell late in the growing season to result in increased revenues for local wine producers.
Which of the following, if true, does most to justify the analysts’ expectation?
(A)

The lower a vineyard’s yield, the less labor is required to harvest the grapes.

(B)

Long, hot, dry spells at the beginning of the grape‐growing season are rare, but they can have a devastating effect on
a vineyard’s yield.

(C)

Grapes grown for wine production are typically made into wine at or near the vineyard in which they were grown.

(D)

When hot, dry spells are followed by heavy rains, the rains frequently destroy grape crops.

(E)

Grapes that have matured in hot, dry weather make signiﬁcantly better wine than ordinary grapes.

Argument Construction
Situation Hot, dry weather at the end of the grape‐growing season reduces yield, so winemakers can

only produce a small quantity of wine. However, analysts expect that this weather will increase
winemakers’ revenues.

Reasoning
What additional piece of information explains the analysts’ expectations? The same conditions that

lead to low quantity also lead to something that increases revenues. What could this be? If
these weather conditions lead to higher‐quality wine that will sell for higher prices, the analysts’
expectations for increased revenues are justiﬁed.

A Lower labor costs mean less expenditure for the winemakers; this does not explain how revenues would
increase.
B

This statement about low yields does not explain an increase in revenues.

C The proximity of production to the vineyard is irrelevant to the question of how hot, dry weather can be
responsible for decreased yield and increased revenues.
D This statement gives another example of weather’s effect on grape crops, but it does not explain how
revenues are increased.
Correct. This statement properly provides the explanation that the weather conditions will lead to better
E
wines. With better wines typically commanding higher prices, the winemakers will gain the increased
revenues that the analysts anticipate.
The correct answer is E.

3.6 Diagnostic Test Verbal Answer Explanations

68.

In the past, most children who went sledding in the winter snow in Verland used wooden sleds with runners and steering
bars. Ten years ago, smooth plastic sleds became popular; they go faster than wooden sleds but are harder to steer and
slow. The concern that plastic sleds are more dangerous is clearly borne out by the fact that the number of children injured
while sledding was much higher last winter than it was 10 years ago.
Which of the following, if true in Verland, most seriously undermines the force of the evidence cited?
(A)

A few children still use traditional wooden sleds.

(B)

Very few children wear any kind of protective gear, such as helmets, while sledding.

(C)

Plastic sleds can be used in a much wider variety of snow conditions than wooden sleds can.

(D)

Most sledding injuries occur when a sled collides with a tree, a rock, or another sled.

(E)

Because the traditional wooden sleds can carry more than one rider, an accident involving a wooden sled can result in
several children being injured.

Argument Evaluation
Situation Ten years ago, wooden sleds began to be replaced by plastic sleds that go faster but are harder

to control. Plastic sleds are more dangerous than wooden sleds because more children suffered
injuries last year than they did 10 years ago.

Reasoning
What weakens this argument? This argument depends on a comparison of two kinds of sleds. Any

evidence that would either strengthen or weaken the argument must indicate a comparison.
Evidence that applies only to one kind of sled or to both kinds of sleds equally cannot weaken this
argument. Consider the implications of the evidence presented in the answer choices. If plastic
sleds can be used in a wider variety of conditions than wooden sleds can, then plastic sleds can be
used more frequently. It is possible that more frequent use, rather than the sleds themselves, has
led to more accidents.

The limited use of some wooden sleds does not weaken the argument.

The absence of protective gear would affect accidents with both kinds of sleds.

C
Correct. This statement weakens the argument by providing an alternate explanation for the increased
accidents.
D

This statement is true of accidents with both kinds of sleds.

E This explains why wooden sleds may be dangerous but does not weaken the argument that plastic sleds are
even more dangerous.
The correct answer is C.

GMAT® Official Guide 2018

69.

Metal rings recently excavated from seventh‐century settlements in the western part of Mexico were made using the
same metallurgical techniques as those used by Ecuadorian artisans before and during that period. These techniques are
sufﬁciently complex to make their independent development in both areas unlikely. Since the people of these two areas
were in cultural contact, archaeologists hypothesize that the metallurgical techniques used to make the rings found in
Mexico were learned by Mexican artisans from Ecuadorian counterparts.
Which of the following would it be most useful to establish in order to evaluate the archaeologists’ hypothesis?
(A)

Whether metal objects were traded from Ecuador to western Mexico during the seventh century

(B)

Whether travel between western Mexico and Ecuador in the seventh century would have been primarily by land or by
sea

(C)

Whether artisans from western Mexico could have learned complex metallurgical techniques from their Ecuadorian
counterparts without actually leaving western Mexico

(D)

Whether metal tools were used in the seventh‐century settlements in western Mexico

(E)

Whether any of the techniques used in the manufacture of the metal rings found in western Mexico are still practiced
among artisans in Ecuador today

Argument Evaluation
Situation Metal rings excavated from seventh‐century settlements in western Mexico were made with the

same complex techniques used in Ecuador before and during a period when the two cultures
were known to be in contact. Mexican artisans are thought to have learned the techniques from
Ecuadorian artisans.

Reasoning
What point could best be applied in evaluating this hypothesis? Consider what speciﬁc information

would help to assess the archaeologists’ theory. It is given that the two areas had some cultural
contact. If it were determined that metal objects were traded from one culture to the other, it
could be possible that the metalworking techniques were passed along as well. Such evidence
would be relevant to the hypothesis that Mexican artisans saw the work of their Ecuadorian
counterparts and, from this exchange, learned the techniques to make the metal rings.

A
Correct. This statement properly identiﬁes information that would be useful in the evaluation of the
archaeologists’ hypothesis.
B

The means of travel is irrelevant to the hypothesis about the source of the techniques.

C The hypothesis is not about where Mexican artisans learned the techniques, but whether they learned
them from the Ecuadorians.
D The existence of metal tools provides no helpful information in establishing whether the Ecuadorians were
the source of the metallurgical techniques.
E

The comparison to the present day is irrelevant to the hypothesis.

The correct answer is A.

3.6 Diagnostic Test Verbal Answer Explanations

70.

Following several years of declining advertising sales, the Greenville Times reorganized its advertising sales force. Before
reorganization, the sales force was organized geographically, with some sales representatives concentrating on city‐center
businesses and others concentrating on different outlying regions. The reorganization attempted to increase the sales
representatives’ knowledge of clients’ businesses by having each sales representative deal with only one type of industry or
of retailing. After the reorganization, revenue from advertising sales increased.
In assessing whether the improvement in advertising sales can properly be attributed to the reorganization, it would be most
helpful to ﬁnd out which of the following?
(A)

What proportion of the total revenue of the Greenville Times is generated by advertising sales?

(B)

Has the circulation of the Greenville Times increased substantially in the last two years?

(C)

Among all the types of industry and retailing that use the Greenville Times as an advertising vehicle, which type
accounts for the largest proportion of the newspaper’s advertising sales?

(D)

Do any clients of the sales representatives of the Greenville Times have a standing order with the Times for a ﬁxed
amount of advertising per month?

(E)

Among the advertisers in the Greenville Times, are there more types of retail business or more types of industrial
business?

Evaluation of a Plan
Situation In the face of declining advertising sales, a newspaper reorganizes its sales force so that sales

representatives have a better understanding of businesses. Revenue from advertising sales increased
after the reorganization.

Reasoning
What additional evidence would help determine the source of the increased revenue? In order to

attribute the increased revenue to the reorganization of the sales force, other possible causes
must be eliminated. Newspaper advertising rates are linked to circulation; when circulation
increases, higher rates can be charged and revenues will increase. An alternate explanation might
be a signiﬁcant rise in circulation, so it would be particularly helpful to know if circulation had
increased.

A The question concerns only increased revenue from advertising sales; the proportion of advertising revenue
to total revenue is outside the scope of the question.
B
Correct. This statement provides another possible explanation for increased revenue of advertising sales,
and so the answer to this question would help to clarify the reason for the increased revenue.
C Knowing how the advertising sales break down by type of business might be useful for other purposes, but
it does not help to show the cause of the increase.
D A ﬁxed amount of advertising would not explain increased revenue, so the answer to this question would
be irrelevant.
E Distinguishing between the types of businesses will not contribute to determining whether the
reorganization was responsible for the increased revenue.
The correct answer is B.

GMAT® Official Guide 2018

71.

Motorists in a certain country frequently complain that trafﬁc congestion is much worse now than it was 20 years ago.
No real measure of how much trafﬁc congestion there was 20 years ago exists, but the motorists’ complaints are almost
certainly unwarranted. The country’s highway capacity has tripled in the last twenty years, thanks to a vigorous highway
construction program, whereas the number of automobiles registered in the country has increased by only 75 percent.
Which of the following, if true, most seriously weakens the argument?
(A)

Most automobile travel is local, and the networks of roads and streets in the country’s settled areas have changed
little over the last twenty years.

(B)

Gasoline prices are high, and miles traveled per car per year have not changed much over the last 20 years.

(C)

The country’s urban centers have well‐developed public transit systems that carry most of the people who commute
into those centers.

(D)

The average age of automobiles registered in the country is lower now than it was 20 years ago.

(E)

Radio stations have long been broadcasting regular trafﬁc reports that inform motorists about trafﬁc congestion.

Argument Evaluation
Situation Motorists complain that trafﬁc congestion in their country is much worse than it was twenty years

ago. But these complaints have no basis since the highway capacity in this country has tripled in
the same period, whereas the number of cars registered has risen by only 75 percent.

Reasoning Which point most undermines the argument that the complaints are unwarranted? Consider that the

response to the generalized complaints about congestion discusses only the topic of highway
capacity. What if the congestion that motorists are complaining about is not on highways but on
local roads? Discovering that travel tends to be local in this country and that the local roads have
not been improved in the last twenty years would seriously weaken the argument.

Correct. This statement properly identiﬁes a weakness in the argument: the response to the broad
A
complaint addresses a different subject, highway capacity, not the issue of trafﬁc congestion encountered by
most motorists.
B If high gas prices actually prevented motorists from driving, and if motorists’ driving habits were the same
as they were twenty years ago, then these points should strengthen the argument that there is no basis for
their complaints.
C The number of commuters who use public transit does not affect the argument that the motorists’
complaints have no basis.
D

The age of registered cars is irrelevant to the argument.

E The radio broadcasts attest to the existence of trafﬁc, but not to its increase, so they do not affect the
argument.
The correct answer is A.

3.6 Diagnostic Test Verbal Answer Explanations

72.

The percentage of households with an annual income of more than $40,000 is higher in Merton County than in any other
county. However, the percentage of households with an annual income of $60,000 or more is higher in Sommer County.
If the statements above are true, which of the following must also be true?
(A)

The percentage of households with an annual income of $80,000 is higher in Sommer County than in Merton County.

(B)

Merton County has the second highest percentage of households with an annual income of $60,000 or more.

(C)

Some households in Merton County have an annual income between $40,000 and $60,000.

(D)

The number of households with an annual income of more than $40,000 is greater in Merton County than in Sommer
County.

(E)

Average annual household income is higher in Sommer County than in Merton County.

Argument Construction
Situation The percentage of households with annual incomes of more than $40,000 is higher in Merton

County than in any other county; the percentage of households with annual incomes of $60,000
or more is higher in Sommer County.

Reasoning
On the basis of this information, what point must be true? The given information makes clear that

Merton County has some households that exceed $40,000 in annual income. Sommer County has
a higher percentage of households with annual incomes at or above $60,000. A higher percentage
of the Merton County households must in turn have annual incomes of $60,000 or less. Thus, the
annual income of some households in Merton County is between $40,000 and $60,000.

A Since it is possible that there are no households with an annual income of $80,000 in Sommer County,
this statement does not follow from the situation.
B It is not possible to make this determination on the basis of the available evidence; Merton County may
have no households at all with an income of more than $60,000.
C
Correct. This statement properly identiﬁes a conclusion that can be drawn from the given information: in
order for the percentage of $40,000‐plus incomes to be higher in Merton county than any other county
while Sommer has the highest percentage of $60,000‐plus incomes, there must be some households in
Merton County that bring in between $40,000 and $60,000 annually.
D On the basis of information about the percentages of households, it is not possible to arrive at this
conclusion about the number of households.
E From the given information, it is not possible to determine where the average income is greater. It is
entirely possible that the number of $60,000‐plus incomes in Sommer County is quite small and that the
number of $40,000‐plus incomes in Merton County is substantial.
The correct answer is C.

GMAT® Official Guide 2018

73.

Tiger beetles are such fast runners that they can capture virtually any nonﬂying insect. However, when running toward
an insect, a tiger beetle will intermittently stop and then, a moment later, resume its attack. Perhaps the beetles cannot
maintain their pace and must pause for a moment’s rest; but an alternative hypothesis is that while running, tiger beetles are
unable to adequately process the resulting rapidly changing visual information and so quickly go blind and stop.
Which of the following, if discovered in experiments using artiﬁcially moved prey insects, would support one of the two
hypotheses and undermine the other?
(A)

When a prey insect is moved directly toward a beetle that has been chasing it, the beetle immediately stops and runs
away without its usual intermittent stopping.

(B)

In pursuing a swerving insect, a beetle alters its course while running and its pauses become more frequent as the
chase progresses.

(C)

In pursuing a moving insect, a beetle usually responds immediately to changes in the insect’s direction, and it pauses
equally frequently whether the chase is up or down an incline.

(D)

If, when a beetle pauses, it has not gained on the insect it is pursuing, the beetle generally ends its pursuit.

(E)

The faster a beetle pursues an insect ﬂeeing directly away from it, the more frequently the beetle stops.

Argument Evaluation
Situation Two hypotheses are offered to explain the sudden stop that tiger beetles make while pursuing

their prey: (1) they cannot maintain the rapid pace and must rest, and (2) they run too quickly to
process visual information and so temporarily go blind.

Reasoning What point would strengthen one of the two hypotheses and weaken the other? Consider the

information provided in each answer choice, remembering that information that supports one
hypothesis must necessarily detract from the other. Any information that is not about pursuit or
that affects the two hypotheses equally may be dismissed from consideration. If the frequency
of stopping increases when the beetle follows a swerving insect and must constantly change its
course, then the second hypothesis is strengthened; the beetle’s pauses increase as the variety of
visual information that it needs to deal with increases.

A The hypotheses concern ongoing pursuit; since this information is not about the beetle’s continuing pursuit
of prey, it neither strengthens nor weakens either hypothesis.
B
Correct. This statement provides information that strengthens the second hypothesis: the swerving pursuit
and the resulting continual course adjustments appear to be forcing the beetle to stop with increasing
frequency to sort out the erratic visual information.
C In this experiment, since neither vision nor tiredness appears to be problematic, the beetle could be
stopping for either reason; this information neither strengthens nor weakens either hypothesis.
D

This information is irrelevant since both the hypotheses are about mid‐pursuit behaviors.

E The correlation of frequency of stops with speed affects both hypotheses equally; the pauses could be
equally due to an inability to maintain the pace or due to a need to process the visual information.
The correct answer is B.

3.6 Diagnostic Test Verbal Answer Explanations

74.

Guillemots are birds of Arctic regions. They feed on ﬁsh that gather beneath thin sheets of ﬂoating ice, and they nest on
nearby land. Guillemots need 80 consecutive snow‐free days in a year to raise their chicks, so until average temperatures
in the Arctic began to rise recently, the guillemots’ range was limited to the southernmost Arctic coast. Therefore, if the
warming continues, the guillemots’ range will probably be enlarged by being extended northward along the coast.
Which of the following, if true, most seriously weakens the argument?
(A)

Even if the warming trend continues, there will still be years in which guillemot chicks are killed by an unusually early
snow.

(B)

If the Arctic warming continues, guillemots’ current predators are likely to succeed in extending their own range
farther north.

(C)

Guillemots nest in coastal areas, where temperatures are generally higher than in inland areas.

(D)

If the Arctic warming continues, much of the thin ice in the southern Arctic will disappear.

(E)

The ﬁsh that guillemots eat are currently preyed on by a wider variety of predators in the southernmost Arctic regions
than they are farther north.

Argument Evaluation
Situation In the southern Arctic, guillemots ﬁnd their prey beneath thin sheets of ice, nest nearby, and

require 80 snow‐free days to raise their young. A warming trend means that their range may be
enlarged by extending northward along the coast.

Reasoning Which point weakens the argument about the enlargement of the guillemots’ range? How could the

birds move northward and simultaneously not enlarge their range? Consider the assumption
implied by the idea of enlargement. If the guillemots lost their southern habitat, then their
northward move would be a displacement rather than an enlargement. If their source of food was
no longer available to them in the southern Arctic, then they would abandon that area as part of
their range.

A An exceptional year is not an argument against an enlarged range because an unusually early snow could
happen in the southern Arctic as well.
B If their current predators also migrate northward, then the guillemots’ situation has not changed, so this is
not an argument against their enlarged range.
C

The argument suggests that they will move not inland, but northward along the coast.

D
Correct. This statement properly identiﬁes a factor that weakens the argument: the guillemots’ move
northward would not enlarge their range if they lost their food source, ﬁsh found under thin ice, in the
southern Arctic.
E The possibility that they may ﬁnd prey more easily in the north does not mean that they would abandon
the southern Arctic, and so this point does not weaken the argument.
The correct answer is D.

GMAT® Official Guide 2018

75.

Some batches of polio vaccine used around 1960 were contaminated with SV40, a virus that in monkeys causes various
cancers. Some researchers now claim that this contamination caused some cases of a certain cancer in humans,
mesothelioma. This claim is not undercut by the fact that a very careful survey made in the 1960s of people who had
received the contaminated vaccine found no elevated incidence of any cancer, since __________.
(A)

most cases of mesothelioma are caused by exposure to asbestos

(B)

in some countries, there was no contamination of the vaccine

(C)

SV40 is widely used in laboratories to produce cancers in animals

(D)

mesotheliomas take several decades to develop

(E)

mesothelioma was somewhat less common in 1960 than it is now

Argument Construction
Situation Researchers claim that contaminated polio vaccine administered in 1960 caused some cases of

mesothelioma, a type of cancer. Their claim is not undermined by the results of a 1960s survey
showing that those who received the contaminated vaccine had no elevated incidence of cancer.

Reasoning Why did the survey results not challenge the researchers’ claim? The survey did not reveal a higher

incidence of mesothelioma. This question then requires completing a sentence that establishes
cause. What could be the reason that the people surveyed in the 1960s showed no signs of the
disease? If the disease takes decades to develop, then those people surveyed would not yet have
shown any signs of it; less than a decade had passed between their exposure to the vaccine and the
survey.

A The contaminated vaccine is said to have caused some cases, not most; the question remains why the survey
results pose no obstacle to the researchers’ claim.
B

The claim is only about contaminated vaccine, not uncontaminated vaccine.

C That the virus can cause cancers in laboratory animals had already been provided as a given; this additional
information is irrelevant to the survey of people who received contaminated vaccine.
Correct. This statement properly identiﬁes the reason that the survey does not call into question the
D
researchers’ claim: the people surveyed in the 1960s showed no signs of disease because the cancer takes
decades to develop.
E The frequency of mesothelioma in the general population is not related to the claim that contaminated
vaccine caused the disease in a speciﬁc population.
The correct answer is D.

3.6 Diagnostic Test Verbal Answer Explanations

76.

Gortland has long been narrowly self‐sufﬁcient in both grain and meat. However, as per capita income in Gortland has risen
toward the world average, per capita consumption of meat has also risen toward the world average, and it takes several
pounds of grain to produce one pound of meat. Therefore, since per capita income continues to rise, whereas domestic
grain production will not increase, Gortland will soon have to import either grain or meat or both.
Which of the following is an assumption on which the argument depends?
(A)

The total acreage devoted to grain production in Gortland will soon decrease.

(B)

Importing either grain or meat will not result in a signiﬁcantly higher percentage of Gortlanders’ incomes being spent
on food than is currently the case.

(C)

The per capita consumption of meat in Gortland is increasing at roughly the same rate across all income levels.

(D)

The per capita income of meat producers in Gortland is rising faster than the per capita income of grain producers.

(E)

People in Gortland who increase their consumption of meat will not radically decrease their consumption of grain.

Argument Construction
Situation A country previously self‐sufﬁcient in grain and meat will soon have to import one or the other or

both. Consumption of meat has risen as per capita income has risen, and it takes several pounds of
grain to produce one pound of meat.

Reasoning What conditions must be true for the conclusion to be true? Meat consumption is rising. What about

grain consumption? A sharp reduction in the amount of grain consumed could compensate for
increased meat consumption, making the conclusion false. If people did radically decrease their
grain consumption, it might not be necessary to import grain or meat or both. Since the argument
concludes that the imports are necessary, it assumes grain consumption will not plunge.

A The argument makes no assumptions about the acreage devoted to grain; it assumes only that the demand
for grain will rise.
B The argument does not discuss the percentage of their income that Gortlanders spend on food, so an
assumption about this topic is not needed.
C

The argument involves only meat consumption in general, not its distribution by income level.

D Since the argument does not refer to the incomes of meat producers and grain producers, it cannot depend
on an assumption about them.
E
Correct. This statement properly identiﬁes the assumption that there will be no great decrease in grain
consumption.
The correct answer is E.

GMAT® Official Guide 2018

77.

The Hazelton coal‐processing plant is a major employer in the Hazelton area, but national environmental regulations will
force it to close if it continues to use old, polluting processing methods. However, to update the plant to use newer, cleaner
methods would be so expensive that the plant will close unless it receives the tax break it has requested. In order to prevent
a major increase in local unemployment, the Hazelton government is considering granting the plant’s request.
Which of the following would be most important for the Hazelton government to determine before deciding whether to grant
the plant’s request?
(A)

Whether the company that owns the plant would open a new plant in another area if the present plant were closed

(B)

Whether the plant would employ far fewer workers when updated than it does now

(C)

Whether the level of pollutants presently being emitted by the plant is high enough to constitute a health hazard for
local residents

(D)

Whether the majority of the coal processed by the plant is sold outside the Hazelton area

(E)

Whether the plant would be able to process more coal when updated than it does now

Evaluation of a Plan
Situation Because of the expenses of mandatory updating, a plant that is a major employer in the local area

will close unless it receives the tax break it has requested from the local government.

Reasoning What point is most critical to the evaluation of the request? Consider the information provided

in the answer choices. The plant is important to the local government primarily because it is a
major employer of local residents. What if updating the plant signiﬁcantly reduced the number
of employees needed? It is crucial for the local government to determine whether the plant will
continue to employ the same number of people once it has updated.

The local government is concerned only with the local area, so a new site outside that area is irrelevant.

B
Correct. This statement properly identiﬁes a factor that is critical to the plant’s argument and the local
government’s decision.
C Updating is mandatory under national environmental regulations, whether the local residents are affected
by the plant’s pollutants or not.
D

At issue is the plant’s role as a major employer; where its product is sold is irrelevant.

E The amount of coal processed by the updated plant is irrelevant to the critical issue of the number of
people employed to process that coal.
The correct answer is B.

3.6 Diagnostic Test Verbal Answer Explanations

78.

A physically active lifestyle has been shown to help increase longevity. In the Wistar region of Bellaria, the average age
at death is considerably higher than in any other part of the country. Wistar is the only mountainous part of Bellaria. A
mountainous terrain makes even such basic activities as walking relatively strenuous; it essentially imposes a physically
active lifestyle on people. Clearly, this circumstance explains the long lives of people in Wistar.
Which of the following, if true, most seriously weakens the argument?
(A)

In Bellaria all medical expenses are paid by the government, so that personal income does not affect the quality of
health care a person receives.

(B)

The Wistar region is one of Bellaria’s least populated regions.

(C)

Many people who live in the Wistar region have moved there in middle age or upon retirement.

(D)

The many opportunities for hiking, skiing, and other outdoor activities that Wistar’s mountains offer make it a favorite
destination for vacationing Bellarians.

(E)

Per capita spending on recreational activities is no higher in Wistar than it is in other regions of Bellaria.

Argument Evaluation
Situation People in one region of a country live longer than people in other areas. The higher average

age at time of death is attributed to the healthy lifestyle of the people in this region, where the
mountainous terrain demands a physically active life.

Reasoning What point weakens the argument? Consider what assumption underlies the argument that the

physically active lifestyle required of living in Wistar is responsible for its residents’ relative
longevity. The mountainous environment necessitates lifelong levels of rigorous physical activity
that build a more robust population. What if a signiﬁcant portion of the population has not been
conditioned since childhood to the demands of the terrain? It is assumed here that the healthy
lifestyle imposed by the terrain has shaped residents from birth and accounts for their longer life
span. If many residents only moved there later in life, the argument is weakened.

A The argument is not about the quality of health care throughout the country, but the length of the
residents’ lives in a particular region.
B

The rate of population density does not affect the argument.

Correct. This statement properly identiﬁes a point that weakens the argument.

The area’s popularity as a vacation destination does not affect the longevity of the local residents.

E The argument establishes that merely living in the region is strenuous; the spending on recreational
activities is irrelevant.
The correct answer is C.

GMAT® Official Guide 2018

79.

Cheever College offers several online courses via remote computer connection, in addition to traditional classroom‐based
courses. A study of student performance at Cheever found that, overall, the average student grade for online courses
matched that for classroom‐based courses. In this calculation of the average grade, course withdrawals were weighted as
equivalent to a course failure, and the rate of withdrawal was much lower for students enrolled in classroom‐based courses
than for students enrolled in online courses.
If the statements above are true, which of the following must also be true of Cheever College?
(A)

Among students who did not withdraw, students enrolled in online courses got higher grades, on average, than
students enrolled in classroom‐based courses.

(B)

The number of students enrolled per course at the start of the school term is much higher, on average, for the online
courses than for the classroom‐based courses.

(C)

There are no students who take both an online and a classroom‐based course in the same school term.

(D)

Among Cheever College students with the best grades, a signiﬁcant majority take online, rather than classroom‐
based, courses.

(E)

Courses offered online tend to deal with subject matter that is less challenging than that of classroom‐based courses.

Argument Construction
Situation A comparison of online and classroom courses showed similar average grades. In determining

average grades, a course withdrawal was weighted as a course failure. The rate of withdrawal was
higher from online than from classroom courses.

Reasoning What conclusion about the courses can be derived from this comparison? Consider the ramiﬁcations

of the methodology used to calculate the grade averages for the two types of courses. Because of
course withdrawals, the online courses experienced a higher rate of failure, but the average grade
for these courses still matched the average grade for classroom courses. From this it is logical to
conclude that, for the two averages to match, the students who remained in the online courses
must have had higher initial average grades than those in classroom courses.

A
Correct. This statement properly identiﬁes the logical conclusion that the higher percentage of
withdrawals from online classes requires higher grades, on average, to compensate for the higher rate of
failure.
B

A number of students cannot be derived from a discussion of average grades and rates of withdrawal.

This conclusion cannot be determined on the basis of the information provided.

D The information is about average grades; the argument does not provide any basis for a conclusion about
best grades.
E

It is impossible to determine the difﬁculty of subject matter from this information.

The correct answer is A.

3.6 Diagnostic Test Verbal Answer Explanations

80.

For years the beautiful Renaissance buildings in Palitito have been damaged by exhaust from the many tour buses that
come to the city. There has been little parking space, so most buses have idled at the curb during each stop on their tour,
and idling produces as much exhaust as driving. The city has now provided parking that accommodates a third of the tour
buses, so damage to Palitito’s buildings from the buses’ exhaust will diminish signiﬁcantly.
Which of the following, if true, most strongly supports the argument?
(A)

The exhaust from Palitito’s few automobiles is not a signiﬁcant threat to Palitito’s buildings.

(B)

Palitito’s Renaissance buildings are not threatened by pollution other than engine exhaust.

(C)

Tour buses typically spend less than one‐quarter of the time they are in Palitito transporting passengers from one site
to another.

(D)

More tourists come to Palitito by tour bus than by any other single means of transportation.

(E)

Some of the tour buses that are unable to ﬁnd parking drive around Palitito while their passengers are visiting a site.

Argument Evaluation
Situation Tour buses have damaged Renaissance buildings with their exhaust fumes because lack of

parking has kept the buses idling at curbs. Providing new parking for a third of the buses should
signiﬁcantly reduce the damage caused by the exhaust.

Reasoning What point strengthens the argument? The argument for reduced damage relies on the reduction of

the vehicles’ exhaust fumes. Any additional evidence regarding the extent to which the vehicular
emissions are likely to be reduced also supports the argument for the beneﬁts of the new parking
spaces. Learning that tour buses spend not just a few minutes but most of their time idling at
the curb strengthens the argument. The new parking spaces will allow a third of the tour buses to
spend 75 percent of their time with their engines off, causing no damage at all.

If automobile exhaust is not a threat, the argument is not affected.

B This statement does not address the question of whether the new parking will reduce the damage caused
by engine exhaust from the buses.
C
Correct. This statement properly cites a factor that supports the argument: since most of the buses’ time
has been spent producing damaging exhaust, the new parking should reduce the damage signiﬁcantly.
D This statement about tourists’ chosen means of transportation is irrelevant to the issue of what the buses do
while in the city.
E It is given that the new parking will only provide space for a third of the buses, and thus some buses will
continue to idle and some to drive around, continuing to contribute equally to the building damage. This
statement does not strengthen the argument.
The correct answer is C.

GMAT® Official Guide 2018

81.

During the 1980s and 1990s, the annual number of people who visited the Sordellian Mountains increased continually, and
many new ski resorts were built. Over the same period, however, the number of visitors to ski resorts who were caught in
avalanches decreased, even though there was no reduction in the annual number of avalanches in the Sordellian Mountains.
Which of the following, if true in the Sordellian Mountains during the 1980s and 1990s, most helps to explain the decrease?
(A)

Avalanches were most likely to happen when a large new snowfall covered an older layer of snow.

(B)

Avalanches destroyed at least some buildings in the Sordellian Mountains in every year.

(C)

People planning new ski slopes and other resort facilities used increasingly accurate information about which
locations are likely to be in the path of avalanches.

(D)

The average length of stay for people visiting the Sordellian Mountains increased slightly.

(E)

Construction of new ski resorts often led to the clearing of wooded areas that had helped prevent avalanches.

Argument Construction
Situation Over a certain period, new ski resorts accommodated an increasing number of visitors at the same

time that fewer visitors were caught in avalanches. Yet there were no fewer avalanches than usual
during this period.

Reasoning What explains the apparent contradiction of increased visitors but fewer visitors caught in avalanches?

More resort visitors would imply more avalanche‐related accidents, but the average has shifted so
that fewer visitors are being caught in the avalanches. It must be that fewer visitors are exposed
to this danger; consider the answer choices to identify a logical reason for this improvement in
their exposure. If the likely paths of avalanches had become better understood, that information
would have been applied to identify safer locations for new ski slopes and ski resorts. The facilities
would thus have been built well out of the way of avalanches, resulting in fewer visitors trapped in
avalanches.

This likelihood would remain true from year to year; it does not explain the decrease.

This point does not explain why fewer visitors were caught in these avalanches.

Correct. This statement properly identiﬁes a factor that explains the decreased number of accidents.

The greater length of stay would seem to expose visitors to greater danger.

E This information points to an expected increase, rather than decrease, in visitors who might be caught by
avalanches.
The correct answer is C.

3.6 Diagnostic Test Verbal Answer Explanations

82.

A year ago, Dietz Foods launched a yearlong advertising campaign for its canned tuna. Last year Dietz sold 12 million cans
of tuna compared to the 10 million sold during the previous year, an increase directly attributable to new customers brought
in by the campaign. Proﬁts from the additional sales, however, were substantially less than the cost of the advertising
campaign. Clearly, therefore, the campaign did nothing to further Dietz’s economic interests.
Which of the following, if true, most seriously weakens the argument?
(A)

Sales of canned tuna account for a relatively small percentage of Dietz Foods’ proﬁts.

(B)

Most of the people who bought Dietz’s canned tuna for the ﬁrst time as a result of the campaign were already loyal
customers of other Dietz products.

(C)

A less expensive advertising campaign would have brought in signiﬁcantly fewer new customers for Dietz’s canned
tuna than did the campaign Dietz Foods launched last year.

(D)

Dietz made money on sales of canned tuna last year.

(E)

In each of the past ﬁve years, there was a steep, industry‐wide decline in sales of canned tuna.

Argument Evaluation
Situation An advertising campaign was responsible for increased sales of canned tuna. Since the proﬁts from

the increased sales were less than the costs of the campaign, the campaign did not contribute to
the company’s economic interests.

Reasoning Which point weakens the argument? Consider the basis of the argument: if proﬁts are lower

than costs, the campaign made no contribution to the company’s ﬁnancial well‐being. In what
case might this be untrue? What if the advertising campaign reversed an industry‐wide trend
of declining sales? If Dietz experienced increasing sales, while other companies experienced
decreased sales, then the campaign did contribute to the economic interests of the company, and
the argument is considerably weakened.

A The issue is not the percentage of proﬁts that canned tuna contributes, but the success of the advertising
campaign.
B If the customers bought the tuna because of the campaign, it is irrelevant to the argument that they also
bought other Dietz products.
C

This information neither strengthens nor weakens the argument.

D The argument is not about proﬁts only, but about whether the advertising campaign contributed to the
economic interests of the company.
E
Correct. This statement properly identiﬁes a factor that weakens the argument: the campaign secured the
beneﬁts of increased sales at a time when the entire industry was experiencing a decline in sales.
The correct answer is E.

GMAT® Official Guide 2018

Sentence Correction
The following discussion is intended to familiarize you with the most efficient and effective approaches
to sentence correction questions. The particular questions in this chapter are generally representative of
the kinds of sentence correction questions you will encounter on the GMAT exam. Remember that it is
the problem solving strategy that is important, not the specific details of a particular question.

83.

Unlike the buildings in Mesopotamian cities, which
were arranged haphazardly, the same basic plan was
followed for all cities of the Indus Valley: with houses
laid out on a north‐south, east‐west grid, and houses
and walls were built of standard‐size bricks.
(A)

the buildings in Mesopotamian cities, which were
arranged haphazardly, the same basic plan was
followed for all cities of the Indus Valley: with
houses

B Illogically contrasts the buildings in
Mesopotamian cities with the same basic plan;
does not clarify what which were haphazard
in arrangement modiﬁes.

(B)

the buildings in Mesopotamian cities, which were
haphazard in arrangement, the same basic plan
was used in all cities of the Indus Valley: houses
were

(C)

the arrangement of buildings in Mesopotamian
cities, which were haphazard, the cities of the
Indus Valley all followed the same basic plan:
houses

C Illogically contrasts the arrangement of
buildings with the cities of the Indus Valley;
not clear whether which were haphazard
modiﬁes buildings or cities; houses not
followed by a verb.

(D)

Mesopotamian cities, in which buildings were
arranged haphazardly, the cities of the Indus Valley
all followed the same basic plan: houses were

(E)

Mesopotamian cities, which had buildings that
were arranged haphazardly, the same basic plan
was used for all cities in the Indus Valley: houses
that were

Comparison-contrast; Modifying clause

The contrast introduced by unlike must be
logical and clear. Contrasting the buildings in
Mesopotamian cities with the same basic plan does
not make sense; Mesopotamian cities should be
contrasted with the cities of the Indus Valley. Also,
it needs to be clear that it was the buildings in the
cities that were arranged haphazardly rather than
the cities. The second half of the sentence needs
houses were laid out to be parallel in structure to
and houses and walls were built.

A Illogically contrasts the buildings in
Mesopotamian cities with the same basic plan;
not clear whether which were arranged
haphazardly modiﬁes cities or buildings; with
houses lacks parallelism and is confusing.

D
Correct. In this sentence, Mesopotamian
cities are properly contrasted with the cities
of the Indus Valley; in which buildings were
arranged haphazardly expresses the idea
clearly; and houses is followed by were as
required.
E Illogically contrasts Mesopotamian cities with
the same basic plan; houses that were lacks
parallelism and is confusing.
The correct answer is D.

3.6 Diagnostic Test Verbal Answer Explanations

84.

A
It has no referent; not be spent is awkward;
on having to extinguish is wordy.

New data from United States Forest Service ecologists
show that for every dollar spent on controlled small‐
scale burning, forest thinning, and the training of
ﬁre‐management personnel, it saves seven dollars that
would not be spent on having to extinguish big ﬁres.
(A)

(B)

(C)

B
Correct. This sentence properly uses seven
dollars as the subject of the clause to balance
every dollar in the introductory phrase; the
phrasing is concise and parallel.

that for every dollar spent on controlled small‐
scale burning, forest thinning, and the training
of ﬁre‐management personnel, it saves seven
dollars that would not be spent on having to
extinguish

Saves does not have a subject; construction
C
is not a complete sentence; not having to
extinguish is wordy and awkward.
D
That introduces a subordinate rather than
main clause, making a sentence fragment;
it has no referent; not having to extinguish is
wordy and awkward.

that for every dollar spent on controlled small‐
scale burning, forest thinning, and the training
of ﬁre‐management personnel, seven dollars
are saved that would have been spent on
extinguishing

E Introductory that makes a sentence
fragment; that would not have been spent on
extinguishing is awkward and illogical.

that for every dollar spent on controlled small‐
scale burning, forest thinning, and the training of
ﬁre‐management personnel saves seven dollars
on not having to extinguish

(D)

for every dollar spent on controlled small‐scale
burning, forest thinning, and the training of
ﬁre‐management personnel, that it saves seven
dollars on not having to extinguish

(E)

for every dollar spent on controlled small‐scale
burning, forest thinning, and the training of
ﬁre‐management personnel, that seven dollars
are saved that would not have been spent on
extinguishing

Logical predication; Rhetorical construction

The pronoun it (it saves seven dollars) has no
referent. Making seven dollars the subject of
the clause eliminates this problem, and it also
fulﬁlls a reader’s expectation that after the
phrase beginning for every dollar another speciﬁc
amount will be given to balance it. This change
in structure also allows the awkward and wordy
clause that would not be spent on having to
extinguish to be rewritten so that spent balances
saved: seven dollars are saved that would have been
spent on extinguishing, and the unnecessary having
to is omitted.

The correct answer is B.
85.

Like the grassy ﬁelds and old pastures that the
upland sandpiper needs for feeding and nesting
when it returns in May after wintering in the Argentine
Pampas, the sandpipers vanishing in the northeastern
United States is a result of residential and industrial
development and of changes in farming practices.
(A)

the sandpipers vanishing in the northeastern
United States is a result of residential and
industrial development and of changes in

(B)

the bird itself is vanishing in the northeastern
United States as a result of residential and
industrial development and of changes in

(C)

that the birds themselves are vanishing in the
northeastern United States is due to residential
and industrial development and changes to

(D)

in the northeastern United States, sandpipers’
vanishing due to residential and industrial
development and to changes in

(E)

in the northeastern United States, the
sandpipers’ vanishing, a result of residential and
industrial development and changing

GMAT® Official Guide 2018

Comparison; Sentence structure

The comparison introduced by like must be
logical and clear; the point of this comparison
is that both the habitat and the bird are
disappearing for similar reasons. The comparison
must use comparable grammatical components;
the bird itself is a noun phrase and matches the
noun phrases grassy ﬁelds and old pastures.
A Illogically compares the sandpipers vanishing
to grassy ﬁelds and old pastures; omits
apostrophe in sandpipers’ vanishing; wordy.
B
Correct. This sentence properly compares
the bird itself to grassy ﬁelds and old pastures;
is vanishing as the verb strengthens the
sentence by making the comparison clearer.
C Does not ﬁnish the comparison begun with
like but instead substitutes a clause (that the
birds themselves are vanishing).
D Illogically compares the sandpipers’ vanishing
to grassy ﬁelds and old pastures; creates a
sentence fragment.
E Illogically compares the sandpipers’ vanishing
to grassy ﬁelds and old pastures; creates a
sentence fragment.
The correct answer is B.

86.

The results of two recent unrelated studies support the
idea that dolphins may share certain cognitive abilities
with humans and great apes; the studies indicate
dolphins as capable of recognizing themselves in
mirrors—an ability that is often considered a sign of
self‐awareness—and to grasp spontaneously the mood
or intention of humans.
(A)

dolphins as capable of recognizing themselves
in mirrors—an ability that is often considered
a sign of self‐awareness—and to grasp
spontaneously

(B)

dolphins’ ability to recognize themselves in
mirrors—an ability that is often considered as a
sign of self‐awareness—and of spontaneously
grasping

(C)

dolphins to be capable of recognizing
themselves in mirrors—an ability that is often
considered a sign of self‐awareness—and to
grasp spontaneously

(D)

that dolphins have the ability of recognizing
themselves in mirrors—an ability that is often
considered as a sign of self‐awareness—and
spontaneously grasping

(E)

that dolphins are capable of recognizing
themselves in mirrors—an ability that is often
considered a sign of self‐awareness—and of
spontaneously grasping

Grammatical construction; Parallelism

In the context of this sentence, the studies indicate
must introduce a clause; the clause must begin
with that and have a subject, dolphins, and a verb,
are (the complete verb phrase would be are capable
of ). The two capabilities should be parallel:
capable of recognizing…and of spontaneously
grasping.

3.6 Diagnostic Test Verbal Answer Explanations

A Context requires a clause, but this
construction is not a clause; capable of
recognizing is not parallel to to grasp
spontaneously.

A In this context, more likely is not a complete
idiomatic expression; was…to begin is not
parallel to was and merged.
B
Correct. In this sentence, more than likely
is the correct comparative construction; the
simple past tense began, parallel to was and
merged, ﬁts grammatically into the sentence.

B Construction is not a clause, and a clause is
required; dolphins’ ability to recognize is not
parallel to of spontaneously grasping.
C A clause is required following the studies
indicate; to be capable of recognizing is not
parallel to to grasp spontaneously.

C Subject should be followed by three verbs;
beginning from is not a verb.
D Use of the pronoun it makes this
construction a main clause, in which case
the comma after communication must be
omitted and began must be used to be
parallel to merged; was…begun is not the
correct tense.

D
Have the ability of is wordy and unidiomatic;
of recognizing and spontaneously grasping are
not parallel.
E
Correct. That introduces the subordinate
clause necessary to complete this sentence
properly; of recognizing and of spontaneously
grasping are parallel.

E In this awkward, unclear, and wordy
construction, the ﬁrst it must be followed
by is, not was, because the theory is
current; the second it acts as the subject
of the subordinate clause, and this usage
requires the omission of the comma after
communication.

The correct answer is E.
87.

According to scholars, the earliest writing was
probably not a direct rendering of speech, but was
more likely to begin as a separate and distinct
symbolic system of communication, and only later
merged with spoken language.
(A)

was more likely to begin as

(B)

more than likely began as

(C)

more than likely beginning from

(D)

it was more than likely begun from

(E)

it was more likely that it began

Idiom; Verb form

This sentence is a comparison in which probably
not x is balanced by but more than likely y. When
more is used in the comparative form of an
adjective (more difﬁcult) or adverb (more likely),
it is followed by than. The words used to show
the comparison between x and y, but more than
likely, must also introduce the correct verb form,
allowing y to ﬁt grammatically into the rest of the
sentence. The subject of the sentence has three
verbs, all of which should be parallel: the earliest
writing was…began…merged. Was…to begin is not
parallel and results in a construction that is not
grammatically correct.

The correct answer is B.
88.

In 1995 Richard Stallman, a well‐known critic of the
patent system, testiﬁed in Patent Ofﬁce hearings that,
to test the system, a colleague of his had managed to
win a patent for one of Kirchhoff’s laws, an observation
about electric current ﬁrst made in 1845 and now
included in virtually every textbook of elementary
physics.
(A)

laws, an observation about electric current ﬁrst
made in 1845 and

(B)

laws, which was an observation about electric
current ﬁrst made in 1845 and it is

(C)

laws, namely, it was an observation about
electric current ﬁrst made in 1845 and

(D)

laws, an observation about electric current ﬁrst
made in 1845, it is

(E)

laws that was an observation about electric
current, ﬁrst made in 1845, and is

GMAT® Official Guide 2018

Logical predication; Parallelism

Agreement; Idiom; Parallelism

The function of the entire long phrase
(observation…physics) that follows one of
Kirchhoff ’s laws is to describe that law. It is a
noun phrase in apposition, which means that it
has the same syntactic relation to all the other
parts of the sentence that the noun phrase one of
Kirchhoff ’s laws does. Within the long modifying
phrase, parallelism is maintained by balancing an
observation…ﬁrst made with and now included.

In long sentences such as this one, the
relationship between parts of the sentence may
be difﬁcult to see. Here, the main clause of
the sentence is excavators…say and the logical
sequence that follows is the discovery…indicates
that. The subject of this ﬁrst subordinate clause
is the singular noun discovery, which should be
followed by the singular verb indicates rather than
by the plural indicate, as is done in the original
sentence. Their, used with either development
or use, has no clear or logical referent in any of
the alternatives. The subject of the following
subordinate (that) clause, which has occurred as its
verb, is a series of three phrases, which must be
parallel, especially in a sentence of this length and
complexity: the development of…, the use of…, and
the standardization of….

Correct. In this sentence, the noun phrase
A
in apposition properly identiﬁes and
explains the law, using parallel structure and
concise expression.

B
Which is ambiguous because it could refer to
one or to laws; it is violates the parallelism of
ﬁrst made and now included.
C
It is ambiguous; the introduction of it
was does not allow this construction to ﬁt
grammatically into the sentence.

D The referent of it is unclear; it is creates a
run‐on sentence and violates the parallelism
of ﬁrst made and now included.

E
That appears to refer to laws rather than
one, but the verb is singular; setting off the
phrase ﬁrst made in 1845 in commas distorts
meaning; is violates parallelism.
The correct answer is A.
89.

Excavators at the Indus Valley site of Harappa in
eastern Pakistan say the discovery of inscribed
shards dating to circa 2800–2600 B.C. indicate their
development of a Harappan writing system, the use
of inscribed seals impressed into clay for marking
ownership, and the standardization of weights for trade
or taxation occurred many decades, if not centuries,
earlier than was previously believed.
(A)

indicate their development of a Harappan writing
system, the use of

(B)

indicate that the development of a Harappan
writing system, using

(C)

indicates that their development of a Harappan
writing system, using

(D)

indicates the development of a Harappan writing
system, their use of

(E)

indicates that the development of a Harappan
writing system, the use of

A
Indicate does not agree with discovery; the
pronoun their has no logical referent, and
their development is not parallel to the use
and the standardization.
B
Indicate does not agree with discovery; using
is not parallel to the development and the
standardization.
Their has no logical referent; the series of
C
three elements should be parallel, but here
all are different.
D The pronoun their has no logical referent,
and their use is not parallel to the development
and the standardization; the preferred
sentence structure would have indicates
followed by that when introducing a clause.
E
Correct. In this sentence, indicates agrees
with discovery and is followed by that to
introduce a clause; the three parallel phrases
begin with an article (the), a noun, and the
preposition of.
The correct answer is E.

3.6 Diagnostic Test Verbal Answer Explanations

90.

The Supreme Court has ruled that public universities can
collect student activity fees even with students’ objections
to particular activities, so long as the groups they give
money to will be chosen without regard to their views.
(A)

with students’ objections to particular activities,
so long as the groups they give money to will be

(B)
(C)
(D)
(E)

91.

Despite the increasing number of women graduating
from law school and passing bar examinations, the
proportion of judges and partners at major law ﬁrms
who are women have not risen to a comparable extent.
(A)

if they have objections to particular activities and
the groups that are given the money are

the proportion of judges and partners at major
law ﬁrms who are women have not risen to a
comparable extent

(B)

if they object to particular activities, but the
groups that the money is given to have to be

the proportion of women judges and partners at
major law ﬁrms have not risen comparably

(C)

from students who object to particular activities,
so long as the groups given money are

the proportion of judges and partners at major law
ﬁrms who are women has not risen comparably

(D)

though students have an objection to particular
activities, but the groups that are given the
money be

yet the proportion of women judges and partners
at major law ﬁrms has not risen to a comparable
extent

(E)

yet the proportion of judges and partners at major
law ﬁrms who are women has not risen comparably

Logical predication; Rhetorical construction

The underlined portion of the sentence fails to
establish a clear relationship among universities,
students, and groups. To which of these three does
they refer? It would appear that the universities must
give the money, but they does not have a referent.
Furthermore, they is followed by their views, and
in this case their must refer to groups. Wordy and
awkward phrasing as well as an unnecessary shift in
verb tense (will be chosen) compound the difﬁculty of
understanding this sentence in its original form.

Agreement; Rhetorical construction

A
With students’ objections…is awkward and
dense; they does not have a referent; the
future will be is incorrect since the Supreme
Court has already ruled.

A Plural verb, have risen, does not agree with
the singular subject, proportion.

B Referent for they is student activity fees,
which cannot possibly have objections…; the
use of and is illogical.
C
They refers to student activity fees rather than
students; but does not have the requisite sense
of with the provision that; have to be is wordy.
D
Correct. In this sentence, from students who
object is clear and idiomatic; so long as is used
appropriately; groups given money eliminates
the problem of a pronoun without a
referent; are is the proper tense.
E
Have an objection is an unnecessarily wordy
way to say object; the verb be does not
complete the latter part of the sentence.
The correct answer is D.

When a number of plural nouns appear in
phrases between a singular subject and the verb,
it can be easy to overlook the true subject of the
verb. Here, judges, partners, ﬁrms, and women all
occur between the singular subject, proportion,
and the verb, which should also be singular, has
risen. Concise expression is particularly important
in a long construction; to a comparable extent may
be more concisely expressed as comparably.

B
Have risen does not agree with proportion;
here, women applies only to judges, not to
partners at major law ﬁrms.

C
Correct. In this sentence, has risen agrees
with proportion, and comparably is more
concise than to a comparable extent. The
modifying clause who are women follows
(1) judges and (2) partners at major law ﬁrms
as closely as is possible given the content of
the sentence; this positioning has the virtue
of being clear in its meaning.

D The contrast has already been introduced by
despite, so the addition of yet is illogical and
ungrammatical; to a comparable extent is wordy.
E
Despite introduces the contrast; adding yet
is illogical and results in an ungrammatical
construction.
The correct answer is C.

GMAT® Official Guide 2018

92.

(A)

B
Correct. This sentence properly has the
single modiﬁer consisting of two contrasting
parts.
C Neither and nor but acts as a logical
connector; the use of connecting results in a
sentence fragment.

Seldom more than 40 feet wide and 12 feet
deep, but it ran 363 miles across the rugged
wilderness of upstate New York, the Erie Canal
connected

(B)

Seldom more than 40 feet wide or 12 feet
deep but running 363 miles across the rugged
wilderness of upstate New York, the Erie Canal
connected

(C)

It was seldom more than 40 feet wide and
12 feet deep, and ran 363 miles across the
rugged wilderness of upstate New York, but the
Erie Canal, connecting

(D)

The Erie Canal was seldom more than 40 feet
wide or 12 feet deep and it ran 363 miles across
the rugged wilderness of upstate New York,
which connected

(E)

The Erie Canal, seldom more than 40 feet wide
and 12 feet deep, but running 363 miles across
the rugged wilderness of upstate New York,
connecting

Logical predication; Grammatical construction

The phrase seldom…deep is the ﬁrst half of a
modiﬁer that describes the Erie Canal. However,
because a comma incorrectly follows deep, this
phrase appears to be the entire modiﬁer, which
must agree with the noun or pronoun that
immediately follows it. This phrase cannot modify
the conjunction but, and it has no referent; but it
ran is not a logical or grammatical construction
following the modifying phrase. Substituting
running for it ran creates an adjective phrase
parallel to the ﬁrst adjective phrase (seldom…
deep).To contrast the small size reported in the
ﬁrst phrase with the great distance reported in the
second, the two phrases may be joined with but;
together they create a single modiﬁer correctly
modifying the Erie Canal. The Erie Canal is then
the subject of the sentence and requires the verb
connected to provide a logical statement.

A
But it ran cannot logically or grammatically
follow the modifying phrase.

Seldom more than 40 feet wide and 12 feet deep,
but it ran 363 miles across the rugged wilderness
of upstate New York, the Erie Canal connected the
Hudson River at Albany to the Great Lakes at Buffalo,
providing the port of New York City with a direct water
link to the heartland of the North American continent.

D The paired concepts of width and depth
should be joined by and, not or; this
construction calls for two main clauses to
be separated by a comma after deep; which is
ambiguous.
E The two halves of the modiﬁer should not
be separated by a comma after deep; the
subject is awkwardly and confusingly placed
at a great distance from the predicate;
the use of connecting rather than connected
creates a sentence fragment.
The correct answer is B.
93.

In 1923, the Supreme Court declared a minimum wage
for women and children in the District of Columbia
as unconstitutional, and ruling that it was a form of
price‐ﬁxing and, as such, an abridgment of the right of
contract.
(A)

the Supreme Court declared a minimum wage
for women and children in the District of
Columbia as unconstitutional, and

(B)

the Supreme Court declared as unconstitutional
a minimum wage for women and children in the
District of Columbia, and

(C)

the Supreme Court declared unconstitutional a
minimum wage for women and children in the
District of Columbia,

(D)

a minimum wage for women and children
in the District of Columbia was declared
unconstitutional by the Supreme Court,

(E)

when the Supreme Court declared a minimum
wage for women and children in the District of
Columbia as unconstitutional,

3.6 Diagnostic Test Verbal Answer Explanations

Idiom; Grammatical construction

This sentence depends on the correct use of
an idiom: the court declares x unconstitutional.
The inverted form should be used here because
of the long phrases involved: the court declares
unconstitutional x. The Supreme Court is the subject
of the sentence; declared is the verb. Ruling…
contract acts as a modiﬁer describing the action
of the main clause; because the modiﬁer is
subordinate to the main clause, the conjunction
and must be omitted. And is used to join two
independent clauses, not a clause and its modiﬁer.
A
Declared…as unconstitutional is not the
correct idiom; the use of and creates an
ungrammatical construction.
B
Declared as unconstitutional is not the
correct idiom; the use of and creates an
ungrammatical construction.
C
Correct. In this sentence, the correct idiom
is used, and the modiﬁer is grammatically
and logically attached to the main clause.
D Passive voice construction is weak and
wordy; its use causes the modiﬁer to be
misplaced and ambiguous.
E
Declared…as unconstitutional is not the
correct idiom; when transforms the main
clause into a subordinate clause, resulting in
a sentence fragment.
The correct answer is C.

94.

Researchers have found that individuals who have been
blind from birth, and who thus have never seen anyone
gesture, nevertheless make hand motions when
speaking just as frequently and in virtually the same
way as sighted people do, and that they will gesture
even when conversing with another blind person.
(A)

who thus have never seen anyone gesture,
nevertheless make hand motions when speaking
just as frequently and in virtually the same way
as sighted people do, and that they will gesture

(B)

who thus never saw anyone gesturing,
nevertheless make hand motions when speaking
just as frequent and in virtually the same way as
sighted people did, and that they will gesture

(C)

who thus have never seen anyone gesture,
nevertheless made hand motions when speaking
just as frequently and in virtually the same way
as sighted people do, as well as gesturing

(D)

thus never having seen anyone gesture,
nevertheless made hand motions when speaking
just as frequent and in virtually the same way as
sighted people did, as well as gesturing

(E)

thus never having seen anyone gesture,
nevertheless to make hand motions when
speaking just as frequently and in virtually the
same way as sighted people do, and to gesture

Parallelism; Verb form; Diction

The researchers have found (1) that individuals…
make hand motions…as sighted people do and
(2) that they will gesture…with another blind
person. In the original sentence, the two ﬁndings
are reported in two parallel subordinate clauses
introduced by that. The verb tenses are logical and
parallel: who have been blind and who have never
seen indicate a condition that began in the past
and continues in the present; make and do refer
to present actions. The verb make (hand motions)
is correctly modiﬁed by the adverb frequently to
show how the action of the verb is carried out.
The emphatic future will gesture is properly used
here with even to emphasize the extreme or the
unexpected.

GMAT® Official Guide 2018

A
Correct. Although the original sentence is
complicated, the parallelism of its structure
and phrasing allows its meaning to be clear
and its expression effective.

Idiom; Grammatical construction

Two constructions create problems in the
original sentence. The ﬁrst is the unidiomatic
construction have the ability of developing; ability
must be followed by an inﬁnitive, to develop, not
a phrase. The second problematic construction
is to develop themselves into. In this biological
context, the verb develop means to progress from
an earlier to a later stage; it is used intransitively,
which means that it cannot take an object. The
pronoun themselves acts as an object, creating a
construction that is not grammatical or logical.
Omitting the pronoun removes the problem.

B Verbs saw and did indicate action completed
in the past; the simple past tense is not
appropriate in either case; the adjective
frequent cannot modify the verb; awkward
and muddy.
C
Made indicates past action, but the
present tense is logically required; as well
as gesturing violates the parallelism of the
two subordinate (that) clauses; choppy and
unclear.

Ability is incorrectly followed by of
A
developing; a pronoun cannot follow
develop, when it is used, as it is here, in its
intransitive sense.

D
Having seen is not parallel to have been;
made and did do not show ongoing action;
frequent incorrectly modiﬁes the verb; as
well as gesturing destroys the parallelism of
the two subordinate (that) clauses; awkward
and unclear.

B
Correct. Ability is properly followed by the
inﬁnitive in this sentence, and the pronoun
themselves is omitted.
C This awkward and wordy construction
violates the parallelism of like embryonic
germ cells…embryonic stem cells….

E Replacing the verb make with the inﬁnitive
to make results in an ungrammatical
construction that fails to complete the
sentence.

D The two parts of the comparison must be
parallel; like embryonic germ cells must be
followed by embryonic stem cells, not the
ability to develop.

The correct answer is A.
95.

Like embryonic germ cells, which are cells that
develop early in the formation of the fetus and that
later generate eggs or sperm, embryonic stem cells
have the ability of developing themselves into different
kinds of body tissue.
(A)

embryonic stem cells have the ability of
developing themselves into different kinds of
body tissue

(B)

embryonic stem cells have the ability to develop
into different kinds of body tissue

(C)

in embryonic stem cells there is the ability to
develop into different kinds of body tissue

(D)

the ability to develop themselves into different
kinds of body tissue characterizes embryonic
stem cells

(E)

the ability of developing into different kinds of
body tissue characterizes embryonic stem cells

E
Ability is followed by the unidiomatic of
developing rather than to develop; the main
clause must begin with embryonic stem cells to
balance and complete like embryonic germ cells.
The correct answer is B.
96.

Critics contend that the new missile is a weapon
whose importance is largely symbolic, more a tool for
manipulating people’s perceptions than to fulﬁll a real
military need.
(A)

for manipulating people’s perceptions than to fulﬁll

(B)

for manipulating people’s perceptions than for
fulﬁlling

(C)

to manipulate people’s perceptions rather than
that it fulﬁlls

(D)

to manipulate people’s perceptions rather than
fulﬁlling

(E)

to manipulate people’s perceptions than for
fulﬁlling

3.6 Diagnostic Test Verbal Answer Explanations

Parallelism

Logical predication

This sentence uses the comparative construction
more x than y where x and y must be parallel.
Here, x is a tool for manipulating people’s
perceptions, and y is to fulﬁll a real military need. A
tool does not need to be repeated in the second
half of the comparison because it is understood,
but the wording of the two phrases does need to
match. There are two acceptable solutions: (1) for
manipulating can be followed by for fulﬁlling or
(2) to manipulate can be followed by to fulﬁll.

The original sentence contains a number of
modiﬁers, but not all of them are correctly
expressed. The clause who trained… describes
Stella Adler, yet a relative clause such as this
one must be placed immediately after the noun
or pronoun it modiﬁes, and this clause follows
theater rather than Adler. Replacing who trained
with training corrects the error because the phrase
training… modiﬁes the whole preceding clause
rather than the single preceding noun. Several
generations of actors including shows the same error
in reverse; including modiﬁes the whole phrase,
but the two actors named are not generations
of actors. The more limiting clause whose ranks
included (referring to actors) is appropriate here.

For manipulating is not parallel to to fulﬁll.

B
Correct. For manipulating and for fulﬁlling
are parallel in this sentence.
C

To manipulate is not parallel to that it fulﬁlls.

To manipulate is not parallel to fulﬁlling.

To manipulate is not parallel to for fulﬁlling.

The correct answer is B.
97.

As an actress and, more importantly, as a teacher
of acting, Stella Adler was one of the most inﬂuential
artists in the American theater, who trained several
generations of actors including Marlon Brando and
Robert De Niro.
(A)

Stella Adler was one of the most inﬂuential
artists in the American theater, who trained
several generations of actors including

(B)

Stella Adler, one of the most inﬂuential artists
in the American theater, trained several
generations of actors who include

(C)

Stella Adler was one of the most inﬂuential
artists in the American theater, training several
generations of actors whose ranks included

(D)

one of the most inﬂuential artists in the American
theater was Stella Adler, who trained several
generations of actors including

(E)

one of the most inﬂuential artists in the American
theater, Stella Adler, trained several generations
of actors whose ranks included

A Relative (who) clause follows theater rather
than Adler; including refers to generations of
actors, when the reference should be to actors
only.
B This construction, in which the subject is
both preceded and followed by modiﬁers, is
awkward; the verbs should be consistently in
the past tense, but include is present tense.
C
Correct. In this sentence, substituting
training for who trained and whose ranks
included for including eliminates the
modiﬁcation errors.
D Introductory modiﬁer must be immediately
followed by Stella Adler, not one…; including
refers to generations of actors rather than to
actors only.
E Introductory modiﬁer must be immediately
followed by Stella Adler, not one.
The correct answer is C.

GMAT® Official Guide 2018

98.

By developing the Secure Digital Music Initiative, the
recording industry associations of North America,
Japan, and Europe hope to create a standardized way
of distributing songs and full‐length recordings on the
Internet that will protect copyright holders and foil the
many audio pirates who copy and distribute digital
music illegally.

A
Correct. The verbs will protect and (will )
foil are parallel in this sentence, as are the
verbs copy and distribute.

(A)

of distributing songs and full‐length recordings
on the Internet that will protect copyright holders
and foil the many audio pirates who copy and
distribute

(B)

of distributing songs and full‐length recordings
on the Internet and to protect copyright holders
and foiling the many audio pirates copying and
distributing

C
Way for should instead be way of; the
pronoun reference in while it protects is
ambiguous; construction suggests that
protection comes from something other
than the standardized way.

(C)

(D)

(E)

D Pronoun they has no referent; use of while
suggests that protection comes from
something other than the standardized way
of distribution.

for distributing songs and full‐length recordings
on the Internet while it protects copyright holders
and foils the many audio pirates who copy and
distribute

And it will protect distorts meaning,
E
suggesting that protection comes in addition
to the standardized way; will protect and
foiling are not parallel.

to distribute songs and full‐length recordings
on the Internet while they will protect copyright
holders and foil the many audio pirates copying
and distributing
to distribute songs and full‐length recordings on
the Internet and it will protect copyright holders
and foiling the many audio pirates who copy and
distribute

Parallelism

The original sentence depends on the parallelism
of its verbs to make its point clearly and
effectively. A standardized way…will protect and
(will understood) foil; pirates…copy and distribute.
In the ﬁrst pair of parallel verbs, will does not
need to be repeated because it is understood.

B
And to protect distorts meaning, suggesting
that protection comes in addition to the
standardized way; foiling is not parallel to to
protect.

The correct answer is A.
99.

Whereas a ramjet generally cannot achieve high
speeds without the initial assistance of a rocket, high
speeds can be attained by scramjets, or supersonic
combustion ramjets, in that they reduce airﬂow
compression at the entrance of the engine and letting
air pass through at supersonic speeds.
(A)

high speeds can be attained by scramjets, or
supersonic combustion ramjets, in that they
reduce

(B)

that high speeds can be attained by scramjets,
or supersonic combustion ramjets, is a result of
their reducing

(C)

the ability of scramjets, or supersonic
combustion ramjets, to achieve high speeds is
because they reduce

(D)

scramjets, or supersonic combustion ramjets,
have the ability of attaining high speeds when
reducing

(E)

scramjets, or supersonic combustion ramjets,
can attain high speeds by reducing

3.6 Diagnostic Test Verbal Answer Explanations

Rhetorical construction

Diction; Parallelism

The underlined portion of the original sentence
is wordy and ineffective. Transforming it from
passive (high speeds can be attained by scramjets)
to active voice (scramjets can attain high speeds)
eliminates much of the problem. As the subject
of the main clause, scramjets correctly parallels a
ramjet, the subject of the subordinate clause; the
contrast is thus clearly and effectively drawn. In
that they reduce is wordy and awkward; it can be
replaced by the more concise phrase by reducing.

The preposition like introduces nouns and noun
phrases; the conjunction as introduces verbs or
clauses, so as is required here. The comparative
construction used here is just as x so y; x and
y must be parallel. The y clause is written in
effective subject‐verb‐object order: melting sea
ice does not increase oceanic volume. The original
wordy, awkward x clause is not parallel. To make
it parallel, melting ice cubes should be the subject
of the clause, do not cause…to overﬂow the verb
phrase, and a glass of water the object.

A Passive voice contributes to a wordy,
awkward, and ineffective construction; in
that they reduce is also wordy and awkward.
B Passive voice and subordinate (that) clause
constructions are wordy, awkward, and
ineffective.
C
The ability…is because is not a grammatical
construction; scramjets, not the ability, should
be parallel to a ramjet.
Have the ability of attaining is wordy; when
D
does not indicate the cause‐and‐effect
relationship.
E
Correct. Scramjets parallels a ramjet for an
effective contrast in this sentence; the active
voice is clear and concise; by reducing shows
how scramjets attain high speeds.

A
Like is used in place of as; the two elements
of comparison are not parallel.
B
Like is used in place of as; that violates
parallelism.
C
As or just as is needed to introduce the
clause; the two clauses are not parallel.
D
That violates the parallelism of the two
clauses and creates an ungrammatical
construction.
E
Correct. This sentence has just as properly
introducing the ﬁrst clause, and the two
clauses are parallel.
The correct answer is E.

The correct answer is E.
100. It will not be possible to implicate melting sea ice in
the coastal ﬂooding that many global warming models
have projected: just like a glass of water that will not
overﬂow due to melting ice cubes, so melting sea ice
does not increase oceanic volume.
(A)

like a glass of water that will not overﬂow due to
melting ice cubes,

(B)

like melting ice cubes that do not cause a glass
of water to overﬂow,

(C)

a glass of water will not overﬂow because of
melting ice cubes,

(D)

as melting ice cubes that do not cause a glass
of water to overﬂow,

(E)

as melting ice cubes do not cause a glass of
water to overﬂow,

4.0 Math Review

100

4.0 Math Review

4.0 Math Review
Although this chapter provides a review of some of the mathematical concepts of arithmetic, algebra,
and geometry, it is not intended to be a textbook. You should use this chapter to familiarize yourself
with the kinds of topics that may be tested in the GMAT® exam. You may wish to consult an
arithmetic, algebra, or geometry book for a more detailed discussion of some of the topics.
Section 4.1, “Arithmetic,” includes the following topics:
1. Properties of Integers

7. Powers and Roots of Numbers

2. Fractions

8. Descriptive Statistics

3. Decimals

9. Sets

4. Real Numbers

10. Counting Methods

5. Ratio and Proportion

11. Discrete Probability

6. Percents
Section 4.2, “Algebra,” does not extend beyond what is usually covered in a first‐year high school algebra
course. The topics included are as follows:
1. Simplifying Algebraic Expressions

6. Solving Quadratic Equations

2. Equations

7. Exponents

3. Solving Linear Equations with One
Unknown

8. Inequalities

4. Solving Two Linear Equations with
Two Unknowns

9. Absolute Value
10. Functions

5. Solving Equations by Factoring
Section 4.3, “Geometry,” is limited primarily to measurement and intuitive geometry or spatial
visualization. Extensive knowledge of theorems and the ability to construct proofs, skills that are usually
developed in a formal geometry course, are not tested. The topics included in this section are the following:
1. Lines

6. Triangles

2. Intersecting Lines and Angles

7. Quadrilaterals

3. Perpendicular Lines

8. Circles

4. Parallel Lines

9. Rectangular Solids and Cylinders

5. Polygons (Convex)

10. Coordinate Geometry

Section 4.4, “Word Problems,” presents examples of and solutions to the following types of word problems:
1. Rate Problems

6. Profit

2. Work Problems

7. Sets

3. Mixture Problems

8. Geometry Problems

4. Interest Problems

9. Measurement Problems

5. Discount

10. Data Interpretation
101

GMAT® Ofﬁcial Guide 2018

4.1 Arithmetic
1. Properties of Integers
An integer is any number in the set {. . . −3, −2, −1, 0, 1, 2, 3, . . .}. If x and y are integers and x ≠ 0, then
x is a divisor (factor) of y provided that y = xn for some integer n. In this case, y is also said to be divisible
by x or to be a multiple of x. For example, 7 is a divisor or factor of 28 since 28 = (7)(4), but 8 is not a
divisor of 28 since there is no integer n such that 28 = 8n.
If x and y are positive integers, there exist unique integers q and r, called the quotient and remainder,
respectively, such that y = xq + r and 0 ≤ r < x. For example, when 28 is divided by 8, the quotient is 3
and the remainder is 4 since 28 = (8)(3) + 4. Note that y is divisible by x if and only if the remainder r
is 0; for example, 32 has a remainder of 0 when divided by 8 because 32 is divisible by 8. Also, note that
when a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller
integer. For example, 5 divided by 7 has the quotient 0 and the remainder 5 since 5 = (7)(0) + 5.
Any integer that is divisible by 2 is an even integer; the set of even integers is
{. . . −4, −2, 0, 2, 4, 6, 8, . . .}. Integers that are not divisible by 2 are odd integers;
{. . . −3, −1, 1, 3, 5, . . .} is the set of odd integers.
If at least one factor of a product of integers is even, then the product is even; otherwise the product is
odd. If two integers are both even or both odd, then their sum and their difference are even. Otherwise,
their sum and their difference are odd.
A prime number is a positive integer that has exactly two different positive divisors, 1 and itself. For
example, 2, 3, 5, 7, 11, and 13 are prime numbers, but 15 is not, since 15 has four different positive
divisors, 1, 3, 5, and 15. The number 1 is not a prime number since it has only one positive divisor. Every
integer greater than 1 either is prime or can be uniquely expressed as a product of prime factors. For
example, 14 = (2)(7), 81 = (3)(3)(3)(3), and 484 = (2)(2)(11)(11).
The numbers −2, −1, 0, 1, 2, 3, 4, 5 are consecutive integers. Consecutive integers can be represented by
n, n + 1, n + 2, n + 3, . . ., where n is an integer. The numbers 0, 2, 4, 6, 8 are consecutive even integers, and 1,
3, 5, 7, 9 are consecutive odd integers. Consecutive even integers can be represented by 2n, 2n + 2, 2n + 4, . . .,
and consecutive odd integers can be represented by 2n + 1, 2n + 3, 2n + 5, . . ., where n is an integer.
Properties of the integer 1. If n is any number, then 1 ⋅ n = n, and for any number n ≠ 0, n ⋅ 1 = 1 .
n
n
The number 1 can be expressed in many ways; for example, = 1 for any number n ≠ 0.
n
Multiplying or dividing an expression by 1, in any form, does not change the value of that expression.
Properties of the integer 0. The integer 0 is neither positive nor negative. If n is any number, then n + 0 = n
and n ⋅ 0 = 0. Division by 0 is not defined.

102

4.1 Math Review Arithmetic

2. Fractions
In a fraction n , n is the numerator and d is the denominator. The denominator of a fraction can never be
d
0, because division by 0 is not defined.
14
are
Two fractions are said to be equivalent if they represent the same number. For example, 8 and
63
36
equivalent since they both represent the number 2 . In each case, the fraction is reduced to lowest terms
9
by dividing both numerator and denominator by their greatest common divisor (gcd). The gcd of 8 and 36
is 4 and the gcd of 14 and 63 is 7.

Addition and subtraction of fractions.
Two fractions with the same denominator can be added or subtracted by performing the required
3 4 3+ 4 = 7
operation with the numerators, leaving the denominators the same. For example, + =
5
5
5
5
5 2 5− 2 = 3
and − =
. If two fractions do not have the same denominator, express them as equivalent
7 7
7
7
3
4
and , multiply the numerator and
fractions with the same denominator. For example, to add
5
7
denominator of the first fraction by 7 and the numerator and denominator of the second fraction by 5,
21
21 + 20 = 41
20
obtaining
, respectively;
and
.
35
35 35 35
35
For the new denominator, choosing the least common multiple (lcm) of the denominators usually lessens
the work. For 2 + 1 , the lcm of 3 and 6 is 6 (not 3 × 6 = 18), so 2 + 1 = 2 × 2 + 1 = 4 + 1 = 5 .
3 6
3 6 3 2 6 6 6 6

Multiplication and division of fractions.
To multiply two fractions, simply multiply the two numerators and multiply the two denominators.
For example, 2 × 4 = 2 × 4 = 8 .
3 7 3 × 7 21
To divide by a fraction, invert the divisor (that is, find its reciprocal ) and multiply. For example,
2 ÷ 4 = 2 × 7 = 14 = 7 .
3 7 3 4 12 6
In the problem above, the reciprocal of 4 is 7 . In general, the reciprocal of a fraction n is d , where n
7
4
d
n
and d are not zero.

Mixed numbers.
A number that consists of a whole number and a fraction, for example, 7 2 , is a mixed number: 7 2
3
3
means 7 + 2 .
3
To change a mixed number into a fraction, multiply the whole number by the denominator of the
fraction and add this number to the numerator of the fraction; then put the result over the denominator
( 3 × 7 ) + 2 = 23 .
of the fraction. For example, 7 2 =
3
3
3

103

GMAT® Ofﬁcial Guide 2018

3. Decimals

Thousandths

Hundredths

Tenths

Ones or units

Tens

Hundreds

Thousands

In the decimal system, the position of the period or decimal point determines the place value of the
digits. For example, the digits in the number 7,654.321 have the following place values:

Some examples of decimals follow.
0.321 = 3 + 2 + 1 = 321
10 100 1, 000 1, 000
0.0321 = 0 + 3 + 2 + 1 = 321
10 100 1, 000 10, 000 10, 000
1.56 = 1 + 5 + 6 = 156
10 100 100
Sometimes decimals are expressed as the product of a number with only one digit to the left of the
decimal point and a power of 10. This is called scientific notation. For example, 231 can be written as
2.31 × 102 and 0.0231 can be written as 2.31 × 10−2. When a number is expressed in scientific notation,
the exponent of the 10 indicates the number of places that the decimal point is to be moved in the
number that is to be multiplied by a power of 10 in order to obtain the product. The decimal point is
moved to the right if the exponent is positive and to the left if the exponent is negative. For example,
2.013 × 104 is equal to 20,130 and 1.91 × 10−4 is equal to 0.000191.

Addition and subtraction of decimals.
To add or subtract two decimals, the decimal points of both numbers should be lined up. If one of the
numbers has fewer digits to the right of the decimal point than the other, zeros may be inserted to the
right of the last digit. For example, to add 17.6512 and 653.27, set up the numbers in a column and add:
17.6512
+ 653.2700
670.9212
Likewise for 653.27 minus 17.6512:
653.2700
−17.6512
635.6188

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Multiplication of decimals.
To multiply decimals, multiply the numbers as if they were whole numbers and then insert the decimal
point in the product so that the number of digits to the right of the decimal point is equal to the sum of
the numbers of digits to the right of the decimal points in the numbers being multiplied. For example:
2.09 (2 digits to the right)
× 1.3 (l digit to the right)
627
2090
2.717 (2 + 1 = 3 digits to the right)

Division of decimals.
To divide a number (the dividend) by a decimal (the divisor), move the decimal point of the divisor to
the right until the divisor is a whole number. Then move the decimal point of the dividend the same
number of places to the right, and divide as you would by a whole number. The decimal point in the
quotient will be directly above the decimal point in the new dividend. For example, to divide 698.12
by 12.4:

will be replaced by:
and the division would proceed as follows:

4. Real Numbers
All real numbers correspond to points on the number line and all points on the number line correspond
to real numbers. All real numbers except zero are either positive or negative.
−

3
2

–6 –5 –4 –3 –2 –1

0.2
0

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On a number line, numbers corresponding to points to the left of zero are negative and numbers
corresponding to points to the right of zero are positive. For any two numbers on the number line, the
number to the left is less than the number to the right; for example, −4 < −3 < − 3 < −1, and 1 < 2 < 2.
2
To say that the number n is between 1 and 4 on the number line means that n > 1 and n < 4, that is, 1
< n < 4. If n is “between 1 and 4, inclusive,” then 1 ≤ n ≤ 4.
The distance between a number and zero on the number line is called the absolute value of the number.
Thus 3 and −3 have the same absolute value, 3, since they are both three units from zero. The absolute
value of 3 is denoted |3|. Examples of absolute values of numbers are
| −5| = | 5| = 5, − 7 = 7 , and | 0 | = 0.
2 2
Note that the absolute value of any nonzero number is positive.
Here are some properties of real numbers that are used frequently. If x, y, and z are real numbers, then
(1) x + y = y + x and xy = yx.
For example, 8 + 3 = 3 + 8 = 11, and (l7)(5) = (5)(l7) = 85.
(2) (x + y) + z = x + (y + z) and (xy)z = x(yz).
For example, (7 + 5) + 2 = 7 + (5 + 2) = 7 + (7) = 14, and 5 3

(

)( 3 ) = ( 5) (

(3) xy + xz = x(y + z).
For example, 718(36) + 718(64) = 718(36 + 64) = 718(l00) = 71,800.

)

3 3 = ( 5 ) ( 3) = 15.

(4) If x and y are both positive, then x + y and xy are positive.
(5) If x and y are both negative, then x + y is negative and xy is positive.
(6) If x is positive and y is negative, then xy is negative.
(7) If xy = 0, then x = 0 or y = 0. For example, 3y = 0 implies y = 0.
(8) |x + y| ≤ |x| + |y|. For example, if x = 10 and y = 2, then |x + y| = |12| = 12 = |x| + |y|;
and if x = 10 and y = −2, then |x + y| = |8| = 8 < 12 = |x| + |y|.

5. Ratio and Proportion
The ratio of the number a to the number b (b ≠ 0) is

a
.
b

A ratio may be expressed or represented in several ways. For example, the ratio of 2 to 3 can be written
as 2 to 3, 2:3, or 2 . The order of the terms of a ratio is important. For example, the ratio of the number
3
of months with exactly 30 days to the number with exactly 31 days is 4 , not 7 .
7
4

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A proportion is a statement that two ratios are equal; for example, 2 = 8 is a proportion. One way to
3 12
solve a proportion involving an unknown is to cross multiply, obtaining a new equality. For example, to
solve for n in the proportion 2 = n , cross multiply, obtaining 24 = 3n; then divide both sides by 3, to
3 12
get n = 8.

6. Percents
Percent means per hundred or number out of 100. A percent can be represented as a fraction with a
denominator of 100, or as a decimal. For example:
37% = 37 = 0.37.
100
To find a certain percent of a number, multiply the number by the percent expressed as a decimal or
fraction. For example:
20% of 90 = 0.2 × 90 = 18
or
20% of 90 = 20 × 90 = 1 × 90 = 18.
100
5

Percents greater than 100%.
Percents greater than 100% are represented by numbers greater than 1. For example:
300% = 300 = 3
100
250% of 80 = 2.5 × 80 = 200.

Percents less than 1%.
The percent 0.5% means 1 of 1 percent. For example, 0.5% of 12 is equal to 0.005 × 12 = 0.06.
2

Percent change.
Often a problem will ask for the percent increase or decrease from one quantity to another quantity.
For example, “If the price of an item increases from $24 to $30, what is the percent increase in price?”
To find the percent increase, first find the amount of the increase; then divide this increase by the
original amount, and express this quotient as a percent. In the example above, the percent increase
would be found in the following way: the amount of the increase is (30 − 24) = 6. Therefore, the
percent increase is 6 = 0.25 = 25% .
24
Likewise, to find the percent decrease (for example, the price of an item is reduced from $30 to $24),
first find the amount of the decrease; then divide this decrease by the original amount, and express this
quotient as a percent. In the example above, the amount of decrease is (30 − 24) = 6.
Therefore, the percent decrease is 6 = 0.20 = 20%.
30
Note that the percent increase from 24 to 30 is not the same as the percent decrease from 30 to 24.
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In the following example, the increase is greater than 100 percent: If the cost of a certain house in 1983
was 300 percent of its cost in 1970, by what percent did the cost increase?
If n is the cost in 1970, then the percent increase is equal to 3n − n = 2n = 2 , or 200%.
n
n

7. Powers and Roots of Numbers
When a number k is to be used n times as a factor in a product, it can be expressed as kn, which means
the nth power of k. For example, 2 2 = 2 × 2 = 4 and 23 = 2 × 2 × 2 = 8 are powers of 2.
Squaring a number that is greater than 1, or raising it to a higher power, results in a larger number;
squaring a number between 0 and 1 results in a smaller number. For example:
32 = 9
2

⎛ 1⎞ = 1
⎝ 3⎠
9
2
( 0.1) = 0.01

(9 > 3)
⎛ 1 < 1⎞
⎝ 9 3⎠
(0.01 < 0.1)

A square root of a number n is a number that, when squared, is equal to n. The square root of a negative
number is not a real number. Every positive number n has two square roots, one positive and the other
negative, but n denotes the positive number whose square is n. For example, 9 denotes 3. The two
square roots of 9 are 9 = 3 and − 9 = −3.
Every real number r has exactly one real cube root, which is the number s such that s3 = r. The real cube
root of r is denoted by 3 r . Since 23 = 8, 3 8 = 2 . Similarly, 3 −8 = −2 , because (−2)3 = −8.

8. Descriptive Statistics
A list of numbers, or numerical data, can be described by various statistical measures. One of the most
common of these measures is the average, or (arithmetic) mean, which locates a type of “center” for the
data. The average of n numbers is defined as the sum of the n numbers divided by n. For example, the
average of 6, 4, 7, 10, and 4 is 6 + 4 + 7 + 10 + 4 = 31 = 6.2.
5
5
The median is another type of center for a list of numbers. To calculate the median of n numbers, first
order the numbers from least to greatest; if n is odd, the median is defined as the middle number,
whereas if n is even, the median is defined as the average of the two middle numbers. In the example
above, the numbers, in order, are 4, 4, 6, 7, 10, and the median is 6, the middle number.
For the numbers 4, 6, 6, 8, 9, 12, the median is 6 + 8 = 7. Note that the mean of these numbers is 7.5.
2
The median of a set of data can be less than, equal to, or greater than the mean. Note that for a large set
of data (for example, the salaries of 800 company employees), it is often true that about half of the data
is less than the median and about half of the data is greater than the median; but this is not always the
case, as the following data show.
3, 5, 7, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 10, 10
Here the median is 7, but only

108

2
of the data is less than the median.
15

4.1 Math Review Arithmetic

The mode of a list of numbers is the number that occurs most frequently in the list. For example, the
mode of 1, 3, 6, 4, 3, 5 is 3. A list of numbers may have more than one mode. For example, the list
1, 2, 3, 3, 3, 5, 7, 10, 10, 10, 20 has two modes, 3 and 10.
The degree to which numerical data are spread out or dispersed can be measured in many ways. The
simplest measure of dispersion is the range, which is defined as the greatest value in the numerical data
minus the least value. For example, the range of 11, 10, 5, 13, 21 is 21 − 5 = 16. Note how the range
depends on only two values in the data.
One of the most common measures of dispersion is the standard deviation. Generally speaking, the
more the data are spread away from the mean, the greater the standard deviation. The standard deviation
of n numbers can be calculated as follows: (1) find the arithmetic mean, (2) find the differences between
the mean and each of the n numbers, (3) square each of the differences, (4) find the average of the
squared differences, and (5) take the nonnegative square root of this average. Shown below is this
calculation for the data 0, 7, 8, 10, 10, which have arithmetic mean 7.

x−7

(x − 7)2

−7

Total

Standard deviation

68 ≈ 3.7
5

Notice that the standard deviation depends on every data value, although it depends most on values
that are farthest from the mean. This is why a distribution with data grouped closely around the mean
will have a smaller standard deviation than will data spread far from the mean. To illustrate this,
compare the data 6, 6, 6.5, 7.5, 9, which also have mean 7. Note that the numbers in the second set of
data seem to be grouped more closely around the mean of 7 than the numbers in the first set. This is
reflected in the standard deviation, which is less for the second set (approximately 1.1) than for the first
set (approximately 3.7).
There are many ways to display numerical data that show how the data are distributed. One simple
way is with a frequency distribution, which is useful for data that have values occurring with varying
frequencies. For example, the 20 numbers
−4
−1

0
−5

0
0

−3
−2

−2
0

−1
−5

−1
−2

0
0

−1
0

−4
−1

are displayed on the next page in a frequency distribution by listing each different value x and the
frequency f with which x occurs.

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Data Value
x

Frequency
f

−5

−4

−3

−2

−1

Total

From the frequency distribution, one can readily compute descriptive statistics:
Mean: =

( −5) ( 2) + ( −4 )( 2) + ( −3)(1) + ( −2)( 3) + ( −1) ( 5) + ( 0 )( 7 ) = −1.6
20

Median: −1 (the average of the 10th and 11th numbers)
Mode: 0 (the number that occurs most frequently)
Range: 0 − (−5) = 5
Standard deviation:

( −5 + 1.6)2 ( 2) + ( −4 + 1.6)2 ( 2 ) +…+ ( 0 + 1.6)2 (7 ) ≈ 1.7
20

9. Sets
In mathematics a set is a collection of numbers or other objects. The objects are called the elements of the
set. If S is a set having a finite number of elements, then the number of elements is denoted by |S|. Such
a set is often defined by listing its elements; for example, S = {−5, 0, l} is a set with |S| = 3.
The order in which the elements are listed in a set does not matter; thus {−5, 0, l} = {0, 1, − 5}.
If all the elements of a set S are also elements of a set T, then S is a subset of T; for example,
S = {−5, 0, 1} is a subset of T = {−5, 0, 1, 4, 10}.
For any two sets A and B, the union of A and B is the set of all elements that are in A or in B or in both.
The intersection of A and B is the set of all elements that are both in A and in B. The union is denoted
by A ∪ B and the intersection is denoted by A ∩ B. As an example, if A = {3, 4} and B = {4, 5, 6}, then
A ∪ B = {3, 4, 5, 6} and A ∩ B = {4}. Two sets that have no elements in common are said to be disjoint
or mutually exclusive.

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The relationship between sets is often illustrated with a Venn diagram in which sets are represented by
regions in a plane. For two sets S and T that are not disjoint and neither is a subset of the other, the
intersection S ∩ T is represented by the shaded region of the diagram below.
S

This diagram illustrates a fact about any two finite sets S and T: the number of elements in their
union equals the sum of their individual numbers of elements minus the number of elements in their
intersection (because the latter are counted twice in the sum); more concisely,
| S ∪ T | = | S | + | T | − | S ∩ T |.
This counting method is called the general addition rule for two sets. As a special case, if S and T are
disjoint, then
| S ∪ T | = | S | + |T |
since | S ∩ T | = 0.

10. Counting Methods
There are some useful methods for counting objects and sets of objects without actually listing the
elements to be counted. The following principle of multiplication is fundamental to these methods.
If an object is to be chosen from a set of m objects and a second object is to be chosen from a different
set of n objects, then there are mn ways of choosing both objects simultaneously.
As an example, suppose the objects are items on a menu. If a meal consists of one entree and one dessert
and there are 5 entrees and 3 desserts on the menu, then there are 5 × 3 = 15 different meals that can
be ordered from the menu. As another example, each time a coin is flipped, there are two possible
outcomes, heads and tails. If an experiment consists of 8 consecutive coin flips, then the experiment has
28 possible outcomes, where each of these outcomes is a list of heads and tails in some order.
A symbol that is often used with the multiplication principle is the factorial. If n is an integer greater
than 1, then n factorial, denoted by the symbol n!, is defined as the product of all the integers from 1 to
n. Therefore,
2! = (1)(2) = 2,
3! = (1)(2)(3) = 6,
4! = (1)(2)(3)(4) = 24, etc.
Also, by definition, 0! = 1! = 1.
The factorial is useful for counting the number of ways that a set of objects can be ordered. If a set of
n objects is to be ordered from 1st to nth, then there are n choices for the 1st object, n − 1 choices for
the 2nd object, n − 2 choices for the 3rd object, and so on, until there is only 1 choice for the nth object.
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Thus, by the multiplication principle, the number of ways of ordering the n objects is
n(n − 1)(n − 2)(3)(2)(1) = n !.
For example, the number of ways of ordering the letters A, B, and C is 3!, or 6:
ABC, ACB, BAC, BCA, CAB, and CBA.
These orderings are called the permutations of the letters A, B, and C.
A permutation can be thought of as a selection process in which objects are selected one by one in a
certain order. If the order of selection is not relevant and only k objects are to be selected from a larger
set of n objects, a different counting method is employed.
Specifically, consider a set of n objects from which a complete selection of k objects is to be made
without regard to order, where 0 ≤ k ≤ n. Then the number of possible complete selections of k objects is
called the number of combinations of n objects taken k at a time and is denoted by ⎛ n ⎞ .
⎝ k⎠
n!
.
The value of ⎛ n ⎞ is given by ⎛ n ⎞ =
⎝ k ⎠ k ! (n − k ) !
⎝ k⎠
Note that ⎛ n ⎞ is the number of k‐element subsets of a set with n elements. For example,
⎝ k⎠
if S = {A, B, C, D, E}, then the number of 2‐element subsets of S, or the number of combinations of
5 letters taken 2 at a time, is ⎛ 5 ⎞ = 5! = 120 = 10 .
⎝ 2 ⎠ 2! 3! ( 2 )( 6 )
The subsets are {A, B}, {A, C}, {A, D}, {A, E}, {B, C}, {B, D}, {B, E}, {C, D}, {C, E}, and {D, E}. Note
that ⎛ 5 ⎞ = 10 = ⎛ 5 ⎞ because every 2‐element subset chosen from a set of 5 elements corresponds to a
⎝ 2⎠
⎝ 3⎠
unique 3‐element subset consisting of the elements not chosen.
In general, ⎛ n ⎞ = ⎛ n ⎞ .
⎝ k⎠ ⎝ n − k⎠

11. Discrete Probability
Many of the ideas discussed in the preceding three topics are important to the study of discrete
probability. Discrete probability is concerned with experiments that have a finite number of outcomes.
Given such an experiment, an event is a particular set of outcomes. For example, rolling a number cube
with faces numbered 1 to 6 (similar to a 6‐sided die) is an experiment with 6 possible outcomes:
1, 2, 3, 4, 5, or 6. One event in this experiment is that the outcome is 4, denoted {4}; another event is
that the outcome is an odd number: {1, 3, 5}.
The probability that an event E occurs, denoted by P (E), is a number between 0 and 1, inclusive.
If E has no outcomes, then E is impossible and P (E) = 0; if E is the set of all possible outcomes of the
experiment, then E is certain to occur and P (E) = 1. Otherwise, E is possible but uncertain, and
0 < P (E) < 1. If F is a subset of E, then P (F) ≤ P (E). In the example above, if the probability of each of
the 6 outcomes is the same, then the probability of each outcome is 1 , and the outcomes are said to be
6

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equally likely. For experiments in which all the individual outcomes are equally likely, the probability of
an event E is
P (E ) =

The number of outcomes in E
.
The total number of possible outcomes

In the example, the probability that the outcome is an odd number is
P ( {1, 3, 5} ) =

|{1, 3, 5}| 3 1
= = .
6
6 2

Given an experiment with events E and F, the following events are defined:
“not E ” is the set of outcomes that are not outcomes in E;
“E or F ” is the set of outcomes in E or F or both, that is, E ∪ F;
“E and F ” is the set of outcomes in both E and F, that is, E ∩ F.
The probability that E does not occur is P (not E) = l − P(E). The probability that “E or F ” occurs is
P (E or F ) = P (E) + P ( F ) − P (E and F ), using the general addition rule at the end of section 4.1.9
(“Sets”). For the number cube, if E is the event that the outcome is an odd number, {1, 3, 5}, and F is
the event that the outcome is a prime number, {2, 3, 5}, then P ( E and F ) = P ( {3, 5} ) = 2 = 1 and so
6 3
3
3
2
4
2
.
P ( E or F ) = P ( E ) + P ( F ) − P ( E and F ) = + − = =
6 6 6 6 3
|{1, 2, 3, 5}| 4 2
= = .
Note that the event “E or F ” is E ∪ F = {1, 2, 3, 5}, and hence P ( E or F ) =
6
6 3
If the event “E and F ” is impossible (that is, E ∩ F has no outcomes), then E and F are said to be
mutually exclusive events, and P(E and F ) = 0. Then the general addition rule is reduced to
P (E or F ) = P (E) + P (F ).
This is the special addition rule for the probability of two mutually exclusive events.
Two events A and B are said to be independent if the occurrence of either event does not alter the
probability that the other event occurs. For one roll of the number cube, let A = {2, 4, 6} and let
| A| 3 1
= = , while, presuming B occurs, the
B = {5, 6}. Then the probability that A occurs is P ( A ) =
6
6 2
probability that A occurs is
| A ∩ B | |{6}|
=
= 1.
|B|
|{5, 6}| 2
|B| 2 1
= = , while, presuming A occurs, the probability
Similarly, the probability that B occurs is P ( B ) =
6
6 3
that B occurs is
|B ∩ A|
|{6}|
=
= 1.
| A|
|{2, 4, 6}| 3
Thus, the occurrence of either event does not affect the probability that the other event occurs.
Therefore, A and B are independent.

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The following multiplication rule holds for any independent events E and F: P (E and F ) = P (E)P (F ).
For the independent events A and B above, P ( A and B ) = P ( A )P ( B ) = ⎛ 1 ⎞ ⎛ 1 ⎞ = ⎛ 1 ⎞ .
⎝ 2⎠ ⎝ 3⎠ ⎝ 6⎠
Note that the event “A and B” is A ∩ B = {6}, and hence P ( A and B ) = P ( {6} ) = 1 . It follows from the
6
general addition rule and the multiplication rule above that if E and F are independent, then
P ( E or F ) = P ( E ) + P ( F ) − P ( E )P ( F ).
For a final example of some of these rules, consider an experiment with events A, B, and C for which
P (A) = 0.23, P (B) = 0.40, and P(C) = 0.85. Also, suppose that events A and B are mutually exclusive and
events B and C are independent. Then
P ( A or B ) = P ( A ) + P ( B ) (since A or B are mutually exclusive)
= 0.23 + 0.40
= 0.63
P ( B or C ) = P ( B ) + P (C ) − P ( B )P (C ) (by independence)
= 0.40 + 0.85 − (0.40)(0.85)
= 0.91
Note that P (A or C) and P (A and C) cannot be determined using the information given. But it can be
determined that A and C are not mutually exclusive since P (A ) + P (C) = 1.08, which is greater than 1,
and therefore cannot equal P (A or C ); from this it follows that P (A and C) ≥ 0.08. One can also deduce
that P (A and C) ≤ P (A) = 0.23, since A ∩ C is a subset of A, and that P (A or C) ≥ P (C) = 0.85 since C
is a subset of A ∪ C. Thus, one can conclude that 0.85 ≤ P (A or C) ≤ 1 and 0.08 ≤ P (A and C) ≤ 0.23.

4.2 Algebra
Algebra is based on the operations of arithmetic and on the concept of an unknown quantity, or
variable. Letters such as x or n are used to represent unknown quantities. For example, suppose Pam
has 5 more pencils than Fred. If F represents the number of pencils that Fred has, then the number of
pencils that Pam has is F + 5. As another example, if Jim’s present salary S is increased by 7%, then his
new salary is 1.07S. A combination of letters and arithmetic operations, such as
2
F + 5, 3x , and 19x2 − 6x + 3, is called an algebraic expression.
2x − 5
The expression 19x2 − 6x + 3 consists of the terms 19x2, −6x, and 3, where 19 is the coefficient of x2,
−6 is the coefficient of x1, and 3 is a constant term (or coefficient of x0 = 1). Such an expression is called
a second degree (or quadratic) polynomial in x since the highest power of x is 2. The expression F + 5 is a
2
first degree (or linear) polynomial in F since the highest power of F is 1. The expression 3x
is not a
2x − 5
polynomial because it is not a sum of terms that are each powers of x multiplied by coefficients.

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1. Simplifying Algebraic Expressions
Often when working with algebraic expressions, it is necessary to simplify them by factoring or
combining like terms. For example, the expression 6x + 5x is equivalent to (6 + 5)x, or 11x.
In the expression 9x − 3y, 3 is a factor common to both terms: 9x − 3y = 3(3x − y). In the expression
5x2 + 6y, there are no like terms and no common factors.
If there are common factors in the numerator and denominator of an expression, they can be divided
out, provided that they are not equal to zero.
For example, if x ≠ 3, then

x−3
is equal to 1; therefore,
x−3
3xy − 9 y 3 y ( x − 3)
=
x−3
x−3
= ( 3 y ) (1)
= 3y

To multiply two algebraic expressions, each term of one expression is multiplied by each term of the
other expression. For example:
(3x − 4)(9 y + x ) = 3x(9 y + x ) − 4(9 y + x )
= ( 3x ) (9 y ) + ( 3x )( x ) + ( −4 ) (9 y ) + ( −4) ( x )
= 27 xy + 3x 2 − 36 y − 4 x
An algebraic expression can be evaluated by substituting values of the unknowns in the expression. For
example, if x = 3 and y = −2, then 3xy − x2 + y can be evaluated as
3(3)( −2) − (3)2 + ( −2) = −18 − 9 − 2 = −29

2. Equations
A major focus of algebra is to solve equations involving algebraic expressions. Some examples of such
equations are
5x − 2 = 9 − x (a linear equation with one unknown)
3x + 1 = y − 2 (a linear equation with two unknowns)
5x 2 + 3x − 2 = 7 x
x ( x − 3) ( x 2 + 5)
=0
x−4

(a quadratic equation with one unknown)
(an equation that is factored on one side with 0 on the other)

The solutions of an equation with one or more unknowns are those values that make the equation true,
or “satisfy the equation,” when they are substituted for the unknowns of the equation. An equation
may have no solution or one or more solutions. If two or more equations are to be solved together, the
solutions must satisfy all the equations simultaneously.

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Two equations having the same solution(s) are equivalent equations. For example, the equations
2+x = 3
4 + 2x = 6
each have the unique solution x = 1. Note that the second equation is the first equation multiplied by 2.
Similarly, the equations
3x − y = 6
6x − 2 y = 12
have the same solutions, although in this case each equation has infinitely many solutions. If any value is
assigned to x, then 3x − 6 is a corresponding value for y that will satisfy both equations; for example,
x = 2 and y = 0 is a solution to both equations, as is x = 5 and y = 9.

3. Solving Linear Equations with One Unknown
To solve a linear equation with one unknown (that is, to find the value of the unknown that satisfies the
equation), the unknown should be isolated on one side of the equation. This can be done by performing
the same mathematical operations on both sides of the equation. Remember that if the same number
is added to or subtracted from both sides of the equation, this does not change the equality; likewise,
multiplying or dividing both sides by the same nonzero number does not change the equality. For
example, to solve the equation 5x − 6 = 4 for x, the variable x can be isolated using the following steps:
3
5x − 6 = 12 (multiplying by 3)
5x = 18 (adding 6)
x = 18 (dividing by 5)
5
The solution, 18 , can be checked by substituting it for x in the original equation to determine whether
5
it satisfies that equation:
5 ⎛ 18 ⎞ − 6
⎝ 5⎠
= 18 − 6 = 12 = 4
3
3
3
Therefore, x = 18 is the solution.
5

4. Solving Two Linear Equations with Two Unknowns
For two linear equations with two unknowns, if the equations are equivalent, then there are infinitely
many solutions to the equations, as illustrated at the end of section 4.2.2 (“Equations”). If the equations
are not equivalent, then they have either one unique solution or no solution. The latter case is illustrated
by the two equations:
3x + 4y = 17
6x + 8 y = 35

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Note that 3x + 4y = 17 implies 6x + 8y = 34, which contradicts the second equation. Thus, no values of x
and y can simultaneously satisfy both equations.
There are several methods of solving two linear equations with two unknowns. With any method, if
a contradiction is reached, then the equations have no solution; if a trivial equation such as 0 = 0 is
reached, then the equations are equivalent and have infinitely many solutions. Otherwise, a unique
solution can be found.
One way to solve for the two unknowns is to express one of the unknowns in terms of the other using
one of the equations, and then substitute the expression into the remaining equation to obtain an
equation with one unknown. This equation can be solved and the value of the unknown substituted into
either of the original equations to find the value of the other unknown. For example, the following two
equations can be solved for x and y.
(1) 3x + 2 y = 11
(2)

x− y = 2

In equation (2), x = 2 + y. Substitute 2 + y in equation (1) for x:
3(2 + y ) + 2 y = 11
6 + 3 y + 2 y = 11
6 + 5 y = 11
5y = 5
y =1
If y = 1, then x − 1 = 2 and x = 2 + 1 = 3.
There is another way to solve for x and y by eliminating one of the unknowns. This can be done by making
the coefficients of one of the unknowns the same (disregarding the sign) in both equations and either
adding the equations or subtracting one equation from the other. For example, to solve the equations
(1) 6x + 5 y = 29
(2) 4 x − 3 y = −6
by this method, multiply equation (1) by 3 and equation (2) by 5 to get
18x + 15 y = 87
20 x − 15 y = −30
Adding the two equations eliminates y, yielding 38x = 57, or x = 3 . Finally, substituting 3 for x in one
2
2
of the equations gives y = 4. These answers can be checked by substituting both values into both of the
original equations.

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5. Solving Equations by Factoring
Some equations can be solved by factoring. To do this, first add or subtract expressions to bring all the
expressions to one side of the equation, with 0 on the other side. Then try to factor the nonzero side into
a product of expressions. If this is possible, then using property (7) in section 4.1.4 (“Real Numbers”)
each of the factors can be set equal to 0, yielding several simpler equations that possibly can be solved.
The solutions of the simpler equations will be solutions of the factored equation. As an example,
consider the equation x3 − 2x2 + x = −5(x − l)2:
x 3 − 2x 2 + x + 5 ( x − 1) = 0
2

x ( x 2 − 2x + 1) + 5 ( x − 1) = 0
2

x ( x − 1) + 5 ( x − 1) = 0
2

( x + 5 ) ( x − 1)2 = 0
2
x + 5 = 0 or ( x − 1) = 0
x = −5 or x = 1.
For another example, consider

x ( x − 3) ( x 2 + 5 )
x−4

equals 0. Thus, x(x − 3)(x2 + 5) = 0:

= 0. A fraction equals 0 if and only if its numerator

x = 0 or x − 3 = 0 or x2 + 5 = 0
x = 0 or x = 3 or x2 + 5 = 0.
But x 2 + 5 = 0 has no real solution because x2 + 5 > 0 for every real number. Thus, the solutions are 0
and 3.
The solutions of an equation are also called the roots of the equation. These roots can be checked by
substituting them into the original equation to determine whether they satisfy the equation.

6. Solving Quadratic Equations
The standard form for a quadratic equation is
ax2 + bx + c = 0,
where a, b, and c are real numbers and a ≠ 0; for example:
x 2 + 6x + 5 = 0
3x 2 − 2x = 0, and
x2 + 4 = 0
Some quadratic equations can easily be solved by factoring. For example:
(1)

x 2 + 6x + 5 = 0
( x + 5)( x + 1) = 0
x + 5 = 0 or x + 1 = 0
x = −5 or x = −1

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3x 2 − 3 = 8x

(2)

3x 2 − 8x − 3 = 0
(3x + 1)( x − 3) = 0
3x + 1 = 0 or x − 3 = 0
x = − 1 or x = 3
3
A quadratic equation has at most two real roots and may have just one or even no real root. For
example, the equation x2 − 6x + 9 = 0 can be expressed as (x − 3)2 = 0, or (x − 3)(x − 3) = 0; thus the
only root is 3. The equation x2 + 4 = 0 has no real root; since the square of any real number is greater
than or equal to zero, x2 + 4 must be greater than zero.
An expression of the form a2 − b2 can be factored as (a − b)(a + b).
For example, the quadratic equation 9x2 − 25 = 0 can be solved as follows.

( 3x − 5)( 3x + 5) = 0
3x − 5 = 0 or 3x + 5 = 0
x = 5 or x = − 5
3
3
If a quadratic expression is not easily factored, then its roots can always be found using the quadratic
formula: If ax2 + bx + c = 0 (a ≠ 0), then the roots are
2
2
x = − b + b − 4 ac and x = − b − b − 4 ac
2a
2a

These are two distinct real numbers unless b2 − 4ac ≤ 0. If b2 − 4ac = 0, then these two expressions for
x are equal to − b , and the equation has only one root. If b2 − 4ac < 0, then b 2 − 4 ac is not a real
2a
number and the equation has no real roots.

7. Exponents
A positive integer exponent of a number or a variable indicates a product, and the positive integer is
the number of times that the number or variable is a factor in the product. For example, x5 means
(x)(x)(x)(x)(x); that is, x is a factor in the product 5 times.
Some rules about exponents follow.
Let x and y be any positive numbers, and let r and s be any positive integers.
(1) (xr)(xs) = x(r + s); for example, (22)(23) = 2(2 + 3) = 25 = 32.
x r = x (r − s ); for example, 4 5 = 4 5 − 2 = 4 3 = 64.
xs
42
(3) (xr)(yr) = (xy)r; for example, (33)(43) = 123 = 1,728.
r
3
r
3
⎛ x⎞
(4) ⎜ ⎟ = x r ; for example, ⎛ 2 ⎞ = 23 = 8 .
⎝ 3⎠
y
3
27
⎝ y⎠

(2)

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(5) (xr)s = xrs = (xs)r; for example, (x3)4 = x12 = (x4)3.
(6) x−r= 1r ; for example, 3−2 = 12 = 1 .
3
9
x
0
0
(7) x = 1; for example, 6 = 1.
r
2
r
2
1
1
1
⎛ 1⎞
⎛ 1⎞
(8) x s = ⎜ x s ⎟ = ( x r ) s = s x r ; for example, 8 3 = ⎜ 8 3 ⎟ = (8 2 ) 3 = 3 8 2 = 3 64 = 4 and 9 2 = 9 = 3.
⎝ ⎠
⎝ ⎠
It can be shown that rules 1 − 6 also apply when r and s are not integers and are not positive, that is,
when r and s are any real numbers.

8. Inequalities
An inequality is a statement that uses one of the following symbols:
≠ not equal to
> greater than
≥ greater than or equal to
< less than
≤ less than or equal to
1
< 3 . Solving a linear inequality with one
2
4
unknown is similar to solving an equation; the unknown is isolated on one side of the inequality. As in
solving an equation, the same number can be added to or subtracted from both sides of the inequality,
or both sides of an inequality can be multiplied or divided by a positive number without changing the
truth of the inequality. However, multiplying or dividing an inequality by a negative number reverses the
order of the inequality. For example, 6 > 2, but (−1)(6) < (−1)(2).

Some examples of inequalities are 5x − 3 < 9, 6x ≥ y, and

To solve the inequality 3x − 2 > 5 for x, isolate x by using the following steps:
3x − 2 > 5
3x > 7 (adding 2 to both sides)
x > 7 (dividing both sides by 3)
3
To solve the inequality 5x − 1 < 3 for x, isolate x by using the following steps:
−2
5x − 1 < 3
−2
5x − 1 > −6 (multiplying both sides by − 2)
5x > −5
(adding 1 to both sides)
x > −1
(dividing both sides by 5)

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9. Absolute Value
The absolute value of x, denoted |x|, is defined to be x if x ≥ 0 and −x if x < 0. Note that
the nonnegative square root of x2, and so x 2 = |x|.

x 2 denotes

10. Functions
An algebraic expression in one variable can be used to define a function of that variable. A function is
denoted by a letter such as f or g along with the variable in the expression. For example, the expression
x3 − 5x2 + 2 defines a function f that can be denoted by
f ( x ) = x 3 − 5x 2 + 2.
The expression

2z + 7
defines a function g that can be denoted by
z +1
g ( z ) = 2z + 7 .
z +1

The symbols “f (x)” or “g (z)” do not represent products; each is merely the symbol for an expression, and
is read “f of x” or “g of z.”
Function notation provides a short way of writing the result of substituting a value for a variable. If x = 1 is
substituted in the first expression, the result can be written f (1) = −2, and f (1) is called the “value of f at
x = 1.” Similarly, if z = 0 is substituted in the second expression, then the value of g at z = 0 is g(0) = 7.
Once a function f (x) is defined, it is useful to think of the variable x as an input and f (x) as the
corresponding output. In any function there can be no more than one output for any given input. However,
more than one input can give the same output; for example, if h (x) = |x + 3|, then h (−4) = 1 = h (−2).
The set of all allowable inputs for a function is called the domain of the function. For f and g defined
above, the domain of f is the set of all real numbers and the domain of g is the set of all numbers greater
than −1. The domain of any function can be arbitrarily specified, as in the function defined by “h (x) = 9x
− 5 for 0 ≤ x ≤ 10.” Without such a restriction, the domain is assumed to be all values of x that result in
a real number when substituted into the function.
The domain of a function can consist of only the positive integers and possibly 0. For example,
a(n ) = n 2 + n for n = 0,1, 2, 3, . . . . .
5
Such a function is called a sequence and a(n) is denoted by an. The value of the sequence an at n = 3 is
a3 = 32 + 3 = 9.60. As another example, consider the sequence defined by bn = (−1)n (n!) for n = 1, 2, 3, .
5
. . . A sequence like this is often indicated by listing its values in the order b1, b2, b3, . . ., bn, . . . as follows:
−1, 2, −6, . . . , (−1)n(n!), . . . , and (−1)n(n!) is called the nth term of the sequence.

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4.3 Geometry
1. Lines
In geometry, the word “line” refers to a straight line that extends without end in both directions.
Q

The line above can be referred to as line PQ or line . The part of the line from P to Q is called a line
segment. P and Q are the endpoints of the segment. The notation PQ is used to denote line segment PQ
and PQ is used to denote the length of the segment.

2. Intersecting Lines and Angles
If two lines intersect, the opposite angles are called vertical angles and have the same measure. In the figure
Q
P

y°
R
x° y° x°

∠PRQ and ∠SRT are vertical angles and ∠QRS and ∠PRT are vertical angles. Also, x + y = 180° since
PRS is a straight line.

3. Perpendicular Lines
An angle that has a measure of 90° is a right angle. If two lines intersect at right angles, the lines are
perpendicular. For example:
1
2

1 and 2 above are perpendicular, denoted by 1 ⊥ 2. A right angle symbol in an angle of intersection
indicates that the lines are perpendicular.

4. Parallel Lines
If two lines that are in the same plane do not intersect, the two lines are parallel. In the figure
1
2

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lines 1 and 2 are parallel, denoted by 1 2. If two parallel lines are intersected by a third line, as
shown below, then the angle measures are related as indicated, where x + y = 180°.
y° x°
x° y°
y° x°
x° y°

1
2

5. Polygons (Convex)
A polygon is a closed plane figure formed by three or more line segments, called the sides of the polygon.
Each side intersects exactly two other sides at their endpoints. The points of intersection of the sides
are vertices. The term “polygon” will be used to mean a convex polygon, that is, a polygon in which each
interior angle has a measure of less than 180°.
The following figures are polygons:

The following figures are not polygons:

A polygon with three sides is a triangle; with four sides, a quadrilateral; with five sides, a pentagon; and
with six sides, a hexagon.
The sum of the interior angle measures of a triangle is 180°. In general, the sum of the interior angle
measures of a polygon with n sides is equal to (n − 2)180°. For example, this sum for a pentagon is
(5 − 2)180° = (3)180° = 540°.

Note that a pentagon can be partitioned into three triangles and therefore the sum of the angle
measures can be found by adding the sum of the angle measures of three triangles.
The perimeter of a polygon is the sum of the lengths of its sides.

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The commonly used phrase “area of a triangle” (or any other plane figure) is used to mean the area of the
region enclosed by that figure.

6. Triangles
There are several special types of triangles with important properties. But one property that all triangles
share is that the sum of the lengths of any two of the sides is greater than the length of the third side, as
illustrated below.
y

z
x + y > z, x + z > y, and y + z > x

An equilateral triangle has all sides of equal length. All angles of an equilateral triangle have equal
measure. An isosceles triangle has at least two sides of the same length. If two sides of a triangle have the
same length, then the two angles opposite those sides have the same measure. Conversely, if two angles
of a triangle have the same measure, then the sides opposite those angles have the same length. In
isosceles triangle PQR below, x = y since PQ = QR.
Q

x°

y°

A triangle that has a right angle is a right triangle. In a right triangle, the side opposite the right angle is
the hypotenuse, and the other two sides are the legs. An important theorem concerning right triangles is
the Pythagorean theorem, which states: In a right triangle, the square of the length of the hypotenuse is
equal to the sum of the squares of the lengths of the legs.
S

6
R

In the figure above, ΔRST is a right triangle, so (RS)2 + (RT )2 = (ST )2. Here, RS = 6 and RT = 8, so
ST = 10, since 62 + 82 = 36 + 64 = 100 = (ST )2 and ST = 100. Any triangle in which the lengths of
the sides are in the ratio 3:4:5 is a right triangle. In general, if a, b, and c are the lengths of the sides of a
triangle and a 2 + b 2 = c 2, then the triangle is a right triangle.

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30º

45º

60º

In 45°− 45°− 90° triangles, the lengths of the sides are in the ratio 1:1: 2 . For example, in ΔJKL,
if JL = 2, then JK = 2 and KL = 2 2. In 30°− 60°− 90° triangles, the lengths of the sides are in the ratio
1: 3:2. For example, in ΔXYZ, if XZ = 3, then XY = 3 3 and YZ = 6.
The altitude of a triangle is the segment drawn from a vertex perpendicular to the side opposite that
vertex. Relative to that vertex and altitude, the opposite side is called the base.
The area of a triangle is equal to:
(the length of the altitude) × (the length of the base)
2
B
E

D
8
BD = 5

In ΔABC, BD is the altitude to base AC and AE is the altitude to base BC . The area of ΔABC is
equal to
BD × AC = 5 × 8 = 20.
2
2
AE
×
BC
. If ΔABC above is isosceles and AB = BC, then altitude BD bisects
The area is also equal to
2
the base; that is, AD = DC = 4. Similarly, any altitude of an equilateral triangle bisects the side to which
it is drawn.
E
6 30º
D

60º

In equilateral triangle DEF, if DE = 6, then DG = 3 and EG = 3 3. The area of ΔDEF is equal to
3 3 × 6 = 9 3.
2
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7. Quadrilaterals
A polygon with four sides is a quadrilateral. A quadrilateral in which both pairs of opposite sides are
parallel is a parallelogram. The opposite sides of a parallelogram also have equal length.
K

L
4

In parallelogram JKLM, JK LM and JK = LM ; KL JM and KL = JM.
The diagonals of a parallelogram bisect each other (that is, KN = NM and JN = NL).
The area of a parallelogram is equal to
(the length of the altitude) × (the length of the base).
The area of JKLM is equal to 4 × 6 = 24.
A parallelogram with right angles is a rectangle, and a rectangle with all sides of equal length is a square.
X

3
W

The perimeter of WXYZ = 2(3) + 2(7) = 20 and the area of WXYZ is equal to 3 × 7 = 21. The diagonals
of a rectangle are equal; therefore WY = XZ = 9 + 49 = 58 .
Q

R
8

A quadrilateral with two sides that are parallel, as shown above, is a trapezoid. The area of trapezoid
PQRS may be calculated as follows:
1 (the sum of the lengths of the bases)(the height) = 1 (QR + PS )(8) = 1 (28 × 8) = 112.
2
2
2

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8. Circles
A circle is a set of points in a plane that are all located the same distance from a fixed point (the center of
the circle).
A chord of a circle is a line segment that has its endpoints on the circle. A chord that passes through the
center of the circle is a diameter of the circle. A radius of a circle is a segment from the center of the circle to a
point on the circle. The words “diameter” and “radius” are also used to refer to the lengths of these segments.
The circumference of a circle is the distance around the circle. If r is the radius of the circle, then the
circumference is equal to 2π r, where π is approximately 22 or 3.14. The area of a circle of radius r is
7
equal to π r2.
K

In the circle above, O is the center of the circle and JK and PR are chords. PR is a diameter and OR
is a radius. If OR = 7, then the circumference of the circle is 2π (7) = 14π and the area of the circle is
π (7)2 = 49π .
The number of degrees of arc in a circle (or the number of degrees in a complete revolution) is 360.
R
O xº

S
T

In the circle with center O above, the length of arc RST is x of the circumference of the circle; for
360
1
of the circumference of the circle.
example, if x = 60, then arc RST has length
6
A line that has exactly one point in common with a circle is said to be tangent to the circle, and that
common point is called the point of tangency. A radius or diameter with an endpoint at the point of
tangency is perpendicular to the tangent line, and, conversely, a line that is perpendicular to a radius or
diameter at one of its endpoints is tangent to the circle at that endpoint.

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The line above is tangent to the circle and radius OT is perpendicular to .
If each vertex of a polygon lies on a circle, then the polygon is inscribed in the circle and the circle is
circumscribed about the polygon. If each side of a polygon is tangent to a circle, then the polygon is
circumscribed about the circle and the circle is inscribed in the polygon.
Q

B
P

E
A

In the figure above, quadrilateral PQRS is inscribed in a circle and hexagon ABCDEF is circumscribed
about a circle.
If a triangle is inscribed in a circle so that one of its sides is a diameter of the circle, then the triangle is
a right triangle.
Y

In the circle above, XZ is a diameter and the measure of ∠XYZ is 90°.

9. Rectangular Solids and Cylinders
A rectangular solid is a three‐dimensional figure formed by 6 rectangular surfaces, as shown below. Each
rectangular surface is a face. Each solid or dotted line segment is an edge, and each point at which the
edges meet is a vertex. A rectangular solid has 6 faces, 12 edges, and 8 vertices. Opposite faces are parallel
rectangles that have the same dimensions. A rectangular solid in which all edges are of equal length is a cube.
The surface area of a rectangular solid is equal to the sum of the areas of all the faces. The volume is equal to
(length) × (width) × (height);
in other words, (area of base) × (height).
U
R

Q
4
P

128

T
8
3

In the rectangular solid above, the dimensions are 3, 4, and 8. The surface area is equal to
2(3 × 4) + 2(3 × 8) + 2(4 × 8) = 136. The volume is equal to 3 × 4 × 8 = 96.

4.3 Math Review Geometry

O
The figure above is a right circular cylinder. The two bases are circles of the same size with centers O and
P, respectively, and altitude (height) OP is perpendicular to the bases. The surface area of a right circular
cylinder with a base of radius r and height h is equal to 2(π r2) + 2π rh (the sum of the areas of the two
bases plus the area of the curved surface).
The volume of a cylinder is equal to π r2h, that is,
(area of base) × (height).

8
5

In the cylinder above, the surface area is equal to
2(25π ) + 2π (5)(8) = 130π ,
and the volume is equal to
25π (8) = 200π .

10. Coordinate Geometry
y
3
2

1
–4

–3

–2
III

–1 O
–1
–2

–3

The figure above shows the (rectangular) coordinate plane. The horizontal line is called the x‐axis and the
perpendicular vertical line is called the y‐axis. The point at which these two axes intersect, designated O,
is called the origin. The axes divide the plane into four quadrants, I, II, III, and IV, as shown.
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Each point in the plane has an x‐coordinate and a y‐coordinate. A point is identified by an ordered pair
(x,y) of numbers in which the x‐coordinate is the first number and the y‐coordinate is the second number.
y
5
4
P

3
2
1
–5 –4 –3 –2 –1 O
–1
–2
Q

1 2

3 4 5 x

–3
–4
–5

In the graph above, the (x,y) coordinates of point P are (2,3) since P is 2 units to the right of the y‐axis
(that is, x = 2) and 3 units above the x‐axis (that is, y = 3). Similarly, the (x,y) coordinates of point Q are
(−4,−3). The origin O has coordinates (0,0).
One way to find the distance between two points in the coordinate plane is to use the Pythagorean
theorem.
y
R

(–2,4)

S
(3,–3)

To find the distance between points R and S using the Pythagorean theorem, draw the triangle as
shown. Note that Z has (x,y) coordinates (−2,−3), RZ = 7, and ZS = 5. Therefore, the distance between
R and S is equal to
7 2 + 5 2 = 74.
For a line in the coordinate plane, the coordinates of each point on the line satisfy a linear equation of
the form y = mx + b (or the form x = a if the line is vertical). For example, each point on the line on the
next page satisfies the equation y = − 1 x + 1 . One can verify this for the points (−2,2), (2,0), and (0,1)
2
by substituting the respective coordinates for x and y in the equation.
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4.3 Math Review Geometry

y
(–2,2)

2
1

–3 –2

(0,1)

–1 O
–1

(2,0)
1

In the equation y = mx + b of a line, the coefficient m is the slope of the line and the constant term b
is the y‐intercept of the line. For any two points on the line, the slope is defined to be the ratio of the
difference in the y‐coordinates to the difference in the x‐coordinates. Using (−2, 2) and (2, 0) above, the
slope is
The difference in the y -coordinates
= 0 − 2 = −2 = − 1 .
The difference in the x -coordinates 2 − ( −2 ) 4
2
The y‐intercept is the y‐coordinate of the point at which the line intersects the y‐axis. For the line
above, the y‐intercept is 1, and this is the resulting value of y when x is set equal to 0 in the equation
y = − 1 x + 1 . The x‐intercept is the x‐coordinate of the point at which the line intersects the x‐axis. The
2
x‐intercept can be found by setting y = 0 and solving for x. For the line y = − 1 x + 1 , this gives
2
− 1 x +1 = 0
2
− 1 x = −1
2
x = 2.
Thus, the x‐intercept is 2.
Given any two points (x1,y1) and (x2,y2) with x1 ≠ x2, the equation of the line passing through these
y − y1
, then using a
points can be found by applying the definition of slope. Since the slope is m = 2
x 2 − x1
y − y1
point known to be on the line, say (x1,y1), any point (x,y) on the line must satisfy
= m , or
x − x1
y − y1 = m(x − x1). (Using (x2,y2) as the known point would yield an equivalent equation.) For example,
consider the points (−2,4) and (3,−3) on the line below.

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y
(–2,4)

5
4
3
2
1

–4

–3 –2

–1 O
–1
–2

–3

4 x
(3,–3)

–4
–5

The slope of this line is −3 − 4 = −7 , so an equation of this line can be found using the point (3,−3)
3 − ( −2 ) 5
as follows:
y − ( −3) = − 7 ( x − 3)
5
7
y + 3 = − x + 21
5
5
y =−7x+6
5
5
The y‐intercept is

6
. The x‐intercept can be found as follows:
5
0=−7x+6
5
5
7x= 6
5
5
x=6
7

Both of these intercepts can be seen on the graph.
If the slope of a line is negative, the line slants downward from left to right; if the slope is positive, the
line slants upward. If the slope is 0, the line is horizontal; the equation of such a line is of the form y = b
since m = 0. For a vertical line, slope is not defined, and the equation is of the form x = a, where a is the
x‐intercept.
There is a connection between graphs of lines in the coordinate plane and solutions of two linear
equations with two unknowns. If two linear equations with unknowns x and y have a unique solution,
then the graphs of the equations are two lines that intersect in one point, which is the solution. If the
equations are equivalent, then they represent the same line with infinitely many points or solutions. If
the equations have no solution, then they represent parallel lines, which do not intersect.

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There is also a connection between functions (see section 4.2.10) and the coordinate plane. If a function
is graphed in the coordinate plane, the function can be understood in different and useful ways.
Consider the function defined by
f (x ) = − 7 x + 6 .
5
5
If the value of the function, f (x), is equated with the variable y, then the graph of the function in the
xy‐coordinate plane is simply the graph of the equation
y =−7x+6
5
5
shown above. Similarly, any function f (x) can be graphed by equating y with the value of the function:
y = f (x ).
So for any x in the domain of the function f, the point with coordinates (x, f (x)) is on the graph of f,
and the graph consists entirely of these points.
As another example, consider a quadratic polynomial function defined by f (x)= x2 −1. One can plot
several points (x, f (x)) on the graph to understand the connection between a function and its graph:
x

f (x)

–2

–1

(–2,3)

(2,3)

3
2

(–1,0)

(1,0)

–2 –1 O
1 2 x
–1 (0,–1)
–2

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If all the points were graphed for −2 ≤ x ≤ 2, then the graph would appear as follows.
y
3
2
1
–2 –1 O
–1

–2

The graph of a quadratic function is called a parabola and always has the shape of the curve above,
although it may be upside down or have a greater or lesser width. Note that the roots of the equation
f (x) = x2 −1 = 0 are x = 1 and x = −1; these coincide with the x‐intercepts since x‐intercepts are
found by setting y = 0 and solving for x. Also, the y‐intercept is f (0) = −1 because this is the value of y
corresponding to x = 0. For any function f, the x‐intercepts are the solutions of the equation f (x) = 0
and the y‐intercept is the value f (0).

4.4 Word Problems
Many of the principles discussed in this chapter are used to solve word problems. The following
discussion of word problems illustrates some of the techniques and concepts used in solving such
problems.

1. Rate Problems
The distance that an object travels is equal to the product of the average speed at which it travels and
the amount of time it takes to travel that distance, that is,
Rate × Time = Distance.
Example 1: If a car travels at an average speed of 70 kilometers per hour for 4 hours, how many
kilometers does it travel?
Solution: Since rate × time = distance, simply multiply 70 km/hour × 4 hours. Thus, the car travels
280 kilometers in 4 hours.
To determine the average rate at which an object travels, divide the total distance traveled by the total
amount of traveling time.
Example 2: On a 400‐mile trip, Car X traveled half the distance at 40 miles per hour (mph) and the
other half at 50 mph. What was the average speed of Car X ?
Solution: First it is necessary to determine the amount of traveling time. During the first 200 miles, the
car traveled at 40 mph; therefore, it took 200 = 5 hours to travel the first 200 miles.
40

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4.4 Math Review Word Problems

During the second 200 miles, the car traveled at 50 mph; therefore, it took 200 = 4 hours to travel the
50
400
4
= 44 mph. Note that the average speed is
second 200 miles. Thus, the average speed of Car X was
9
9
40
+
50
= 45.
not
2
Some rate problems can be solved by using ratios.
Example 3: If 5 shirts cost $44, then, at this rate, what is the cost of 8 shirts?
Solution: If c is the cost of the 8 shirts, then 5 = 8 . Cross multiplication results in the equation
44 c
5c = 8 × 44 = 352
c = 352 = 70.40
5
The 8 shirts cost $70.40.

2. Work Problems
In a work problem, the rates at which certain persons or machines work alone are usually given, and it is
necessary to compute the rate at which they work together (or vice versa).
The basic formula for solving work problems is 1 + 1 = 1 , where r and s are, for example, the number
r s h
of hours it takes Rae and Sam, respectively, to complete a job when working alone, and h is the number
of hours it takes Rae and Sam to do the job when working together. The reasoning is that in 1 hour Rae
does 1 of the job, Sam does 1 of the job, and Rae and Sam together do 1 of the job.
r
s
h
Example 1: If Machine X can produce 1,000 bolts in 4 hours and Machine Y can produce 1,000 bolts in
5 hours, in how many hours can Machines X and Y, working together at these constant rates, produce
1,000 bolts?
Solution:
1+1=1
4 5 h
5 + 4 =1
20 20 h
9 =1
20 h
9h = 20
h = 20 = 2 2
9
9
Working together, Machines X and Y can produce 1,000 bolts in 2 2 hours.
9

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Example 2: If Art and Rita can do a job in 4 hours when working together at their respective constant
rates and Art can do the job alone in 6 hours, in how many hours can Rita do the job alone?
Solution:
1+ 1 =1
6 R 4
R+6 = 1
6R
4
4 R + 24 = 6 R
24 = 2 R
12 = R
Working alone, Rita can do the job in 12 hours.

3. Mixture Problems
In mixture problems, substances with different characteristics are combined, and it is necessary to
determine the characteristics of the resulting mixture.
Example 1: If 6 pounds of nuts that cost $1.20 per pound are mixed with 2 pounds of nuts that cost
$1.60 per pound, what is the cost per pound of the mixture?
Solution: The total cost of the 8 pounds of nuts is
6($1.20) + 2($1.60) = $10.40.
The cost per pound is

$10.40 = $1.30.
8

Example 2: How many liters of a solution that is 15 percent salt must be added to 5 liters of a solution
that is 8 percent salt so that the resulting solution is 10 percent salt?
Solution: Let n represent the number of liters of the 15% solution. The amount of salt in the 15%
solution [0.15n] plus the amount of salt in the 8% solution [(0.08)(5)] must be equal to the amount of
salt in the 10% mixture [0.10(n + 5)]. Therefore,
0.15n + 0.08(5) = 0.10(n + 5)
15n + 40 = 10n + 50
5n = 10
n = 2 liters
Two liters of the 15% salt solution must be added to the 8% solution to obtain the 10% solution.

4. Interest Problems
Interest can be computed in two basic ways. With simple annual interest, the interest is computed on
the principal only and is equal to (principal) × (interest rate) × (time). If interest is compounded, then
interest is computed on the principal as well as on any interest already earned.
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4.4 Math Review Word Problems

Example 1: If $8,000 is invested at 6 percent simple annual interest, how much interest is earned after
3 months?
Solution: Since the annual interest rate is 6%, the interest for 1 year is

( 0.06)($8, 000 ) = $480.
The interest earned in 3 months is

3 $480 = $120.
(
)
12

Example 2: If $10,000 is invested at 10 percent annual interest, compounded semiannually, what is the
balance after 1 year?
Solution: The balance after the first 6 months would be
10, 000 + (10, 000)(0.05) = $10, 500.
The balance after one year would be

10, 500 + (10, 500)(0.05) = $11, 025.

Note that the interest rate for each 6‐month period is 5%, which is half of the 10% annual rate. The
balance after one year can also be expressed as
2

10, 000 ⎛ 1 + 0.10 ⎞ dollars.
⎝
2 ⎠

5. Discount
If a price is discounted by n percent, then the price becomes (100 − n) percent of the original price.
Example 1: A certain customer paid $24 for a dress. If that price represented a 25 percent discount on
the original price of the dress, what was the original price of the dress?
Solution: If p is the original price of the dress, then 0.75p is the discounted price and 0.75p = $24, or
p = $32. The original price of the dress was $32.
Example 2: The price of an item is discounted by 20 percent and then this reduced price is discounted by
an additional 30 percent. These two discounts are equal to an overall discount of what percent?
Solution: If p is the original price of the item, then 0.8p is the price after the first discount. The price
after the second discount is (0.7)(0.8) p = 0.56p. This represents an overall discount of 44 percent
(l00% − 56%).

6. Proﬁt
Gross profit is equal to revenues minus expenses, or selling price minus cost.
Example: A certain appliance costs a merchant $30. At what price should the merchant sell the
appliance in order to make a gross profit of 50 percent of the cost of the appliance?
Solution: If s is the selling price of the appliance, then s − 30 = (0.5)(30), or s = $45. The merchant
should sell the appliance for $45.

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7. Sets
If S is the set of numbers 1, 2, 3, and 4, you can write S = {l, 2, 3, 4}. Sets can also be represented by
Venn diagrams. That is, the relationship among the members of sets can be represented by circles.
Example 1: Each of 25 people is enrolled in history, mathematics, or both. If 20 are enrolled in history
and 18 are enrolled in mathematics, how many are enrolled in both history and mathematics?
Solution: The 25 people can be divided into three sets: those who study history only, those who study
mathematics only, and those who study history and mathematics. Thus a Venn diagram may be drawn
as follows, where n is the number of people enrolled in both courses, 20 − n is the number enrolled in
history only, and 18 − n is the number enrolled in mathematics only.
History

Mathematics
20 – n

18 – n

Since there is a total of 25 people, (20 − n) + n + (18 − n) = 25, or n = 13. Thirteen people are enrolled
in both history and mathematics. Note that 20 + 18 − 13 = 25, which is the general addition rule for
two sets (see section 4.1.9).
Example 2: In a certain production lot, 40 percent of the toys are red and the remaining toys are green.
Half of the toys are small and half are large. If 10 percent of the toys are red and small, and 40 toys are
green and large, how many of the toys are red and large?
Solution: For this kind of problem, it is helpful to organize the information in a table:
Red
Small

Green

10%

50%

Large
Total

Total

50%
40%

60%

100%

The numbers in the table are the percentages given. The following percentages can be computed on the
basis of what is given:
Red

Green

Total

Small

10%

40%

50%

Large

30%

20%

50%

Total

40%

60%

100%

Since 20% of the number of toys (n) are green and large, 0.20n = 40 (40 toys are green and large), or
n = 200. Therefore, 30% of the 200 toys, or (0.3)(200) = 60, are red and large.

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4.4 Math Review Word Problems

8. Geometry Problems
The following is an example of a word problem involving geometry.

200 meters

Example:

200 meters
The figure above shows an aerial view of a piece of land. If all angles shown are right angles, what is the
perimeter of the piece of land?
Solution: For reference, label the figure as
Q

R
S

T
U

V
W

If all the angles are right angles, then QR + ST + UV = PW, and RS + TU + VW = PQ. Hence, the
perimeter of the land is 2PW + 2PQ = 2 × 200 + 2 × 200 = 800 meters.

9. Measurement Problems
Some questions on the GMAT involve metric units of measure, whereas others involve English units of
measure. However, except for units of time, if a question requires conversion from one unit of measure
to another, the relationship between those units will be given.
Example: A train travels at a constant rate of 25 meters per second. How many kilometers does it travel
in 5 minutes? (1 kilometer = 1,000 meters)
Solution: In 1 minute the train travels (25)(60) = 1,500 meters, so in 5 minutes it travels 7,500 meters.
7, 500
, or 7.5 kilometers.
Since 1 kilometer = 1,000 meters, it follows that 7,500 meters equals
1, 000

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10. Data Interpretation
Occasionally a question or set of questions will be based on data provided in a table or graph. Some
examples of tables and graphs are given below.
Example 1:
Population by Age Group
(in thousands)
Age

Population

17 years and under

63,376

18−44 years

86,738

45−64 years

43,845

65 years and over

24,054

How many people are 44 years old or younger?
Solution: The figures in the table are given in thousands. The answer in thousands can be obtained by
adding 63,376 thousand and 86,738 thousand. The result is 150,114 thousand, which is 150,114,000.
Example 2:

30
25
20
15
10
5
0

Precipitation
(right-hand scale)

Temperature
(left-hand scale)
January April

July

October

12
10
8
6
4
2
0

Precipitation in
Centimeters

Temperature in
Degrees Celsius

AVERAGE TEMPERATURE AND
PRECIPITATION IN CITY X

What are the average temperature and precipitation in City X during April?
Solution: Note that the scale on the left applies to the temperature line graph and the one on the right
applies to the precipitation line graph. According to the graph, during April the average temperature is
approximately 14° Celsius and the average precipitation is approximately 8 centimeters.

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4.4 Math Review Word Problems

Example 3:
DISTRIBUTION OF AL’S WEEKLY NET SALARY

Fuel and
Insurance
for Automobile
10%
Savings
25%

Food
15%

Rent and
Utilities
30%

Other
20%

Al’s weekly net salary is $350. To how many of the categories listed was at least $80 of Al’s weekly net
salary allocated?
Solution: In the circle graph, the relative sizes of the sectors are proportional to their corresponding
values and the sum of the percents given is 100%. Note that 80 is approximately 23%, so at least $80
350
was allocated to each of 2 categories—Rent and Utilities, and Savings—since their allocations are each
greater than 23%.

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5.0 Problem Solving

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5.0 Problem Solving

5.0 Problem Solving
The Quantitative section of the GMAT® exam uses problem solving and data sufﬁciency questions to
gauge your skill level. This chapter focuses on problem solving questions. Remember that quantitative
questions require knowledge of the following:
• Arithmetic
• Elementary algebra
• Commonly known concepts of geometry
Problem solving questions are designed to test your basic mathematical skills and understanding of
elementary mathematical concepts, as well as your ability to reason quantitatively, solve quantitative
problems, and interpret graphic data. The mathematics knowledge required to answer the questions
is no more advanced than what is generally taught in secondary school (or high school) mathematics
classes.
In these questions, you are asked to solve each problem and select the best of the ﬁve answer choices
given. Begin by reading the question thoroughly to determine exactly what information is given and
to make sure you understand what is being asked. Scan the answer choices to understand your options.
If the problem seems simple, take a few moments to see whether you can determine the answer. Then,
check your answer against the choices provided.
If you do not see your answer among the choices, or if the problem is complicated, take a closer look at
the answer choices and think again about what the problem is asking. See whether you can eliminate
some of the answer choices and narrow down your options. If you are still unable to narrow the answer
down to a single choice, reread the question. Keep in mind that the answer will be based solely on the
information provided in the question—don’t allow your own experience and assumptions to interfere
with your ability to ﬁnd the correct answer to the question.
If you ﬁnd yourself stuck on a question or unable to select the single correct answer, keep in mind that
you have about two minutes to answer each quantitative question. You may run out of time if you take
too long to answer any one question; you may simply need to pick the answer that seems to make the
most sense. Although guessing is generally not the best way to achieve a high GMAT score, making
an educated guess is a good strategy for answering questions you are unsure of. Even if your answer to
a particular question is incorrect, your answers to other questions will allow the test to accurately gauge
your ability level.
The following pages include test‐taking strategies, directions that will apply to questions of this type,
sample questions, an answer key, and explanations for all the problems. These explanations present
problem solving strategies that could be helpful in answering the questions.

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5.1 Test-Taking Strategies
1.

Pace yourself.

Consult the on‐screen timer periodically. Work as carefully as possible, but do not spend
valuable time checking answers or pondering problems that you ﬁnd difﬁcult.
2.

Use the erasable notepad provided at the test center.

Working a problem out may help you avoid errors in solving the problem. If diagrams or ﬁgures
are not presented, it may help to draw your own.
3.

Read each question carefully to determine what is being asked.

For word problems, take one step at a time, reading each sentence carefully and translating the
information into equations or other useful mathematical representations.
4.

Scan the answer choices before attempting to answer a question.

Scanning the answers can prevent you from putting answers in a form that is not given
(e.g., ﬁnding the answer in decimal form, such as 0.25, when the choices are given in fractional
1
form, such as ). Also, if the question requires approximations, a shortcut could serve well
4
(e.g., you may be able to approximate 48 percent of a number by using half ).
5.

Don’t waste time trying to solve a problem that is too difﬁcult for you.

Make your best guess and then move on to the next question.

5.2 The Directions
These directions are very similar to those you will see for problem solving questions when you take the
GMAT exam. If you read them carefully and understand them clearly before sitting for the GMAT
exam, you will not need to spend too much time reviewing them once the test begins.
Solve the problem and indicate the best of the answer choices given.
Numbers: All numbers used are real numbers.
Figures: A ﬁgure accompanying a problem solving question is intended to provide information useful
in solving the problem. Figures are drawn as accurately as possible. Exceptions will be clearly noted.
Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of
points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All
ﬁgures lie in a plane unless otherwise indicated.

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5.2 Problem Solving The Directions

To register for the GMAT exam go to www.mba.com
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5.3 Practice Questions
Solve the problem and indicate the best of the answer choices given.
Numbers: All numbers used are real numbers.
Figures: A ﬁgure accompanying a problem solving question is intended to provide information useful
in solving the problem. Figures are drawn as accurately as possible. Exceptions will be clearly
noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The
positions of points, angles, regions, etc., exist in the order shown, and angle measures are
greater than zero. All ﬁgures lie in a plane unless otherwise indicated.

146

The price of a coat in a certain store is $500. If the
price of the coat is to be reduced by $150, by what
percent is the price to be reduced?
(A)

10%

(B)

15%

(C)

20%

(D)

25%

(E)

30%

1 1 5 1 1 9
+ − + + −
=
3 2 6 5 4 20
(A)
(B)

On a vacation, Rose exchanged $500.00 for euros
at an exchange rate of 0.80 euro per dollar and
3
spent of the euros she received. If she exchanged
4
the remaining euros for dollars at an exchange rate
of $1.20 per euro, what was the dollar amount she
received?

0
2
15

(C)

2
5

(D)

9
20

(E)

5
6

Bouquets are to be made using white tulips and red
tulips, and the ratio of the number of white tulips to
the number of red tulips is to be the same in each
bouquet. If there are 15 white tulips and 85 red tulips
available for the bouquets, what is the greatest number
of bouquets that can be made using all the tulips
available?

(A)

$60.00

(B)

$80.00

(C)

$100.00

(A)

(D)

$120.00

(B)

(E)

$140.00

(C)

(D)

(E)

Each of the 12 squares shown is labeled x, y, v, or w.
What is the ratio of the number of these squares labeled
x or y to the number of these squares labeled v or w ?

125% of 5 =
(A)

5.125

(B)

5.25

(C)

(A)

1:2

(D)

6.125

(B)

2:3

(E)

6.25

(C)

4:3

(D)

3:2

(E)

2:1

5.3 Problem Solving Practice Questions

Today Rebecca, who is 34 years old, and her daughter,
who is 8 years old, celebrate their birthdays. How
many years will pass before Rebecca’s age is twice her
daughter’s age?
(A)

(B)

(C)

(D)

(E)

When traveling at a constant speed of 32 miles per hour,
a certain motorboat consumes 24 gallons of fuel
per hour. What is the fuel consumption of this boat at
this speed measured in miles traveled per gallon of fuel?
(A)
(B)
(C)
(D)
(E)

2
3
3
4
4
5
4
3
3
2

12.

A case contains c cartons. Each carton contains
b boxes, and each box contains 100 paper clips.
How many paper clips are contained in 2 cases?
(A)
(B)
(C)

10.

11.

100bc
100b
c
200bc

(D)

200b
c

(E)

200
bc

A technician makes a round‐trip to and from a certain
service center by the same route. If the technician
completes the drive to the center and then completes
10 percent of the drive from the center, what percent
of the round‐trip has the technician completed?
(A)

(B)

10%

(C)

25%

(D)

40%

(E)

55%

13.

Rafﬂe tickets numbered consecutively from 101
through 350 are placed in a box. What is the
probability that a ticket selected at random will have a
number with a hundreds digit of 2 ?
(A)

2
5

(B)

2
7

(C)

33
88

(D)

99
250

(E)

100
249

When Leo imported a certain item, he paid a 7 percent
import tax on the portion of the total value of the item
in excess of $1,000. If the amount of the import tax
that Leo paid was $87.50, what was the total value of
the item?
(A)

$1,600

(B)

$1,850

(C)

$2,250

(D)

$2,400

(E)

$2,750

A collection of 16 coins, each with a face value of
either 10 cents or 25 cents, has a total face value of
$2.35. How many of the coins have a face value of
25 cents?
(A)

(B)

(C)

(D)

(E)

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14.

15.

16.

17.

148

The numbers of cars sold at a certain dealership on
six of the last seven business days were 4, 7, 2, 8, 3,
and 6, respectively. If the number of cars sold on the
seventh business day was either 2, 4, or 5, for which
of the three values does the average (arithmetic mean)
number of cars sold per business day for the seven
business days equal the median number of cars sold
per day for the seven days?
I.

II.

III.

(A)

II only

(B)

III only

(C)

I and II only

(D)

II and III only

(E)

I, II, and III

If it is assumed that 60 percent of those who receive
a questionnaire by mail will respond and 300 responses
are needed, what is the minimum number of
questionnaires that should be mailed?
(A)

400

(B)

420

(C)

480

(D)

500

(E)

600

18.

19.

20.

If 1 < x < y < z, which of the following has the greatest
value?

A rectangular ﬂoor that measures 8 meters by
10 meters is to be covered with carpet squares that
each measure 2 meters by 2 meters. If the carpet
squares cost $12 apiece, what is the total cost for the
number of carpet squares needed to cover the ﬂoor?
(A)

$200

(B)

$240

(C)

$480

(D)

$960

(E)

$1,920

If 893 × 78 = p, which of the following is equal to
893 × 79 ?
(A)

p+1

(B)

p + 78

(C)

p + 79

(D)

p + 893

(E)

p + 894

Thabo owns exactly 140 books, and each book is
either paperback fiction, paperback nonfiction, or
hardcover nonfiction. If he owns 20 more paperback
nonfiction books than hardcover nonfiction books, and
twice as many paperback fiction books as paperback
nonfiction books, how many hardcover nonfiction
books does Thabo own?
(A)

(B)

(C)

(A)

z(x + 1)

(D)

(B)

z(y + 1)

(E)

(C)

x(y + z)

(D)

y(x + z)

(E)

z(x + y)

A rectangular garden is to be twice as long as it is
wide. If 360 yards of fencing, including the gate, will
completely enclose the garden, what will be the length
of the garden, in yards?
(A)

120

(B)

140

(C)

160

(D)

180

(E)

200

21.

If the average (arithmetic mean) of the four numbers
3, 15, 32, and (N + 1) is 18, then N =
(A)

(B)

(C)

(D)

(E)

5.3 Problem Solving Practice Questions

22.

23.

24.

Abdul, Barb, and Carlos all live on the same straight
road, on which their school is also located. The school
is halfway between Abdul’s house and Barb’s house.
Barb’s house is halfway between the school and
Carlos’s house. If the school is 4 miles from Carlos’s
house, how many miles is Abdul’s house from Carlos’s
house?
(A)

1
1
3

(B)

(C)

(D)

(E)

25.

If y −

1 11
< , which of the following could be a value
2
2

of y ?
(A)

–11
11
−
2

(B)
(C)

11
2

(D)

(E)

During a certain time period, Car X traveled north
along a straight road at a constant rate of 1 mile per
minute and used fuel at a constant rate of 5 gallons
every 2 hours. During this time period, if Car X used
exactly 3.75 gallons of fuel, how many miles did Car X
travel?

120°
A

26.

In the figure shown, AC = 2 and BD = DC = 1. What is
the measure of angle ABD ?

(A)

(B)

37.5

(C)

(A)

15°

(D)

(B)

20°

(E)

(C)

30°

(D)

40°

(E)

45°

Cheryl purchased 5 identical hollow pine doors and
6 identical solid oak doors for the house she is
building. The regular price of each solid oak door
was twice the regular price of each hollow pine door.
However, Cheryl was given a discount of 25% off the
regular price of each solid oak door. If the regular price
of each hollow pine door was $40, what was the total
price of all 11 doors?
(A)

$320

(B)

$540

(C)

$560

(D)

$620

(E)

$680

Note: Not drawn to scale.

27.

28.

If k2 = m2, which of the following must be true?
(A)

k=m

(B)

k = –m

(C)

k = |m|

(D)

k = –|m|

(E)

|k| = |m|

Makoto, Nishi, and Ozuro were paid a total of $780
for waxing the floors at their school. Each was paid in
proportion to the number of hours he or she worked. If
Makoto worked 15 hours, Nishi worked 20 hours, and
Ozuro worked 30 hours, how much was Makoto paid?
(A)

$52

(B)

$117

(C)

$130

(D)

$180

(E)

$234
149

GMAT® Ofﬁcial Guide 2018

32.

6
R
29.

The figure above shows a path around a triangular
piece of land. Mary walked the distance of 8 miles
from P to Q and then walked the distance of 6 miles
from Q to R. If Ted walked directly from P to R, by what
percent did the distance that Mary walked exceed the
distance that Ted walked?

150

(B)

(C)

(D)

(E)

100

(A)

30%

(B)

40%

(C)

50%

(A)

(D)

60%

4
27

(E)

80%

(B)

1
3

(C)

(D)

(E)

27
4

1
At a supermarket, John spent of his money on fresh
2
1
1
on
fruits and vegetables, on meat products, and
3
10
bakery products. If he spent the remaining $6 on
candy, how much did John spend at the supermarket?
34.

31.

(A)

2
The quotient when a certain number is divided by is
3
9
. What is the number?
2

33.

30.

A carpenter constructed a rectangular sandbox with
a capacity of 10 cubic feet. If the carpenter were to
make a similar sandbox twice as long, twice as wide,
and twice as high as the first sandbox, what would be
the capacity, in cubic feet, of the second sandbox?

(A)

$60

(B)

$80

(C)

$90

(D)

$120

(E)

$180

If a sphere with radius r is inscribed in a cube with
edges of length e, which of the following expresses the
relationship between r and e ?
1
e
2

(A)

(B)

r=e

(C)

r = 2e

If (1 – 1.25)N = 1, then N =

(D)

r= e

(A)

–400

(E)

(B)

–140

(C)

–4

(D)

(E)

400

35.

1 2
e
4

If 2x + y = 7 and x + 2y = 5, then
(A)

(B)

4
3

(C)

17
5

(D)

18
5

(E)

x+y
=
3

5.3 Problem Solving Practice Questions

36.

City X has a population 4 times as great as the
population of City Y, which has a population twice as
great as the population of City Z. What is the ratio of
the population of City X to the population of City Z ?
(A)

1:8

(B)

1:4

(C)

2:1

(D)

4:1

(E)

8:1

40ft
15ft

35ft
house
20ft

39.

Tides at Bay Cove on July 13

height of tide

11:30 a.m.
2.2 ft
3

4:30 a.m.
0.3 ft

2
1
0

time

–1

6 a.m. noon

midnight

40.

6 p.m.
–0.5 ft
37.

38.

The graph above shows the height of the tide, in feet,
above or below a baseline. Which of the following is
closest to the difference, in feet, between the heights
of the highest and lowest tides on July 13 at Bay Cove?
(A)

1.7

(B)

1.9

(C)

2.2

(D)

2.5

(E)

2.7

patio

A manufacturer of a certain product can expect that
between 0.3 percent and 0.5 percent of the units
manufactured will be defective. If the retail price is
$2,500 per unit and the manufacturer offers a full
refund for defective units, how much money can the
manufacturer expect to need to cover the refunds on
20,000 units?
(A)

Between $15,000 and $25,000

(B)

Between $30,000 and $50,000

(C)

Between $60,000 and $100,000

(D)

Between $150,000 and $250,000

(E)

Between $300,000 and $500,000

41.

A flat patio was built alongside a house as shown in the
figure above. If all angles are right angles, what is the
area of the patio in square feet?
(A)

800

(B)

875

(C)

1,000

(D)

1,100

(E)

1,125

The sum of the weekly salaries of 5 employees is
$3,250. If each of the 5 salaries is to increase by
10 percent, then the average (arithmetic mean) weekly
salary per employee will increase by
(A)

$52.50

(B)

$55.00

(C)

$57.50

(D)

$62.50

(E)

$65.00

A student’s average (arithmetic mean) test score on
4 tests is 78. What must be the student’s score
on a 5th test for the student’s average score on the
5 tests to be 80 ?
(A)

(B)

(C)

(D)

(E)

151

GMAT® Ofﬁcial Guide 2018

42.

Last week Chris earned x dollars per hour for the first
40 hours worked plus 22 dollars per hour for each
hour worked beyond 40 hours. If last week Chris
earned a total of 816 dollars by working 48 hours,
what is the value of x ?
(A)

(B)

(C)

(D)

(E)

46.

47.
B
y°

(y + 10)°
A
43.

44.

In the figure above, what is the ratio of the measure of
angle B to the measure of angle A ?

(B)

(C)

(D)

(E)

In a certain sequence of 8 numbers, each number
after the first is 1 more than the previous number. If
the first number is –5, how many of the numbers in the
sequence are positive?
(A)

None

(B)

One

(C)

Two

(D)

Three

(E)

Four

2 to 3

(B)

3 to 4

(C)

3 to 5

(D)

4 to 5

A total of s oranges are to be packaged in boxes that
will hold r oranges each, with no oranges left over.
When n of these boxes have been completely filled,
what is the number of boxes that remain to be filled?

(E)

5 to 6

(A)

s – nr

(B)

s−

(C)

rs – n

(D)

s
−r
n

(E)

s
−n
r

48.

If n is a prime number greater than 3, what is the
remainder when n2 is divided by 12 ?
(A)

(B)

(C)

(D)

(E)

1
1
1+
3
(A)
(B)
(C)
(D)
(E)

152

(A)

49.
45.

The positive two‐digit integers x and y have the same
digits, but in reverse order. Which of the following must
be a factor of x + y ?

−

1
1
1+
2

1
3
1
−
6
1
−
12
1
12
1
3
−

n
r

If 0 < a < b < c, which of the following statements
must be true?
I.

2a > b + c

II.

c–a>b–a

III.

c b
<
a a

(A)

I only

(B)

II only

(C)

III only

(D)

I and II

(E)

II and III

5.3 Problem Solving Practice Questions

50.

51.

In the xy-plane, the origin O is the midpoint of line
segment PQ. If the coordinates of P are (r,s), what are
the coordinates of Q ?
(A)

(r,s)

(B)

(s,–r)

(C)

(–s,–r)

(D)

(–r,s)

(E)

(–r,–s)

53.

Which of the following equations is NOT equivalent to
10y2 = (x + 2)(x – 2) ?

Company Q plans to make a new product next year
and sell each unit of this new product at a selling price
of $2. The variable costs per unit in each production
run are estimated to be 40% of the selling price, and
the fixed costs for each production run are estimated
to be $5,040. Based on these estimated costs, how
many units of the new product will Company Q need to
make and sell in order for their revenue to equal their
total costs for each production run?
(A)

4,200

(B)

3,150

(C)

2,520

(A)

30y2 = 3x2 – 12

(D)

2,100

(B)

20y2 = (2x – 4)(x + 2)

(E)

1,800

(C)

10y2 + 4 = x2

(D)

5y 2 = x 2 – 2

(E)

y2 =

x2 − 4
10

Monday

Tuesday

Wednesday

Thursday

Company A

Company B

Company C

Company D

The dial shown above is divided into equal‐sized
intervals. At which of the following letters will the
pointer stop if it is rotated clockwise from S through
1,174 intervals?

Company E

(A)

(B)

(C)

(D)

(E)

52.

The table shows the numbers of packages shipped
daily by each of five companies during a 4‐day period.
The standard deviation of the numbers of packages
shipped daily during the period was greatest for which
of the five companies?
(A)

(B)

(C)

(D)

(E)

B
C

54.

153

GMAT® Ofﬁcial Guide 2018

Estimated Number of Home-Schooled
Students by State, January 2001

55.

State

Number (in thousands)

181

125

103

57.

According to the table shown, the estimated number
of home-schooled students in State A is approximately
what percent greater than the number in State D ?
(A)

25%

(B)

55%

(C)

100%

(D)

125%

(E)

155%

When n liters of fuel were added to a tank that was
1
7
already full, the tank was filled to of its capacity. In
3
9
terms of n, what is the capacity of the tank, in liters?
(A)

10
n
9

(B)

4
n
3

(C)

3
n
2

(D)

9
n
4

(E)

7
n
3
w ft

100 ft
150 ft
w ft
Note: Not drawn to scale.

y
II

I
x

O
III

56.

154

58.

The graph of the equation xy = k, where k < 0, lies in
which two of the quadrants shown above?
(A)

I and II

(B)

I and III

(C)

II and III

(D)

II and IV

(E)

III and IV

59.

The smaller rectangle in the figure above represents
the original size of a parking lot before its length
and width were each extended by w feet to make
the larger rectangular lot shown. If the area of the
enlarged lot is twice the area of the original lot, what is
the value of w ?
(A)

(B)

(C)

(D)

100

(E)

200

1
=
0.75 − 1
(A)

–4

(B)

–0.25

(C)

0.25

(D)

0.75

(E)

5.3 Problem Solving Practice Questions

60.

Kevin invested $8,000 for one year at a simple annual
interest rate of 6 percent and invested $10,000 for
one year at an annual interest rate of 8 percent
compounded semiannually. What is the total amount of
interest that Kevin earned on the two investments?

64.

(A)

320

(A)

$880

(B)

480

(B)

$1,088

(C)

520

(C)

$1,253

(D)

730

(D)

$1,280

(E)

920

(E)

$1,296
65.

61.

62.

63.

Car X averages 25.0 miles per gallon of gasoline
and Car Y averages 11.9 miles per gallon. If each car
is driven 12,000 miles, approximately how many more
gallons of gasoline will Car Y use than Car X ?

The harvest yield from a certain apple orchard was
350 bushels of apples. If x of the trees in the orchard
each yielded 10 bushels of apples, what fraction of the
harvest yield was from these x trees?
(A)

x
35

(B)

1−

(C)

10x

(D)

35 – x

(E)

350 – 10x

x
35
66.

If n is an integer, which of the following must be even?

If y is an integer, then the least possible value of
23 − 5y is
(A)

(B)

(C)

(D)

(E)

80 + 125 =
(A)

9 5

(B)

20 5

(C)

41 5
205

(A)

n+1

(D)

(B)

n+2

(E)

(C)

(D)

2n + 1

(E)

The sum
(A)
(B)

7 1
+ is between
8 9

1
3
and
2
4
3
and 1
4
1 and 1

(D)

(E)

y = kx + 3
67.

(C)

100

1
4

In the equation above, k is a constant. If y = 17 when
x = 2, what is the value of y when x = 4 ?
(A)

(B)

(C)

(D)

(E)

1
1
and 1
4
2
1
1 and 2
2

155

GMAT® Ofﬁcial Guide 2018

68.

69.

70.

156

Which of the following is greatest?
(A)

10 3

(B)

9 4

(C)

8 5

(D)

7 6

(E)

6 7

Al and Ben are drivers for SD Trucking Company. One
snowy day, Ben left SD at 8:00 a.m. heading east and
Al left SD at 11:00 a.m. heading west. At a particular
time later that day, the dispatcher retrieved data
from SD’s vehicle tracking system. The data showed
that, up to that time, Al had averaged 40 miles per
hour and Ben had averaged 20 miles per hour. It also
showed that Al and Ben had driven a combined total
of 240 miles. At what time did the dispatcher retrieve
data from the vehicle tracking system?
(A)

1:00 p.m.

(B)

2:00 p.m.

(C)

3:00 p.m.

(D)

5:00 p.m.

(E)

6:00 p.m.

71.

72.

73.

Of the land owned by a farmer, 90 percent was
cleared for planting. Of the cleared land, 40 percent
was planted with soybeans and 50 percent of the
cleared land was planted with wheat. If the remaining
720 acres of cleared land was planted with corn, how
many acres did the farmer own?
(A)

5,832

(B)

6,480

(C)

7,200

(D)

8,000

(E)

8,889

74.

At the start of an experiment, a certain population
consisted of 3 animals. At the end of each month after
the start of the experiment, the population size was
double its size at the beginning of that month. Which of
the following represents the population size at the end
of 10 months?
(A)

(B)

(C)

2(310)

(D)

3(210)

(E)

3(102)

1
1⎞
⎛ 1 1 1 1⎞
⎛1 1
If ⎜ + + + ⎟ = r ⎜ +
+
+
, then r =
⎝ 3 4 5 6⎠
⎝ 9 12 15 18 ⎟⎠
(A)

1
3

(B)

4
3

(C)

(D)

(E)

If x and y are positive integers such that y is a multiple
of 5 and 3x + 4y = 200, then x must be a multiple of
which of the following?
(A)

(B)

(C)

(D)

(E)

Which of the following expressions can be written as
an integer?
I.

⎛
⎜⎝ 82 + 82

II.

(82) (

)

III.

( 82 )( 82 )

(A)

None

(B)

I only

(C)

III only

(D)

I and II

(E)

I and III

5.3 Problem Solving Practice Questions

75.

76.

77.

78.

Four staff members at a certain company worked
on a project. The amounts of time that the four staff
members worked on the project were in the ratio 2 to
3 to 5 to 6. If one of the four staff members worked
on the project for 30 hours, which of the following
CANNOT be the total number of hours that the four
staff members worked on the project?
(A)

(B)

(C)

160

(D)

192

(E)

240

Pumping alone at their respective constant rates, one
1
inlet pipe fills an empty tank to of capacity in 3 hours
2
and a second inlet pipe fills the same empty tank to
2
of capacity in 6 hours. How many hours will it take
3
both pipes, pumping simultaneously at their respective
constant rates, to fill the empty tank to capacity?
(A)

3.25

(B)

3.6

(C)

4.2

(D)

4.4

(E)

5.5

In the xy‐coordinate plane, which of the following points
must lie on the line kx + 3y = 6 for every possible value
of k ?

79.

Book number

Pages in book

Total pages read

253

110

363

117

480

170

650

155

805

855

205

1,060

1,130

165

1,295

105

1,400

143

1,543

207

1,750

Shawana made a schedule for reading books during
4 weeks (28 days) of her summer vacation. She has
checked out 12 books from the library. The number of
pages in each book and the order in which she plans to
read the books are shown in the table above. She will
read exactly 50 pages each day. The only exception
will be that she will never begin the next book on the
same day that she finishes the previous one, and
therefore on some days she may read fewer than
50 pages. At the end of the 28th day, how many books
will Shawana have finished?
(A)

(A)

(1,1)

(B)

(0,2)

(C)

(2,0)

(D)

(3,6)

(E)

(6,3)

If x 2 – 2 < 0, which of the following specifies all the
possible values of x ?
(A)

0< x <2

(B)

0 512 ?

400,000
300,000
200,000
100,000
0

1990

(B)

(A)

250,000

(C)

(B)

280,000

(D)

(C)

310,000

(E)

(D)

325,000

(E)

340,000

1994

1996
year

1998

2000

165. For the positive integers a, b, and k, a k b means that
a k is a divisor of b, but a k +1 is not a divisor of b. If k is
a positive integer and 2k 72, then k is equal to

(A)

0.10

(A)

(B)

0.15

(B)

(C)

0.27

(C)

(D)

0.33

(D)

(E)

0.45

(E)

163. Each year for 4 years, a farmer increased the number
1
of trees in a certain orchard by of the number of
4
trees in the orchard the preceding year. If all of the
trees thrived and there were 6,250 trees in the orchard
at the end of the 4‐year period, how many trees were
in the orchard at the beginning of the 4‐year period?

1992

164. According to the chart shown, which of the following is
closest to the median annual number of shipments of
manufactured homes in the United States for the years
from 1990 to 2000, inclusive?

(A)

162. Sixty percent of the members of a study group are
women, and 45 percent of those women are lawyers.
If one member of the study group is to be selected
at random, what is the probability that the member
selected is a woman lawyer?

170

Number of Shipments of Manufactured Homes
in the United States, 1990–2000

number of shipments

160. The present ratio of students to teachers at a certain
school is 30 to 1. If the student enrollment were to
increase by 50 students and the number of teachers
were to increase by 5, the ratio of students to
teachers would then be 25 to 1. What is the present
number of teachers?

166. A certain characteristic in a large population has a
distribution that is symmetric about the mean m.
If 68 percent of the distribution lies within one
standard deviation d of the mean, what percent of the
distribution is less than m + d ?
(A)

16%

(B)

32%

(A)

1,250

(C)

48%

(B)

1,563

(D)

84%

(C)

2,250

(E)

92%

(D)

2,560

(E)

2,752

5.3 Problem Solving Practice Questions

167. Four extra‐large sandwiches of exactly the same size
were ordered for m students, where m > 4. Three
of the sandwiches were evenly divided among the
students. Since 4 students did not want any of the
fourth sandwich, it was evenly divided among the
remaining students. If Carol ate one piece from each
of the four sandwiches, the amount of sandwich that
she ate would be what fraction of a whole extra‐large
sandwich?

170. In a box of 12 pens, a total of 3 are defective. If a
customer buys 2 pens selected at random from the
box, what is the probability that neither pen will be
defective?
(A)

1
6

(B)

2
9

(A)

m+4
m (m − 4)

(C)

6
11

(B)

2m − 4
m (m − 4)

(D)

9
16

(C)

4m − 4
m (m − 4)

(E)

3
4

(D)

4m − 8
m (m − 4)

(E)

4m − 12
m (m − 4)

168. Which of the following equations has 1 + 2 as one of
its roots?

171. At a certain fruit stand, the price of each apple is
40 cents and the price of each orange is 60 cents.
Mary selects a total of 10 apples and oranges from the
fruit stand, and the average (arithmetic mean) price of
the 10 pieces of fruit is 56 cents. How many oranges
must Mary put back so that the average price of the
pieces of fruit that she keeps is 52 cents?

(A)

x 2 + 2x – 1 = 0

(A)

(B)

– 2x + 1 = 0

(B)

(C)

+ 2x + 1 = 0

(C)

(D)

– 2x – 1 = 0

(D)

(E)

–x–1=0

(E)

169. In Country C, the unemployment rate among
construction workers dropped from 16 percent on
September 1, 1992, to 9 percent on September 1,
1996. If the number of construction workers was
20 percent greater on September 1, 1996, than
on September 1, 1992, what was the approximate
percent change in the number of unemployed
construction workers over this period?

172. A pharmaceutical company received $3 million in
royalties on the first $20 million in sales of the generic
equivalent of one of its products and then $9 million
in royalties on the next $108 million in sales. By
approximately what percent did the ratio of royalties to
sales decrease from the first $20 million in sales to the
next $108 million in sales?
(A)

(A)

50% decrease

(B)

15%

(B)

30% decrease

(C)

45%

(C)

15% decrease

(D)

52%

(D)

30% increase

(E)

56%

(E)

55% increase

171

GMAT® Ofﬁcial Guide 2018

Times at Which the Door
Opened from 8:00 to 10:00
8:00

8:06

8:30

9:05

8:03

8:10

8:31

9:11

8:04

8:18

8:54

9:29

8:04

8:19

8:57

9:31

173. The light in a restroom operates with a 15-minute timer
that is reset every time the door opens as a person
goes in or out of the room. Thus, after someone
enters or exits the room, the light remains on for only
15 minutes unless the door opens again and resets the
timer for another 15 minutes. If the times listed above
are the times at which the door opened from 8:00 to
10:00, approximately how many minutes during this
two-hour period was the light off?
(A)

(B)

(C)

(D)

(E)

60°
174. The parallelogram shown has four sides of equal
length. What is the ratio of the length of the shorter
diagonal to the length of the longer diagonal?
(A)
(B)
(C)
(D)
(E)

172

1
2
1
2
1
2 2
1
3
1
2 3

175. If p is the product of the integers from 1 to 30,
inclusive, what is the greatest integer k for which
3k is a factor of p ?
(A)

(B)

(C)

(D)

(E)

176. If n = 38 – 28, which of the following is NOT a factor
of n ?
(A)

(B)

(C)

(D)

(E)

177. In the figure shown, if the area of the shaded region is
3 times the area of the smaller circular region, then the
circumference of the larger circle is how many times
the circumference of the smaller circle?
(A)

(B)

(C)

(D)

(E)

5.3 Problem Solving Practice Questions

178. Club X has more than 10 but fewer than 40 members.
Sometimes the members sit at tables with 3 members
at one table and 4 members at each of the other tables,
and sometimes they sit at tables with 3 members at
one table and 5 members at each of the other tables.
If they sit at tables with 6 members at each table
except one and fewer than 6 members at that one
table, how many members will be at the table that has
fewer than 6 members?
(A)

(B)

(C)

(D)

(E)

182. A researcher plans to identify each participant in a
certain medical experiment with a code consisting of
either a single letter or a pair of distinct letters written
in alphabetical order. What is the least number of
letters that can be used if there are 12 participants,
and each participant is to receive a different code?
(A)

(B)

(C)

(D)

(E)

8
y

179. In order to complete a reading assignment on time,
Terry planned to read 90 pages per day. However,
she read only 75 pages per day at first, leaving
690 pages to be read during the last 6 days before the
assignment was to be completed. How many days in
all did Terry have to complete the assignment on time?
(A)

(B)

(C)

(D)

(E)

180. If s > 0 and
(A)

(C)

s s

(D)

(E)

s2 − s

r
= s, what is r in terms of s ?
s

x
181. If 3 < x < 100, for how many values of x is the
3
square of a prime number?
(A)

Two

(B)

Three

(C)

Four

(D)

Five

(E)

Nine

1
O

183. The graph of which of the following equations is a
straight line that is parallel to line ℓ in the figure above?

1
s

(B)

(A)

3y – 2x = 0

(B)

3y + 2x = 0

(C)

3y + 2x = 6

(D)

2y – 3x = 6

(E)

2y + 3x = –6

184. An object thrown directly upward is at a height of h feet
after t seconds, where h = –16(t – 3)2 + 150. At what
height, in feet, is the object 2 seconds after it reaches
its maximum height?
(A)

(B)

(C)

134

(D)

150

(E)

166

173

GMAT® Ofﬁcial Guide 2018

185. Which of the following is equivalent to the pair of
inequalities x + 6 > 10 and x – 3 ≤ 5 ?
(A)

2 ≤ x < 16

(B)

2≤x<4

(C)

2 y. Two packages weighing 3 pounds and 5 pounds,
respectively, can be mailed separately or combined as
one package. Which method is cheaper, and how much
money is saved?
(A)

Combined, with a savings of x – y cents

(B)

Combined, with a savings of y – x cents

(C)

Combined, with a savings of x cents

(D)

Separately, with a savings of x – y cents

(E)

Separately, with a savings of y cents

191. If money is invested at r percent interest, compounded
annually, the amount of the investment will double
70
in approximately
years. If Pat’s parents invested
r
$5,000 in a long‐term bond that pays 8 percent
interest, compounded annually, what will be the
approximate total amount of the investment 18 years
later, when Pat is ready for college?
(A)

$20,000

(B)

$15,000

(C)

$12,000

(D)

$10,000

(E)

$9,000

5.3 Problem Solving Practice Questions

192. On a recent trip, Cindy drove her car 290 miles,
rounded to the nearest 10 miles, and used 12 gallons
of gasoline, rounded to the nearest gallon. The actual
number of miles per gallon that Cindy’s car got on this
trip must have been between

195. If y is the smallest positive integer such that
3,150 multiplied by y is the square of an integer,
then y must be
(A)

(B)

(A)

290
290
and
12.5
11.5

(C)

295
285
and
12
11.5

(D)

(B)

(E)

(C)

285
295
and
12
12

(D)

285
295
and
12.5
11.5

(E)

295
285
and
12.5
11.5

–5 –4 –3 –2 –1

196. If [x] is the greatest integer less than or equal to x,
what is the value of [–1.6] + [3.4] + [2.7] ?

193. Which of the following inequalities is an algebraic
expression for the shaded part of the number line
above?

(A)

(B)

(C)

(D)

(E)

197. In the first week of the year, Nancy saved $1. In each
of the next 51 weeks, she saved $1 more than she
had saved in the previous week. What was the total
amount that Nancy saved during the 52 weeks?

(A)

|x|≤3

(B)

|x|≤5

(A)

$1,326

(C)

|x–2|≤3

(B)

$1,352

(D)

|x–1|≤4

(C)

$1,378

(E)

|x+1|≤4

(D)

$2,652

(E)

$2,756

194. In a small snack shop, the average (arithmetic mean)
revenue was $400 per day over a 10‐day period.
During this period, if the average daily revenue was
$360 for the first 6 days, what was the average daily
revenue for the last 4 days?

198. In a certain sequence, the term xn is given by the
1
formula x n = 2x n − 1 − ( x n − 2 ) for all n ≥ 2. If x 0 = 3
2
and x1 = 2, what is the value of x 3 ?

(A)

$420

(A)

2.5

(B)

$440

(B)

3.125

(C)

$450

(C)

(D)

$460

(D)

(E)

$480

(E)

6.75

175

GMAT® Ofﬁcial Guide 2018

199. During a trip, Francine traveled x percent of the total
distance at an average speed of 40 miles per hour
and the rest of the distance at an average speed of
60 miles per hour. In terms of x, what was Francine’s
average speed for the entire trip?
(A)

180 − x
2

(B)

x + 60
4

(C)

300 − x
5

(D)

600
115 − x

(E)

12,000
x + 200

200. If n =

(33)43

(A)

One

(B)

Two

(C)

Three

(D)

Seven

(E)

Ten

204. For any positive integer n, the sum of the first
n(n + 1)
n positive integers equals
. What is the sum of
2
all the even integers between 99 and 301 ?

(43)33,

what is the units digit of n ?

(A)

10,100

(B)

20,200

(C)

22,650

(A)

(D)

40,200

(B)

(E)

45,150

(C)

(D)

(E)

205. How many prime numbers between 1 and 100 are
factors of 7,150 ?

201. Team A and Team B are competing against each other
in a game of tug-of-war. Team A, consisting of 3 males
and 3 females, decides to line up male, female, male,
female, male, female. The lineup that Team A chooses
will be one of how many different possible lineups?
(A)

(B)

(C)

(D)

(E)

720

202. A border of uniform width is placed around a
rectangular photograph that measures 8 inches by
10 inches. If the area of the border is 144 square
inches, what is the width of the border, in inches?

176

1
is expressed as a terminating decimal,
2 × 57
how many nonzero digits will d have?

203. If d =

(A)

(B)

(C)

(D)

(E)

(A)

One

(B)

Two

(C)

Three

(D)

Four

(E)

Five

206. A sequence of numbers a1, a2, a3, . . . is defined as
follows: a1 = 3, a2 = 5, and every term in the sequence
after a2 is the product of all terms in the sequence
preceding it, e.g., a3 = (a1)(a2) and a4 = (a1)(a2)(a3).
If an = t and n > 2, what is the value of an + 2 in terms
of t ?
(A)

(B)

(C)

(D)

(E)

5.3 Problem Solving Practice Questions

207. Last year the price per share of Stock X increased
by k percent and the earnings per share of Stock X
increased by m percent, where k is greater than m.
By what percent did the ratio of price per share to
earnings per share increase, in terms of k and m ?

210. A photography dealer ordered 60 Model X cameras to
be sold for $250 each, which represents a 20 percent
markup over the dealer’s initial cost for each camera.
Of the cameras ordered, 6 were never sold and
were returned to the manufacturer for a refund of
50 percent of the dealer’s initial cost. What was the
dealer’s approximate profit or loss as a percent of the
dealer’s initial cost for the 60 cameras?

(A)

k
%
m

(B)

(k – m)%
100(k − m )
%
100 + k

(A)

7% loss

(C)

(B)

13% loss

(D)

100(k − m )
%
100 + m

(C)

7% profit

(D)

13% profit

(E)

100(k − m )
%
100 + k + m

(E)

15% profit

208. Of the 300 subjects who participated in an experiment
using virtual‐reality therapy to reduce their fear of
heights, 40 percent experienced sweaty palms,
30 percent experienced vomiting, and 75 percent
experienced dizziness. If all of the subjects
experienced at least one of these effects and
35 percent of the subjects experienced exactly two of
these effects, how many of the subjects experienced
only one of these effects?

211. Seven pieces of rope have an average (arithmetic
mean) length of 68 centimeters and a median length of
84 centimeters. If the length of the longest piece of rope
is 14 centimeters more than 4 times the length of the
shortest piece of rope, what is the maximum possible
length, in centimeters, of the longest piece of rope?
(A)

(B)

118

(C)

120

(A)

105

(D)

134

(B)

125

(E)

152

(C)

130

(D)

180

(E)

195

1
209. If m −1 = − , then m −2 is equal to
3

212. What is the difference between the sixth and the fifth
terms of the sequence 2, 4, 7, … whose nth term is
n + 2n − 1 ?
(A)

(B)

(A)

−9

(C)

(B)

−3

(D)

(C)

−

1
9

(E)

(D)

1
9

(E)

213. From the consecutive integers –10 to 10, inclusive,
20 integers are randomly chosen with repetitions
allowed. What is the least possible value of the product
of the 20 integers?
(A)

(−10)20

(B)

(−10)10

(C)

(D)

−(10)19

(E)

−(10)20
177

GMAT® Ofﬁcial Guide 2018

214. The letters D, G, I, I, and T can be used to form 5‐letter
strings such as DIGIT or DGIIT. Using these letters, how
many 5‐letter strings can be formed in which the two
occurrences of the letter I are separated by at least
one other letter?

217. For the past n days, the average (arithmetic mean)
daily production at a company was 50 units. If today’s
production of 90 units raises the average to 55 units
per day, what is the value of n ?
(A)

(A)

(B)

(C)

(D)

(E)

48
y

215. Last Sunday a certain store sold copies of Newspaper A
for $1.00 each and copies of Newspaper B for $1.25
each, and the store sold no other newspapers that
day. If r percent of the store’s revenue from newspaper
sales was from Newspaper A and if p percent of
the newspapers that the store sold were copies of
Newspaper A, which of the following expresses r in
terms of p ?

216.

178

(A)

100p
125 − p

(B)

150p
250 − p

(C)

300p
375 − p

(D)

400p
500 − p

(E)

500p
625 − p

0.99999999 0.99999991
−
=
1.0001
1.0003
(A)

10−8

(B)

3(10−8 )

(C)

3(10−4 )

(D)

2(10−4 )

(E)

10−4

1
O

218. In the coordinate system above, which of the following
is the equation of line ?
(A)

2x − 3y = 6

(B)

2x + 3y = 6

(C)

3x + 2y = 6

(D)

2x − 3y = −6

(E)

3x − 2y = −6

219. If a two‐digit positive integer has its digits reversed,
the resulting integer differs from the original by 27. By
how much do the two digits differ?
(A)

(B)

(C)

(D)

(E)

5.3 Problem Solving Practice Questions

220. In an electric circuit, two resistors with resistances x
and y are connected in parallel. In this case, if r is the
combined resistance of these two resistors, then the
reciprocal of r is equal to the sum of the reciprocals of
x and y. What is r in terms of x and y ?

⎛ 1⎞
223. ⎜ ⎟
⎝ 2⎠

−3

⎛ 1⎞
⎜⎝ ⎟⎠
4

−2

⎛ 1⎞
⎜⎝ ⎟⎠
2

−48

(A)

⎛ 1⎞
⎜⎝ ⎟⎠
2

−11

(B)

(A)

(B)

x+y

(C)

⎛ 1⎞
⎜⎝ ⎟⎠
2

−6

1
x+y

(D)

⎛ 1⎞
⎜⎝ ⎟⎠
8

−11

xy
x+y

(E)

⎛ 1⎞
⎜⎝ ⎟⎠
8

−6

x+y
xy

11
8

(B)

7
8

(C)

9
64

(D)

5
64

(E)

3
64

222. If

1
1
1
−
=
, then x could be
x x +1 x + 4

(A)

(B)

–1

(C)

–2

(D)

–3

(E)

–4

−1

a°

221. Xavier, Yvonne, and Zelda each try independently to
solve a problem. If their individual probabilities for
1 1
5
success are , , and , respectively, what is the
4 2
8
probability that Xavier and Yvonne, but not Zelda, will
solve the problem?
(A)

⎛ 1⎞
⎜⎝ ⎟⎠
16

224. The figure shown above consists of a shaded 9-sided
polygon and 9 unshaded isosceles triangles. For each
isosceles triangle, the longest side is a side of the
shaded polygon and the two sides of equal length are
extensions of the two adjacent sides of the shaded
polygon. What is the value of a ?
(A)

100

(B)

105

(C)

110

(D)

115

(E)

120

179

GMAT® Ofﬁcial Guide 2018

225. List T consists of 30 positive decimals, none of which
is an integer, and the sum of the 30 decimals is S. The
estimated sum of the 30 decimals, E, is defined as
follows. Each decimal in T whose tenths digit is even
is rounded up to the nearest integer, and each decimal
in T whose tenths digit is odd is rounded down to the
nearest integer; E is the sum of the resulting integers.
1
If of the decimals in T have a tenths digit that is
3
even, which of the following is a possible value of
E–S?

(A)

(B)

(C)

(D)

(E)

229. Of the 150 houses in a certain development,
60 percent have air-conditioning, 50 percent have a
sunporch, and 30 percent have a swimming pool. If
5 of the houses have all three of these amenities and
5 have none of them, how many of the houses have
exactly two of these amenities?

–16

II.

III.

(A)

I only

(B)

I and II only

(C)

I and III only

(A)

(D)

II and III only

(B)

(E)

I, II, and III

(C)

(D)

(E)

6
226. If 5 − = x , then x has how many possible values?
x
(A)

None

(B)

One

(C)

Two

(D)

A finite number greater than two

(E)

An infinite number

227. Seed mixture X is 40 percent ryegrass and 60 percent
bluegrass by weight; seed mixture Y is 25 percent
ryegrass and 75 percent fescue. If a mixture of X and
Y contains 30 percent ryegrass, what percent of the
weight of the mixture is X ?

180

228. How many of the integers that satisfy the inequality
( x + 2)( x + 3)
≥ 0 are less than 5 ?
x −2

(A)

10%

(B)

1
33 %
3

(C)

40%

(D)

50%

(E)

2
66 %
3

2−14 + 2−15 + 2−16 + 2−17
is how many
5
times the value of 2( −17) ?

230. The value of

(A)

3
2

(B)

5
2

(C)

(D)

(E)

5.4 Problem Solving Answer Key

5.4 Answer Key
1.

31.

61.

91.

32.

62.

92.

33.

63.

93.

34.

64.

94.

35.

65.

95.

36.

66.

96.

37.

67.

97.

38.

68.

98.

39.

69.

99.

10.

40.

70.

100.

11.

41.

71.

101.

12.

42.

72.

102.

13.

43.

73.

103.

14.

44.

74.

104.

15.

45.

75.

105.

16.

46.

76.

106.

17.

47.

77.

107.

18.

48.

78.

108.

19.

49.

79.

109.

20.

50.

80.

110.

21.

51.

81.

111.

22.

52.

82.

112.

23.

53.

83.

113.

24.

54.

84.

114.

25.

55.

85.

115.

26.

56.

86.

116.

27.

57.

87.

117.

28.

58.

88.

118.

29.

59.

89.

119.

30.

60.

90.

120.

181

GMAT® Ofﬁcial Guide 2018

121.

151.

181.

211.

122.

152.

182.

212.

123.

153.

183.

213.

124.

154.

184.

214.

125.

155.

185.

215.

126.

156.

186.

216.

127.

157.

187.

217.

128.

158.

188.

218.

129.

159.

189.

219.

130.

160.

190.

220.

131.

161.

191.

221.

132.

162.

192.

222.

133.

163.

193.

223.

134.

164.

194.

224.

135.

165.

195.

225.

136.

166.

196.

226.

137.

167.

197.

227.

138.

168.

198.

228.

139.

169.

199.

229.

140.

170.

200.

230.

141.

171.

201.

142.

172.

202.

143.

173.

203.

144.

174.

204.

145.

175.

205.

146.

176.

206.

147.

177.

207.

148.

178.

208.

149.

179.

209.

150.

180.

210.

182

5.5 Problem Solving Answer Explanations

5.5 Answer Explanations
The following discussion is intended to familiarize you with the most efﬁcient and effective approaches
to the kinds of problems common to problem solving questions. The particular questions in this chapter
are generally representative of the kinds of problem solving questions you will encounter on the GMAT
exam. Remember that it is the problem solving strategy that is important, not the speciﬁc details of a
particular question.

Arithmetic Operations with rational numbers

The price of a coat in a certain store is $500. If the
price of the coat is to be reduced by $150, by what
percent is the price to be reduced?
(A)

10%

(B)

15%

(C)

20%

(D)

25%

(E)

30%

At the exchange rate of 0.80 euro per dollar,
Rose exchanged $500.00 for (0.80)(500) =
3
400 euros. She spent (400) = 300 euros, had
4
400 – 300 = 100 euros left, and exchanged them
for dollars at the exchange rate of $1.20 per euro.
Therefore, the dollar amount she received was
(1.20)(100) = $120.00.
The correct answer is D.

Arithmetic Percents

A reduction of $150 from $500 represents a
⎛ 150
⎞
× 100 ⎟ % = 30% .
percent decrease of ⎜
⎝ 500
⎠
Therefore, the price of the coat was reduced
by 30%.
The correct answer is E.
2.

$60.00

(B)

$80.00

(C)

$100.00

(D)

$120.00

(E)

$140.00

Each of the 12 squares shown is labeled x, y, v, or
w. What is the ratio of the number of these squares
labeled x or y to the number of these squares labeled v
or w ?
(A)

1:2

(B)

2:3

(C)

4:3

(D)

3:2

(E)

2:1

Arithmetic Ratio and proportion

By a direct count, there are 8 squares labeled
x or y (5 labeled x, 3 labeled y) and there are
4 squares labeled v or w (2 labeled v, 2 labeled
w). Therefore, the ratio of the number of squares
labeled x or y to the number of squares labeled v
or w is 8:4, which reduces to 2:1.
The correct answer is E.

183

GMAT® Ofﬁcial Guide 2018

tulips, or 85. Thus, the number of bouquets must
be a factor of 85, and so the number must be 1, 5,
17, or 85. Since the number of bouquets must be
1, 3, 5, or 15, and the number of bouquets must
be 1, 5, 17, or 85, it follows that the number of
bouquets must be 1 or 5, and thus the greatest
number of bouquets that can be made is 5. Note
that each of the 5 bouquets will have 3 white
tulips, because (5)(3) = 15, and each of the
5 bouquets will have 17 red tulips, because (5)
(17) = 85.

1 1 5 1 1 9
+ − + + −
=
3 2 6 5 4 20
(A)

(B)

2
15

(C)

2
5

(D)

9
20

(E)

5
6

Arithmetic Operations with rational numbers

A number that is divisible by each of the
denominators is 60. Therefore, 60 can be used as a
common denominator, which gives the following:

The correct answer is B.
6.

20 30 50 12 15 27
+
−
+
+
−
=
60 60 60 60 60 60
20 + 30 − 50 + 12 + 15 − 27 0
=
= 0.
60
60

(B)

(C)

(D)

(E)

Arithmetic Applied problems; Properties of
numbers

Because all the tulips are to be used and the same
number of white tulips will be in each bouquet,
the number of white tulips in each bouquet times
the number of bouquets must equal the total
number of white tulips, or 15. Thus, the number
of bouquets must be a factor of 15, and so the
number must be 1, 3, 5, or 15. Also, the number
of red tulips in each bouquet times the number
of bouquets must equal the total number of red
184

(A)

5.125

(B)

5.25

(C)

(D)

6.125

(E)

6.25

Arithmetic Percents

The correct answer is A.
5.

125% of 5 =

125% of 5 represents

125
× 5, or 1.25 × 5 = 6.25.
100

The correct answer is E.
7.

Today Rebecca, who is 34 years old, and her daughter,
who is 8 years old, celebrate their birthdays. How
many years will pass before Rebecca’s age is twice her
daughter’s age?
(A)

(B)

(C)

(D)

(E)

Algebra Applied problems

Let x be the desired number of years. In x years,
Rebecca will be 34 + x years old and her
daughter will be 8 + x years old. From the given
information, it follows that 34 + x = 2(8 + x). The
last equation is equivalent to 34 + x = 16 + 2x,
which has solution x = 18.
The correct answer is C.

5.5 Problem Solving Answer Explanations

When traveling at a constant speed of 32 miles per
hour, a certain motorboat consumes 24 gallons of fuel
per hour. What is the fuel consumption of this boat at
this speed measured in miles traveled per gallon of
fuel?

10.

(A)

2
3

(A)

(B)

10%

(B)

3
4

(C)

25%

(D)

40%

(C)

4
5

(E)

55%

(D)

4
3

(E)

3
2

Arithmetic Percents

In completing the drive to the service center, the
technician has completed 50% of the round‐trip.
The drive from the center is the other 50% of
the round‐trip. In completing 10% of the drive
from the center, the technician has completed
an additional 10% of 50%, or 5% of the
round‐trip. Thus, the technician has completed
50% + 5% = 55% of the round‐trip.

Arithmetic Operations with rational numbers

If the motorboat consumes 24 gallons of fuel
in 1 hour, then it consumes 1 gallon of fuel in
1
hour. If the motorboat travels 32 miles in
24
1
32 4
hour,
1 hour, then it travels
= miles in
24
24 3
which is the length of time it takes to consume
1 gallon of fuel. Thus, the motorboat travels
4
miles per gallon of fuel.
3

The correct answer is E.
11.

Rafﬂe tickets numbered consecutively from 101
through 350 are placed in a box. What is the
probability that a ticket selected at random will have a
number with a hundreds digit of 2 ?

The correct answer is D.
(A)
9.

A case contains c cartons. Each carton contains
b boxes, and each box contains 100 paper clips.
How many paper clips are contained in 2 cases?

(B)
(C)

(A)

100bc

(B)

100b
c

(D)

(C)

200bc

(E)

(D)

200b
c

(E)

200
bc

Algebra Simplifying algebraic expressions

Each case has bc boxes, each of which has
100 paper clips. The total number of paper clips
in 2 cases is thus 2( bc )(100) = 200 bc .
The correct answer is C.

2
5
2
7
33
88
99
250
100
249

Arithmetic Probability

There are 250 integers from 101 to 350 inclusive,
100 of which (that is, 200 through 299) have a
hundreds digit of 2. Therefore, the probability that
a ticket selected from the box at random will have
100 2
a hundreds digit of 2 can be expressed as
= .
250 5
The correct answer is A.

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GMAT® Ofﬁcial Guide 2018

12.

When Leo imported a certain item, he paid a 7 percent
import tax on the portion of the total value of the item in
excess of $1,000. If the amount of the import tax that
Leo paid was $87.50, what was the total value of the item?

14.

(A)

$1,600

(B)

$1,850

(C)

$2,250

(D)

$2,400

(E)

$2,750

II.

Algebra First-degree equations

III.

Letting x represent the total value of the item,
convert the words to symbols and solve the equation.

(A)

II only

(B)

III only

7% of value in excess of $1, 000 = 87.50

(C)

I and II only

(D)

II and III only

(E)

I, II, and III

0.07( x − 1, 000) = 87.50
x − 1, 000 = 1, 250
x = 2, 250

Arithmetic Statistics

Listed in numerical order, the given numbers are
2, 3, 4, 6, 7, and 8. If the seventh number were 2
or 4, then the numbers in numerical order would
be 2, 2, 3, 4, 6, 7, and 8 or 2, 3, 4, 4, 6, 7, and 8.
In either case the median would be 4 and the
2 + 2 + 3 + 4 + 6 + 7 + 8 32
=
or
average would be
7
7
2 + 3 + 4 + 4 + 6 + 7 + 8 34
=
, neither of which
7
7
equals 4. So, for neither of the values in I or
II does the average equal the median. If the
seventh number were 5, then the numbers in
numerical order would be 2, 3, 4, 5, 6, 7, and 8.
The median would be 5 and the average would
2 + 3 + 4 + 5 + 6 + 7 + 8 35
=
= 5. Thus, for the
be
7
7
value in III, the average equals the median.

The correct answer is C.
13.

A collection of 16 coins, each with a face value of either
10 cents or 25 cents, has a total face value of $2.35.
How many of the coins have a face value of 25 cents?
(A)

(B)

(C)

(D)

(E)

Algebra First-degree equations

Let x represent the number of coins each with
a face value of 25 cents. Then, since there are
16 coins in all, 16 – x represents the number of
coins each with a face value of 10 cents. The total
face value of the coins is $2.35 or 235 cents so,
25x + 10(16 – x)
25x + 160 – 10x
15x + 160
15x
x

186

=
=
=
=
=

235
235
235
75
5

given
distributive property
combine like terms
subtract 160 from both sides
divide both sides by 15

The correct answer is B.
15.

If it is assumed that 60 percent of those who receive a
questionnaire by mail will respond and 300 responses
are needed, what is the minimum number of
questionnaires that should be mailed?
(A)

400

(B)

420

Therefore, 5 of the coins have a face value of
25 cents.

(C)

480

(D)

500

The correct answer is B.

(E)

600

5.5 Problem Solving Answer Explanations

Arithmetic Percents

expression in C cannot be the greatest. Finally,
the value of the expression in E is greater than
the value of the expression in D, since z > y
implies xz > xy, which implies xz + yz > xy + yz,
which implies z(x + y) > y(x + z), and so the value
of the expression in D cannot be the greatest.
Since none of the values of the expressions in A,
B, C, or D can be the greatest, it follows that the
value of the expression in E is the greatest.

From the given information, 60% of the
minimum number of questionnaires is equal to
300. This has the form “60% of what number
equals 300,” and the number can be determined
by dividing 300 by 60%. Performing the
100
60
= 300 ×
= 500.
calculation gives 300 ÷
100
60
The correct answer is D.

The correct answer is E.
16.

If 1 < x < y < z, which of the following has the greatest
value?
(A)

z(x + 1)

(B)

z(y + 1)

(C)

x(y + z)

(D)

y(x + z)

(E)

z(x + y)

17.

A rectangular garden is to be twice as long as it is
wide. If 360 yards of fencing, including the gate, will
completely enclose the garden, what will be the length
of the garden, in yards?
(A)

120

(B)

140

(C)

160

Algebra Inequalities

(D)

180

This problem can be solved by calculating each
of the options for a fixed and appropriate choice
of values for each of the variables. For example, if
x = 2, y = 3, and z = 4, then 1 < x < y < z and the
values of the options are as follows:

(E)

200

Geometry Quadrilaterals; Perimeter

Let the width of the rectangle be x. Then the
length is 2x. Since the perimeter of a rectangle is
twice the sum of the length and width, it follows
that
360 = 2 ( x + 2x )

z(x + 1) = 4(2 + 1) = 12

z(y + 1) = 4(3 + 1) = 16

x(y + z) = 2(3 + 4) = 14

y(x + z) = 3(2 + 4) = 18

360 = 6x
120 = 2x

z(x + y) = 4(2 + 3) = 20

So, the length is 120.

This problem can also be solved by the use of
algebraic ordering principles in a way that does
not assume the same answer option is true
for each choice of values of x, y, and z such
that 1 < x < y < z. First, note that neither the
value of the expression in A nor the value of
the expression in B can ever have the greatest
value, since the value of the expression in E is
greater than the value of the expression in A
(because y > 1) and the value of the expression
in E is greater than the value of the expression
in B (because x > 1). Also, the value of the
expression in D is greater than the value of the
expression in C, since y > x implies yz > xz,
which implies xy + yz > xy + xz, which implies
y(x + z) > x(y + z), and so the value of the

The correct answer is A.
18.

$200

(B)

$240

(C)

$480

(D)

$960

(E)

$1,920

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GMAT® Ofﬁcial Guide 2018

given that F + Np + Nh = 140, Np = Nh + 20, and
F = 2Np = 2(Nh + 20). It follows that

Geometry Area (Rectangles)

The area of the floor is (10 m)(8 m) = 80
and
the area of each carpet square is (2 m)(2 m) = 4 m2.
Therefore, the number of carpet squares needed
80
to cover the floor is
= 20, and these 20 carpet
4
squares have a total cost of (20)($12) = $240.
m2

F + Np + Nh = 140 given
2(Nh + 20) + (Nh + 20) + Nh = 140 by substitution
4Nh + 60 = 140 combine like
terms
4Nh = 80 subtract 60
from both sides
Nh = 20 divide both
sides by 4

The correct answer is B.
19.

If 893 × 78 = p, which of the following is equal to
893 × 79 ?
(A)

p+1

(B)

p + 78

(C)

p + 79

(D)

p + 893

(E)

p + 894

The correct answer is B.
21.

Arithmetic Properties of numbers

893 × 79 = 893 × (78 + 1)
since 79 = 78 + 1
= (893 × 78) + 893 distributive property
= p + 893
since p = 893 × 78

(B)

(C)

(D)

(E)

Algebra Simultaneous ﬁrst-degree equations

Let F represent the number of paperback fiction
books that Thabo owns; Np, the number of
paperback nonfiction books; and Nh, the number
of hardcover nonfiction books. It is

188

(B)

(C)

(D)

(E)

From the given information and the
definition of average, it follows that
3 + 15 + 32 + ( N + 1)
51 + N
= 18, or
= 18.
4
4
Multiplying both sides of the last equation by 4
gives 51 + N = 72. Therefore, N = 72 – 51 = 21.

(A)

Arithmetic Statistics

The correct answer is D.
20.

If the average (arithmetic mean) of the four numbers 3,
15, 32, and (N + 1) is 18, then N =

The correct answer is C.
22.

(A)

(B)

(C)

(D)

(E)

5.5 Problem Solving Answer Explanations

Geometry Applied problems

In the diagram, A represents the location of
Abdul’s house, S represents the location of the
school, B represents the location of Barb’s house,
and C represents the location of Carlos’s house.
Because the school is halfway between Abdul’s
house and Barb’s house, S is the midpoint of AB ,
and because Barb’s house is halfway between
the school and Carlos’s house, B is the midpoint
of SC. Therefore, AS = SB = BC. Finally, since
SC = 4, it follows that AS = SB = BC = 2 and
hence AC = 2 + 2 + 2 = 6.

shows that dividing these two known rates
leads to the appropriate unit cancellations:
mi 2 hr
mi
mi 5 gal
60
÷
= 60
×
= 24
.
hr 2 hr
hr 5 gal
gal
Therefore, after using 3.75 gal of fuel, the car has
mi
traveled 24
× 3.75 gal = 90 miles.
gal
The correct answer is E.
24.

The correct answer is D.
23.

$320

(B)

$540

(C)

$560

(D)

$620

(E)

$680

(A)

(B)

37.5

Algebra Applied problems; Percents

(C)

(D)

(E)

The price of each pine door is $40, so the price of
5 pine doors is 5($40) = $200. The price of each
oak door is twice that of a pine door, and thus
$80, which becomes (0.75)($80) = $60 when the
25% discount is applied. Therefore, the price of
6 oak doors at the 25% discount is 6($60) = $360,
and hence the total price of all 11 doors is
$200 + $360 = $560.

Arithmetic Applied problems

The car traveled at a rate of
mi ⎛ mi ⎞ ⎛
min ⎞
mi
1
= ⎜1
60
= 60
using fuel
⎟
⎜
min ⎝ min ⎠ ⎝
hr ⎟⎠
hr
5 gal
. If
at a rate of 5 gallons every 2 hours, or
2 hr
the usage rate in miles per gallon was known,
then that usage rate times 3.75 gallons would give
the number of miles.
One approach to finding this usage rate
is to consider whether multiplying or
mi
5 gal
and
dividing the known rates 60
hr
2 hr
will lead to unit cancellations that result in
miles per gallon. The following calculation

The correct answer is C.
25.

If y −

1 11
< , which of the following could be a value
2
2

of y ?
(A)

−11

(B)

−

11
2

(C)

11
2

(D)

(E)

189

GMAT® Ofﬁcial Guide 2018

Algebra Inequalities; Absolute value

Algebra Simplifying algebraic expressions

1 11
<
is equivalent to
2
2
11
1 11
10
12
− < y − < , or −
< y < , select the
2
2 2
2
2
10
12
= 6.
value that lies between − = −5 and
2
2
11
That value is .
2

One method of solving this is to first take
the nonnegative square root of both sides of
the equation k 2 = m 2 and then make use of the
fact that u 2 = u . Doing this gives |k| = |m|.
Alternatively, if (k, m) is equal to either of
the pairs (1,1) or (–1,1), then k 2 = m 2 is true.
However, each of the answer choices except
|k| = |m| is false for at least one of these two
pairs.

Since y −

The correct answer is C.

The correct answer is E.
B
28.

120°
A
26.

In the figure shown, AC = 2 and BD = DC = 1. What is
the measure of angle ABD ?

$52

(B)

$117

(C)

$130

(D)

$180

(E)

$234

15°

(B)

20°

Arithmetic Ratio and proportion

(C)

30°

(D)

40°

(E)

45°

Makoto, Nishi, and Ozuro worked a total of
15 + 20 + 30 = 65 hours and were paid a total of
$780. Each was paid in proportion to the number
of hours he or she worked. Therefore, Makoto was
15
paid ($780) = $180.
65

Since AC = AD + DC, and it is given that AC = 2
and DC = 1, it follows that AD = 1. Therefore,
AD = BD = 1 and ΔABD is an isosceles triangle
where the measure of ∠ABD is equal to the
measure of ∠BAD. Letting x° be the common
degree measure of these two angles, it follows that
x° + x° + 120° = 180°, or 2x° = 60°, or x° = 30°.
The correct answer is C.

190

(A)

Geometry Triangles

27.

If k 2 = m 2, which of the following must be true?
(A)

k=m

(B)

k = −m

(C)

k = |m|

(D)

k = −|m|

(E)

|k| = |m|

The correct answer is D.

5.5 Problem Solving Answer Explanations

1
of the total. It follows that
15
the total is (15)($6) = $90.

so the $6 left was

The correct answer is C.
R
31.
29.

30%

(B)

40%

(C)

50%

(D)

60%

(E)

80%

The correct answer is B.
30.

1
At a supermarket, John spent of his money on fresh
2
1
1
fruits and vegetables, on meat products, and
3
10
on bakery products. If he spent the remaining $6 on
candy, how much did John spend at the supermarket?
(A)

$60

(B)

$80

(C)

$90

(D)

$120

(E)

$180

(A)

–400

(B)

–140

(C)

–4

(D)

(E)

400

Algebra Operations with rational numbers

1
Since (1 – 1.25)N = –0.25N = − N, the equation
4
1
becomes – N = 1, which has solution N = –4.
4
The correct answer is C.

Geometry Pythagorean theorem

Mary walked a distance of 6 + 8 = 14 miles.
The distance that Ted walked, PR, can be
found by using the Pythagorean theorem
6 2 + 8 2 = ( PR )2 , or ( PR )2 = 100. Taking square
roots, it follows that Ted walked 10 miles.
Therefore, the distance Mary walked exceeded the
distance Ted walked by 14 – 10 = 4 miles and 4 is
40% of 10.

If (1 – 1.25)N = 1, then N =

32.

(B)

(C)

(D)

(E)

100

Geometry Volume

When all the dimensions of a three‐dimensional
object are changed by a factor of 2, the
capacity, or volume, changes by a factor of
(2)(2)(2) = 23 = 8. Thus the capacity of the second
sandbox is 10(8) = 80 cubic feet.
The correct answer is D.

Arithmetic Fractions

The amount spent was
1 1 1 15 10 3 28 14
+ +
=
+
+
=
=
of the total
2 3 10 30 30 30 30 15
191

GMAT® Ofﬁcial Guide 2018

33.

2
The quotient when a certain number is divided by is
3
9
. What is the number?
2

35.

(A)

4
27

(B)

4
3

(B)

1
3

(C)

17
5

(C)

(D)

18
5

(D)

(E)

27
4

Algebra Simultaneous equations

Recall the definition of division: a ÷ b = c means
a = bc. For example, if a certain number divided
by 2 equals 3, then the number is 2 × 3 = 6. For
2
this problem, a certain number divided by
3
2 9
9
equals , so the number is × = 3.
3 2
2
This problem can also be solved by algebra. Let N
2 9
N 9
be the number. Then
= , or N = × = 3.
2 2
3 2
3
The correct answer is C.
If a sphere with radius r is inscribed in a cube with
edges of length e, which of the following expresses the
relationship between r and e ?
1
e
2

(A)

(B)

r=e

(C)

r = 2e

(D)

r= e

(E)

1 2
e
4

Adding the equations 2x + y = 7 and x + 2y = 5
gives 3x + 3y = 12, or x + y = 4. Dividing both
x+ y 4
sides of the last equation by 3 gives
= .
3
3
The correct answer is B.
36.

1:8

(B)

1:4

(C)

2:1

(D)

4:1

(E)

8:1

Arithmetic Ratio and Proportion

Let X, Y, and Z be the populations of Cities X,
Y, and Z, respectively. It is given that X = 4Y,
4Y
Y
X
and Y = 2Z or Z = . Then,
=
=
Y
2
Z
⎛ 2⎞ 8
(4Y)⎜ ⎟ = .
2
⎝Y ⎠ 1
The correct answer is E.

Geometry Volume

A sphere inscribed in a cube touches, but does
not extend beyond, each of the 6 sides of the
cube, and thus the diameter of the sphere is equal
to the distance between a pair of opposite sides of
the cube, which is equal to the edge length of the
cube. Therefore, the radius of the sphere is equal
1
to half the edge length of the cube, or r = e.
2
The correct answer is A.
192

x+y
=
3

(A)

Arithmetic Operations with rational numbers

34.

If 2x + y = 7 and x + 2y = 5, then

5.5 Problem Solving Answer Explanations

Arithmetic Applied problems

Tides at Bay Cove on July 13

The expected number of defective units is
between 0.3% and 0.5% of 20,000, or between
(0.003)(20,000) = 60 and (0.005)(20,000) = 100.
Since each unit has a retail price of $2,500, the
amount of money needed to cover the refunds
for the expected number of defective units is
between 60($2,500) and 100($2,500), or between
$150,000 and $250,000.

height of tide

11:30 a.m.
2.2 ft
3

4:30 a.m.
0.3 ft

2
1
0

time

–1

6 a.m. noon

midnight
6 p.m.
–0.5 ft

37.

The correct answer is D.
40ft

15ft

patio
35ft

house

38.

(A)

1.7

(B)

1.9

(C)

2.2

(D)

2.5

(E)

2.7

20ft

39.

A ﬂat patio was built alongside a house as shown in the
figure above. If all angles shown are right angles, what
is the area of the patio in square feet?

Arithmetic Interpretation of graphs and tables

(A)

800

From the graph, the highest tide is 2.2 ft above
the baseline and the lowest tide is 0.5 ft below the
baseline. Therefore, the difference between the
heights of the highest tide and the lowest tide
is [2.2 – (–0.5)] ft = (2.2 + 0.5) ft = 2.7 ft.

(B)

875

(C)

1,000

(D)

1,100

(E)

1,125

The correct answer is E.

Geometry Area

Between $15,000 and $25,000

(B)

Between $30,000 and $50,000

(C)

Between $60,000 and $100,000

(D)

Between $150,000 and $250,000

(E)

Between $300,000 and $500,000

40
15

20
20

The area of the patio can be calculated by
imagining the patio to be a rectangle of dimensions
40 ft by 35 ft that has a lower‐left square corner of
dimensions 20 ft by 20 ft covered up, as shown in
the figure above. The area of the patio will be the
area of the uncovered part of the rectangle, and
therefore the area of the patio, in square feet, is
(40)(35) – (20)(20) = 1,400 – 400 = 1,000.
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GMAT® Ofﬁcial Guide 2018

41.

Alternatively, the area of the patio can be
calculated by dividing the patio into two
rectangles and adding the areas of the two
rectangles. This can be done in two ways:
by using rectangles of dimensions 20 ft by
35 ft and 20 ft by 15 ft, as shown in the figure
above on the left (for a total area of
700 ft2 + 300 ft2 = 1,000 ft2), or by using
rectangles of dimensions 40 ft by 15 ft
and 20 ft by 20 ft, as shown in the figure
above on the right (for a total area of
600 ft2 + 400 ft2 = 1,000 ft2).

194

$52.50

(B)

$55.00

(C)

$57.50

(D)

$62.50

(E)

$65.00

(B)

(C)

(D)

(E)

The average of the student’s first 4 test scores
is 78, so the sum of the first 4 test scores is
4 ( 78 ) = 312. If x represents the fifth test score,
then the sum of all 5 test scores is 312 + x and
312 + x
. But the
the average of all 5 test scores is
average of all 5 test scores is 80 so 5
312 + x
= 80
5
312 + x = 400
x = 88

The sum of the weekly salaries of 5 employees is
$3,250. If each of the 5 salaries is to increase by
10 percent, then the average (arithmetic mean) weekly
salary per employee will increase by
(A)

(A)

Arithmetic Statistics

The correct answer is C.
40.

A student’s average (arithmetic mean) test score on
4 tests is 78. What must be the student’s score on a
5th test for the student’s average score on the 5 tests
to be 80 ?

The correct answer is E.
42.

Arithmetic Applied problems; Percents

(A)

Let S1, S2, S3, S4, and S5 be the salaries,
in dollars, of the 5 employees. Since
the sum of the 5 salaries is 3,250, then
S1 + S2 + S3 + S4 + S5 = 3,250 and the average
S + S 2 + S3 + S4 + S5 3, 250
=
salary is 1
= 650.
5
5
After each salary is increased by 10%, the salaries
will be (1.1)S1, (1.1)S2, (1.1)S3, (1.1)S4, and
(1.1)S5 and the average salary, in dollars, will be
(1.1)S1 + (1.1)S 2 + (1.1)S3 + (1.1)S4 + (1.1)S5
=
5
⎛ S + S + S3 + S4 + S5 ⎞
1.1 × ⎜ 1 2
⎟⎠ = 1.1 × 650 = 715.
⎝
5
Therefore, the increase in the average salary is
$715 – $650 = $65.

(B)

(C)

(D)

(E)

The correct answer is E.

The correct answer is D.

Algebra Applied problems

Chris worked 40 hours at a rate of $x per hour,
48 – 40 = 8 hours at a rate of $22 per hour, and
earned a total of $816.
40x + 8(22)
40x + 176
40x
x

=
=
=
=

816
816
640
16

given information
multiply 8 and 22
subtract 176 from both sides
divide both sides by 40

5.5 Problem Solving Answer Explanations

For the more mathematically inclined, consider the
remainder when each prime number n greater than
3 is divided by 6. The remainder cannot be 0 because
that would imply that n is divisible by 6, which is
impossible since n is a prime number. The remainder
cannot be 2 or 4 because that would imply that n is
even, which is impossible since n is a prime number
greater than 3. The remainder cannot be 3 because
that would imply that n is divisible by 3, which is
impossible since n is a prime number greater than
3. Therefore, the only possible remainders when a
prime number n greater than 3 is divided by 6 are
1 and 5. Thus, n has the form 6q + 1 or 6q + 5,
where q is an integer, and, therefore, n2 has the
form 36q2 + 12q + 1 = 12(3q2 + q) + 1 or
36q2 + 60q + 25 = 12(3q2 + 5q + 2) + 1. In either
case, n2 has a remainder of 1 when divided by 12.

B
y°

(y + 10)°
A
43.

In the figure above, what is the ratio of the measure of
angle B to the measure of angle A ?
(A)

2 to 3

(B)

3 to 4

(C)

3 to 5

(D)

4 to 5

(E)

5 to 6

Geometry Angles

Because the sum of the degree measures of the
three interior angles of a triangle is 180, it follows
that y + (y + 10) + 90 = 180. Therefore, 2y = 80,
and hence y = 40. The ratio of the measure of
angle B to the measure of angle A can now be
y
40 4
=
= .
determined:
y + 10 50 5

The correct answer is B.
45.

1
1
1+
3

−

1
3

(B)

−

1
6

−

(B)

(C)

(D)

(E)

Arithmetic Properties of numbers

The simplest way to solve this problem is to
choose a prime number greater than 3 and divide
its square by 12 to see what the remainder is. For
example, if n = 5, then n2 = 25, and the remainder
is 1 when 25 is divided by 12. A second prime
number can be used to check the result. For
example, if n = 7, then n2 = 49, and the remainder
is 1 when 49 is divided by 12. Because only one of
the answer choices can be correct, the remainder
must be 1.

1
1+
2

(A)

The correct answer is D.
44.

−

1
12

(D)

1
12

(E)

1
3

Arithmetic Operations with rational numbers

Perform the arithmetic calculations as follows:
1
1+

1
3

−

1 1
−
1 4 3
1+
2 3 2
3 2
= −
4 3
9 8
=
−
12 12
9−8
=
12
1
=
12
=

The correct answer is D.
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GMAT® Ofﬁcial Guide 2018

46.

Algebra Algebraic expressions

The positive two‐digit integers x and y have the same
digits, but in reverse order. Which of the following must
be a factor of x + y ?
(A)

(B)

(C)

If s oranges are packed r oranges to a box with no
oranges left over, then the number of boxes that
s
will be filled is . If n of these boxes are already
r
s
filled, then – n boxes remain to be filled.
r

(D)

The correct answer is E.

(E)

14
49.

Arithmetic Properties of numbers

Let m and n be digits. If x = 10m + n, then
y = 10n + m. Adding x and y gives
x + y = (10m + n) + (10n + m) = 11m + 11n =
11(m + n), and therefore 11 is a factor of x + y.

If 0 < a < b < c, which of the following statements must
be true?
I.

2a > b + c

II.

c–a>b–a

III.

c b
<
a a

(A)

I only

(B)

II only

(C)

III only

(D)

I and II

(E)

II and III

The correct answer is D.
47.

In a certain sequence of 8 numbers, each number
after the first is 1 more than the previous number. If
the first number is –5, how many of the numbers in the
sequence are positive?
(A)

None

(B)

One

Algebra Inequalities

(C)

Two

(D)

Three

(E)

Four

Given 0 < a < b < c, Statement I is not necessarily
true. If, for example, a = 1, b = 2, and c = 3, then
0 < a < b < c, but 2a = 2(1) < 2 + 3 = b + c.
Given 0 < a < b < c, then c > b, and subtracting
a from both sides gives c – a > b – a. Therefore,
Statement II is true.

Arithmetic Sequences

The sequence consists of eight consecutive
integers beginning with –5:
In this sequence exactly two of the numbers are
positive.

Given 0 < a < b < c, Statement III is not
necessarily true. If, for example, a = 1, b = 2, and
c 3 2 b
c = 3, then 0 < a < b < c, but = > = .
a 1 1 a

The correct answer is C.

The correct answer is B.

−5, − 4, − 3, − 2, − 1, 0, 1, 2

48.

196

s – nr
n
s–
r
rs – n
s
–r
n
s
–n
r

50.

In the xy-plane, the origin O is the midpoint of line
segment PQ. If the coordinates of P are (r,s), what are
the coordinates of Q ?
(A)

(r,s)

(B)

(s,–r)

(C)

(–s,–r)

(D)

(–r,s)

(E)

(–r,–s)

5.5 Problem Solving Answer Explanations

Algebra Coordinate geometry

Monday Tuesday Wednesday Thursday

Let (x,y) be the coordinates of Q. The midpoint
⎛r + x s + y⎞
,
of (r,s) and (x,y) is ⎜
⎟ . Since the
⎝ 2
2 ⎠
r +x
= 0 and
midpoint is (0,0), it follows that
2
s+ y
= 0. Therefore, r + x = 0 and s + y = 0, or
2
x = –r and y = –s, or (x,y) = (–r,–s).
This problem can also be solved by observing that
Q is the reflection of P about the origin, and so
Q = (–r,–s) (i.e., change the sign of each of the
coordinates of P to obtain the coordinates of Q).
The correct answer is E.
51.

Which of the following equations is NOT equivalent to
10y2 = (x + 2)(x – 2) ?
(A)

30y2 = 3x2 – 12

(B)

20y2 = (2x – 4)(x + 2)

(C)

10y2 + 4 = x2

(D)

5y2 = x2 – 2

(E)

y2 =

x −4
10

Algebra Simplifying algebraic expressions

When x = 2 or x = –2, the equation becomes
10y2 = 0, or y = 0. Since, in the equation given in
(D), y does not become 0 when x = 2, it follows
that the equation given in (D) is not equivalent
to the given equation. Alternatively, when each of
the equations given in (A) through (E) is solved
for 10y2 in terms of x, only the resulting equation
in (D) fails to give an expression in terms of x
that is equivalent to (x + 2)(x – 2) = x2 – 4.
The correct answer is D.

52.

Company A

Company B

Company C

Company D

Company E

(B)

(C)

(D)

(E)

Arithmetic Statistics

Since the standard deviation of a data set is a
measure of how widely the data are scattered
about their mean, find the mean number of
packages shipped by each company and then
determine the company for which the data is
most widely scattered about its mean.
For Company A, the mean number of packages
45 + 55 + 2(50)
= 50. Two of the data
shipped is
4
points are each 50 and the other two each differ
from 50 by 5.
For Company B, the mean number of packages
2(10) + 2(30)
shipped is
= 20. Each of the data
4
points differs from 20 by 10. Thus, the data for
Company B is more widely scattered about its
mean of 20 than the data for Company A is
about its mean of 50.
For Company C, the mean number of packages
34 + 2(28) + 30
shipped is
= 30. One data point
4
is 30, two others each differ from 30 by only 2,
and the fourth data point differs from 30 by only 4.
Therefore, the data for Company C is not as
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GMAT® Ofﬁcial Guide 2018

widely scattered about its mean of 30 as the data
for Company B is about its mean of 20.
For Company D, the mean number of packages
39 + 42 + 41 + 38
= 40. Two of
shipped is
4
the data points each differ from 40 by only 1
and the other two each differ from 40 by only
2. Therefore, the data for Company D is not as
widely scattered about its mean of 40 as the data
for Company B is about its mean of 20.
For Company E, the mean number of packages
50 + 2(60) + 70
shipped is
= 60. Two of the
4
data points are each 60 and the other two each
differ from 60 by 10. Therefore, the data for
Company E is not as widely scattered about its
mean of 60 as the data for Company B is about
its mean of 20.
Thus, the data for Company B is most widely
scattered about its mean and, therefore, the
standard deviation of the number of packages
shipped daily by the five companies is greatest for
Company B.

(50 − 60)2 + 2(60 − 60)2 + (70 − 60)2
4

For E:
=

200
= 50 .
4

The correct answer is B.
53.

4,200

(B)

3,150

(C)

2,520

(D)

2,100

(E)

1,800

Algebra Applied problems

For those interested, the standard deviations for
the five companies can be calculated as follows:
(45 − 50)2 + (55 − 50)2 + 2(50 − 50)2
4

For A:
=

For B:

2(25)
=
4

25
= 12.5.
2

2(10 − 20)2 + 2(30 − 20)2
=
4

4(100)
4

= 100.
(34 − 30)2 + 2(28 − 30)2 + (30 − 30)2
For C:
4
24
=
= 6.
4
For D:
(39 − 40)2 + (42 − 40)2 + (41 − 40)2 + (38 − 40)2
4
10
=
= 2.5 .
4
198

Let x be the desired number of units to be sold
at a price of $2 each. Then the revenue for selling
these units is $2x, and the total cost for selling
these units is (40%)($2.00)x = $0.80x plus a fixed
cost of $5,040.
revenue = total cost
given requirement
2x = 0.8x + 5,040 given information
1.2x = 5,040
subtract 0.8x from both
sides
x = 4,200
divide both sides by 1.2
The correct answer is A.

5.5 Problem Solving Answer Explanations

Arithmetic Percents

The percent increase from the number in State D
to the number in State A is
E

A
D

⎛ 181, 000 − 79, 000
⎞
× 100 ⎟ % expression for
= ⎜⎝
⎠
79, 000
percent increase

B
C

54.

The dial shown above is divided into equal‐sized intervals.
At which of the following letters will the pointer stop if it
is rotated clockwise from S through 1,174 intervals?
(A)

(B)

(C)

(D)

(E)

⎛ 181 − 79
⎞
× 100 ⎟ %
=⎜
⎝ 79
⎠

reduce fraction

⎛ 102
⎞
=⎜
× 100 ⎟ %
⎝ 79
⎠

subtract

⎛ 100
⎞
× 100 ⎟ %
≈⎜
⎝ 80
⎠

approximate

= (1.25 × 100)%
= 125%

divide
multiply

The correct answer is D.

Arithmetic Properties of numbers

There are 8 intervals in each complete revolution.
Dividing 8 into 1,174 gives 146 with remainder
6. Therefore, 1,174 intervals is equivalent
to 146 complete revolutions followed by an
additional 6 intervals measured clockwise
from S, which places the pointer at E.

Estimated Number of Home-Schooled
Students by State, January 2001

I
x

O
III

56.

The correct answer is E.

55.

The graph of the equation xy = k, where k < 0, lies in
which two of the quadrants shown above?
(A)

I and II

(B)

I and III

State

Number (in thousands)

(C)

II and III

181

(D)

II and IV

125

(E)

III and IV

103

Algebra Coordinate geometry

According to the table shown, the estimated number
of home-schooled students in State A is approximately
what percent greater than the number in State D ?

If a point lies on the graph of xy = k, then the
product of the point’s x‐ and y‐coordinates is k.
Since k is negative, it follows that for any such point,
the product of the point’s x‐ and y‐coordinates is
negative. Therefore, for any such point, the point’s
x‐ and y‐coordinates have opposite signs, and hence
the point must be in quadrant II or in quadrant IV.

(A)

25%

The correct answer is D.

(B)

55%

(C)

100%

(D)

125%

(E)

155%
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GMAT® Ofﬁcial Guide 2018

57.

Geometry Area

When n liters of fuel were added to a tank that was
1
7
already full, the tank was filled to of its capacity. In
3
9
terms of n, what is the capacity of the tank, in liters?
(A)
(B)
(C)
(D)
(E)

From the given information it follows that
(100 + w)(150 + w) = 2(100)(150), or
(100 + w)(150 + w) = (200)(150). This is a
quadratic equation that can be solved by several
methods. One method is by inspection. The left
side is clearly equal to the right side when w = 50.
Another method is by factoring. Expanding the left
side gives (100)(150) + 250w + w2 = (200)(150), or
w2 + 250w – (100)(150) = 0. Factoring the left
side gives (w – 50)(w + 300) = 0, which has
w = 50 as its only positive solution.

10
n
9
4
n
3
3
n
2
9
n
4
7
n
3

The correct answer is B.
59.

Algebra Applied problems

Let C represent the capacity of the tank, in liters.
It follows that
1
7
C+n= C
3
9
7
1
n= C– C
9
3
n=

4
C
9

9
n=C
4

given
1
subtract C from both
3
sides

4
9

(B)

–0.25

(C)

0.25

(D)

0.75

(E)

1
1
=
0.75 − 1 3 − 1
4
1
=
1
−
4
= −4

w ft

100 ft
150 ft
w ft
Note: Not drawn to scale.

200

–4

Perform the arithmetic calculations as follows:

The correct answer is D.

58.

(A)

Arithmetic Operations with rational numbers

combine like terms
divide both sides by

1
=
0.75 − 1

The smaller rectangle in the figure above represents
the original size of a parking lot before its length and
width were each extended by w feet to make the larger
rectangular lot shown. If the area of the enlarged lot is
twice the area of the original lot, what is the value of w ?

The correct answer is A.
60.

$880

(B)

$1,088

(A)

(C)

$1,253

(B)

(D)

$1,280

(C)

(E)

$1,296

(D)

100

(E)

200

5.5 Problem Solving Answer Explanations

Arithmetic Applied problems

Since only one answer choice can be correct,
the other answer choices need not be checked.
However, for completeness:

The amount of interest after one year is the total
value of the investment after one year minus the
total initial value of the investment. The total
value of the investment after one year is
$8,000(1.06) + $10,000(1.04)2 = $8,480 + $10,816,
so the amount of interest is ($8,480 + $10,816) –
($8,000 + $10,000) = $480 + $816 = $1,296.

n + 1 is odd if n is even and even if n is odd.
Therefore, it is not true that n + 1 must be
even.

n + 2 is even if n is even and odd if n is odd.
Therefore, it is not true that n + 2 must be
even.

2n + 1 is odd whether n is even or odd.
Therefore, it is not true that 2n + 1 must be
even.

n2 is even if n is even and odd if n is odd.
Therefore, it is not true that n2 must be
even.

The correct answer is E.
61.

x
35

(B)

1−

(C)

10x

(D)

35 – x

(E)

350 – 10x

Algebra Algebraic expressions

Since each of the x trees yielded 10 bushels, the
total number of bushels yielded by these trees
was 10x. Since the yield of the entire orchard was
x
10 x
350 bushels,
was the fraction of the
=
350 35
total yield from these x trees.
The correct answer is A.
62.

The correct answer is C.

x
35

If n is an integer, which of the following must be even?
(A)

n+1

(B)

n+2

(C)

(D)

2n + 1

(E)

63.

The sum

(A)
(B)
(C)
(D)
(E)

7 1
+ is between
8 9

1
3
and
2
4
3
and 1
4
1
1 and 1
4
1
1
1 and 1
4
2
1
1 and 2
2

Arithmetic Operations with rational numbers

Since 1 < 1 , 7 + 1 < 7 + 1 = 1, and answer
9 8 8 9 8 8
choices C, D, and E can be eliminated. Since
7 > 6 = 3 , 7 + 1 > 3 , and answer choice A can
8 8 4 8 9 4
be eliminated. Thus, 3 < 7 + 1 < 1.
4 8 9
The correct answer is B.

Arithmetic Properties of integers

A quick look at the answer choices reveals the
expression 2n in answer choice C. 2n is a multiple
of 2 and hence must be even.

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GMAT® Ofﬁcial Guide 2018

64.

Car X averages 25.0 miles per gallon of gasoline and
Car Y averages 11.9 miles per gallon. If each car is
driven 12,000 miles, approximately how many more
gallons of gasoline will Car Y use than Car X ?
(A)

320

(B)

480

(C)

520

(D)

730

(E)

920

below, it is clear that the least possible integer
value of |23 − 5y| is 2, which occurs when y = 5.
y |23 – 5y|
0
23
1
18
2
13
3
8
4
3
5
2
6
7
7
12

Arithmetic Applied problems

Car X uses 1 gallon of gasoline for every 25 miles
1
it is driven, so Car X uses
of a gallon for
25
every 1 mile it is driven. Therefore Car X will
⎛ 1⎞
use (12, 000 ) ⎜ ⎟ = 480 gallons of gasoline
⎝ 25 ⎠
when it is driven 12,000 miles. Car Y uses
119
miles
1 gallon of gasoline for every 11.9 or
10
10
it is driven, so Car Y uses
of a gallon for
119
every 1 mile it is driven. Therefore Car Y will use
10
10
(12, 000 ) ⎛⎜⎝ ⎞⎟⎠ ≈ (12, 000 ) ⎛⎜⎝ ⎞⎟⎠ = 1, 000 gallons
119
120
of gasoline when it is driven 12,000 miles. Thus,
Car Y will use approximately 1,000 – 480 = 520
more gallons of gasoline than Car X.
The correct answer is C.
65.

The correct answer is B.
66.

80 + 125 =
(A)

9 5

(B)

20 5

(C)

41 5

(D)

205

(E)

100

If y is an integer, then the least possible value of
|23 – 5y | is

Arithmetic Operations with radical expressions

(A)

(B)

Rewrite each radical in the form a b , where a
and b are positive integers and b is as small as
possible, and then add.

(C)

(D)

(E)

Arithmetic Absolute value; Operations with
integers

Since y is an integer, 23 − 5y is also an integer.
The task is to find the integer y for which |23 − 5y|
is the least. If y ≥ 0, −5y ≤ 0, and 23 − 5y ≤ 23. On
the other hand, if y ≤ 0, −5y ≥ 0, and 23 − 5y ≥ 23.
Therefore, the least possible value of |23 – 5y|
occurs at a nonnegative value of y. From the chart

202

Alternatively, since |23 – 5y| ≥ 0, the minimum
possible real value of |23 – 5y| is 0. The integer
value of y for which |23 – 5y| is least is the
integer closest to the solution of the equation
23
= 4.6 and the
23 – 5y = 0. The solution is y =
5
integer closest to 4.6 is 5.

80 + 125 = 16(5) + 25(5)
=

(

)( 5 ) + (

= 4 5+5 5
=9 5
The correct answer is A.

)( 5 )

5.5 Problem Solving Answer Explanations

y = kx + 3
67.

69.

In the equation above, k is a constant. If y = 17 when
x = 2, what is the value of y when x = 4 ?
(A)

(B)

(C)

(D)

(E)

7
(A)

1:00 p.m.

Algebra First-degree equations

(B)

2:00 p.m.

If y = kx + 3 and y = 17 when x = 2, then

(C)

3:00 p.m.

17 = 2k + 3

(D)

5:00 p.m.

14 = 2k

(E)

6:00 p.m.

7=k

Algebra Applied problems

Therefore, y = 7 x + 3. When x = 4,
y = 7(4) + 3 = 31.

Let t be the number of hours after 8:00 a.m.
that Ben drove. Then, in t hours, Ben
drove 20t miles and Al, who began driving
3 hours after Ben began driving, drove
40(t – 3) miles. Therefore, their combined
total distance at that time can be expressed as
(20t + 40t – 120) miles = (60t – 120) miles.
It follows that 60t – 120 = 240, or 60t = 360,
or t = 6, which corresponds to 6 hours after
8:00 a.m. or 2:00 p.m.

The correct answer is B.
68.

Which of the following is greatest?
(A)

10 3

(B)

9 4

(C)

8 5

(D)

7 6

(E)

6 7

Arithmetic Operations on radical expressions

Since all the expressions represent positive
numbers, the expression that has the greatest
squared value will be the expression that has the
greatest value.

(10 3 ) = 100 × 3 = 300 (not greatest)
(9 4 ) = 81 × 4 = 324 (greatest)
(8 5 ) = 64 × 5 = 320 (not greatest)
(7 6 ) = 49 × 6 = 294 (not greatest)
(6 7 ) = 36 × 7 = 252 (not greatest)

The correct answer is B.
70.

(A)

5,832

(B)

6,480

(C)

7,200

(D)

8,000

(E)

8,889

2
2

The correct answer is B.

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GMAT® Ofﬁcial Guide 2018

71.

Arithmetic Applied problems; Percents

Arithmetic Operations with rational numbers

Corn was planted on 100% – (40% + 50%) = 10%
of the cleared land, and the cleared land
represents 90% of the farmer’s land. Therefore,
corn was planted on 10% of 90%, or
(0.10)(0.90) = 0.09 = 9%, of the farmer’s land. It
is given that corn was planted on 720 acres, so if
x is the number of acres the farmer owns, then
720
= 8, 000 .
0.09x = 720 and x =
0.09

1
1⎞
⎛ 1 1 1 1⎞
⎛1 1
+
+ ⎟
⎜⎝ + + + ⎟⎠ = r ⎜⎝ +
3 4 5 6
9 12 15 18 ⎠

The correct answer is D.

Alternatively,

⎛1 1 1
⎜⎝ + + +
3 4 5

(B)

(C)

2(310)

(D)

3(210)

(E)

3(102)

3=r

⎛1 1 1
⎜⎝ + + +
3 4 5

72.

204

1
1⎞
1⎞
⎛1 1
+
+ ⎟
⎟⎠ = r ⎜⎝ +
9 12 15 18 ⎠
6

12
10 ⎞
⎛ 20 15 12 10 ⎞
⎛ 20 15
+
+
+ ⎟ = r⎜
+
+
+
⎜⎝
⎝ 180 180 180 180 ⎟⎠
60 60 60 60 ⎠
⎛ 20 + 15 + 12 + 10 ⎞
⎛ 20 + 15 + 12 + 10 ⎞
⎜⎝
⎟⎠ = r ⎜⎝
⎟⎠
60
180
⎞
180
⎛ 20 + 15 + 12 + 10 ⎞ ⎛
=r
⎜⎝
⎟⎠ ⎜
60
⎝ 20 + 15 + 12 + 10 ⎟⎠
180
=r
60
3=r

Arithmetic Applied problems; Sequences

The population doubles each month, so multiply
the previous month’s population by 2 to get
the next month’s population. Thus, at the end
of the 1st month the population will be (3)(2),
at the end of the 2nd month the population
will be (3)(2)(2), at the end of the 3rd month
the population will be (3)(2)(2)(2), and so on.
Therefore, at the end of the 10th month the
population will be the product of 3 and ten
factors of 2, which equals 3(210).

The correct answer is C.
73.

If x and y are positive integers such that y is a multiple
of 5 and 3x + 4y = 200, then x must be a multiple of
which of the following?
(A)

(B)

(C)

(D)

The correct answer is D.

(E)

1
1⎞
⎛1 1
⎛ 1 1 1 1⎞
If ⎜ + + + ⎟ = r ⎜ +
+
+
, then r =
⎝ 3 4 5 6⎠
⎝ 9 12 15 18 ⎟⎠

Arithmetic Properties of numbers

(A)

1
3

(B)

4
3

(C)

(D)

(E)

1⎞
⎟
6⎠

⎛ 1⎞
1 = r⎜ ⎟
⎝ 3⎠

1⎞
⎛ 1⎞ ⎛ 1 1 1
⎟⎠ = r ⎜⎝ ⎟⎠ ⎜⎝ + + +
6
3 3 4 5

Since it is given that y is a multiple of 5,
let y = 5q, where q is a positive integer. It is
given that 3x + 4y = 200, so 3x + 4(5q) = 200
or 3x + 20q = 200. It follows that
3x = 200 – 20q = 20(10 – q). Since 3 is prime and
is not a factor of 20, then 3 must be a factor of
10 – q, which means the only possible values

5.5 Problem Solving Answer Explanations

of q are 1, 4, and 7. This is summarized in the
following table.

75.

q 10 – q 20(10 – q) 3x x
1
9
180
180 60
4
6
120
120 40
7
3
60
60 20

(A)

(B)

Alternatively, since 3x = 20(10 – q), the factors
of the product of 3 and x must correspond to the
factors of the product of 20 and (10 – q). Since
the factors of 20 include 10, then x must have a
factor of 10, which means that x is a multiple of 10.

(C)

160

(D)

192

(E)

240

The correct answer is E.

For a certain value of x, the numbers of hours
worked on the project by the four staff members are
2x, 3x, 5x, and 6x, for a total of 16x. It is given that
one of these four numbers is equal to 30. If 2x = 30,
then x = 15 and 16x = 16(15) = 240, which is (E).
If 3x = 30, then x = 10 and 16x = 16(10) = 160,
which is (C). If 5x = 30, then x = 6 and
16x = 16(6) = 96, which is (B). If 6x = 30, then
x = 5 and 16x = 16(5) = 80, which is (A).

In each case, the value of x is a multiple of 10.

74.

Arithmetic Ratio and proportion

Which of the following expressions can be written as
an integer?
I.

(

II.

(82)

III.

(

(A)

None

(B)

I only

(C)

III only

(D)

I and II

(E)

I and III

82 + 82

(

)(

)

The correct answer is D.
76.

Arithmetic Operations with radical expressions

Expression I represents an integer because

(

82 + 82

) = (2
2

)

= (4)(82).

3.25

Expression II does not represent an integer

(B)

3.6

because (82) 82 = 823 and 823 = 23 × 413 is
not a perfect square. Regarding this last assertion,
note that the square of any integer has the
property that each of its distinct prime factors is
repeated an even number of times. For example,
242 = (23 × 3)2 = 26 × 32 has the prime factor 2
repeated 6 times and the prime factor 3 repeated
twice. Expression III represents an integer,
82
82
82
because
=
= 1.
82
82

(C)

4.2

(D)

4.4

(E)

5.5

(

)(

)

The correct answer is E.
205

GMAT® Ofﬁcial Guide 2018

Arithmetic Applied problems

1
of the tank in 3 hours,
2
which is equivalent to the rate of filling
1
1
÷ 3 = of the tank per hour. The second
2
6
2
pipe can fill of the tank in 6 hours, which is
3
2
1
equivalent to the rate of filling ÷ 6 = of the
3
9
tank per hour. Together, they can fill the tank at
1 1
5
a rate of + =
of the tank per hour. Thus,
6 9 18
when both pipes are used at the same time, they
18
will fill the tank in
= 3.6 hours.
5
The correct answer is B.

78.

If x2 − 2 < 0, which of the following specifies all the
possible values of x ?

The first pipe can fill

77.

In the xy‐coordinate plane, which of the following points
must lie on the line kx + 3y = 6 for every possible value
of k ?
(A)

(1,1)

(B)

(0,2)

(C)

(2,0)

(D)

(3,6)

(E)

(6,3)

Algebra Coordinate geometry

Substituting the various answer choices for (x,y)
into kx + 3y = 6 gives the following equations:
A

k+3=6

0 + 3(2) = 6

2k + 3(0) = 6

3k + 3(6) = 6

6k + 3(3) = 6

Each of these, except for the equation in B, holds
for only one value of k. The equation in B does
not include k and therefore holds for every value
of k.
The correct answer is B.

206

(A)

0 2 . Choose x = –2, x = 0, and x = 2 from
these intervals, respectively, to test whether the
inequality holds. Then for these choices, the
inequality becomes (–2)2 – 2 < 0 (False),
(0)2 – 2 < 0 (True), and (2)2 – 2 < 0 (False).
Therefore, the solution consists of only the
interval − 2 < x < 2 . Alternatively, the graph
of y = x2 – 2 is easily seen to be a parabola that
opens upward with vertex at (0, –2) and
x‐intercepts at x = 2 and x = − 2 . The solution
to the inequality is the set of the x‐coordinates
of the portion of this parabola that lies below the
x‐axis, which is − 2 < x < 2 .
The correct answer is C.

5.5 Problem Solving Answer Explanations

Book 5: 4 days—50 pages on each of Days 17–19,
5 pages on Day 20 [3(50) + 5 = 155]

Book number

Pages in book

Total pages read

253

110

363

117

480

170

650

155

805

855

205

1,060

At this point, Shawana has read on a total of
28 days and has finished 8 books.

1,130

The correct answer is B.

165

1,295

105

1,400

143

1,543

207

1,750

Book 6: 1 day—50 pages on Day 21 [1(50) = 50]
Book 7: 5 days—50 pages on each of Days 22–25,
5 pages on Day 26 [4(50) + 5 = 205]
Book 8: 2 days—50 pages on Day 27, 20 pages on
Day 28 [50 + 20 = 70]

80.

79.

In Western Europe, x bicycles were sold in each of
the years 1990 and 1993. The bicycle producers of
Western Europe had a 42 percent share of this market
in 1990 and a 33 percent share in 1993. Which of the
following represents the decrease in the annual number
of bicycles produced and sold in Western Europe from
1990 to 1993?
(A)

9% of

x
100

(B)

14% of

x
100

(C)

75% of

x
100

(D)

9% of x

(E)

14% of x

Arithmetic Percents
(A)

(B)

(C)

(D)

(E)

Arithmetic Operations with integers

Book 1: 6 days—50 pages on each of Days 1–5, 3
pages on Day 6 [5(50) + 3 = 253]
Book 2: 3 days—50 pages on each of Days 7 and
8, 10 pages on Day 9 [2(50) + 10 = 110]
Book 3: 3 days—50 pages on each of Days 10 and
11, 17 pages on Day 12 [2(50) + 17 = 117]

Of the x bicycles sold in Western Europe in
1990, 42% of them were produced in Western
Europe. It follows that the number of bicycles
produced and sold in Western Europe in 1990
was 0.42x. Similarly, of the x bicycles sold in
Western Europe in 1993, 33% were produced in
Western Europe. It follows that the number of
bicycles produced and sold in Western Europe
in 1993 was 0.33x. Therefore, the decrease in
the annual number of bicycles produced and
sold in Western Europe from 1990 to 1993 was
0.42x – 0.33x = 0.09x, which is 9% of x.
The correct answer is D.

Book 4: 4 days—50 pages on each of Days 13–15,
20 pages on Day 16 [3(50) + 20 = 170]
207

GMAT® Ofﬁcial Guide 2018

81.

Arithmetic Fractions

If k is a positive integer, what is the remainder when
(k + 2)(k3 – k) is divided by 6 ?
(A)

(B)

(C)

(D)

(E)

1
Find the distance between each fraction and :
2
4 1
8
7
1
− =
−
=
7 2
14 14 14
5 1
10 9
1
− =
−
=
9 2
18 18 18
6 1
12 11
1
− =
−
=
11 2
22 22 22

Algebra Properties of numbers

Since k can be any positive integer, the remainder
must be the same regardless of the value of k. If
k = 2, for example, then (k + 2)(k 3 – k) =
(2 + 2)(23 – 2) = (4)(6), which is a multiple of 6,
and therefore, the remainder when divided by
6 is 0.

7 1
14 13
1
− =
−
=
13 2
26 26 26
9 1
9 8
1
− =
−
=
16 2
16 16 16
Each distance is the reciprocal of a positive
1
integer, the greatest of which is 26. Therefore,
26
7
is the least of the distances and
is the fraction
13
1
closest to .
2
Alternatively, asking which fraction is closest to
1
is equivalent to asking which of the fractions,
2
when doubled, is closest to 1. The fractions after
8 10 12 14
18 9
doubling are , , , , and
= . Each of
7 9 11 13
16 8
n+1
1
=1+
these fractions can be expressed as
n
n
for some integer n. Of these fractions, the one
1
closest to 1 is the one for which
is least, or in
n
other words, the one for which n is greatest. That
14
fraction is , and therefore the closest of the
13
1 ⎛ 1 ⎞ ⎛ 14 ⎞ 7
original fractions to is ⎜ ⎟ ⎜ ⎟ = .
2 ⎝ 2 ⎠ ⎝ 13 ⎠ 13

Alternatively, factor the given expression:
(k + 2)(k 3 – k) = (k + 2)(k)(k 2 – 1)
= (k + 2)(k)(k + 1)(k – 1)
Now, rearrange the factors in ascending order
(k – 1)(k)(k + 1)(k + 2), and observe that for any
positive integer k, the factors are 4 consecutive
integers, two of which are even and one of which
is divisible by 3. Therefore, (k + 2)(k 3 – k) is
divisible by both 2 and 3. Thus, (k + 2)(k 3 – k) is
divisible by 6 with 0 remainder.
The correct answer is A.
82.

Which of the following fractions is closest to
(A)
(B)
(C)
(D)
(E)

208

4
7
5
9
6
11
7
13
9
16

1
?
2

The correct answer is D.
83.

If p ≠ 0 and p –

1 − p2 r
= , then r =
p
p

(A)

p+1

(B)

2p – 1

(C)

p2 + 1

(D)

2p2 – 1

(E)

p2 + p – 1

5.5 Problem Solving Answer Explanations

Algebra Simplifying algebraic expressions

p–

1 − p2
r
=
p
p

p2 – (1 – p2) = r
2p2 – 1 = r

Algebra First-degree equations

Let x be the number that is 108 more than two2
thirds of itself. Then, 108 + x = x. Solve for x as
3
follows:

given
multiply both sides by p
combine like terms

2
108 + x = x
3
1
108 = x
3
324 = x

The correct answer is D.
84.

85.

If the range of the six numbers 4, 3, 14, 7, 10, and
x is 12, what is the difference between the greatest
possible value of x and the least possible value of x ?
(A)

(B)

(C)

(D)

(E)

The correct answer is E.
86.

A doctor prescribed 18 cubic centimeters of a certain
drug to a patient whose body weight was 120 pounds.
If the typical dosage is 2 cubic centimeters per
15 pounds of body weight, by what percent was the
prescribed dosage greater than the typical dosage?

Arithmetic Statistics

(A)

The range of the six numbers 3, 4, 7, 10, 14, and
x is 12. If x were neither the greatest nor the
least of the six numbers, then the greatest and
least of the six numbers would be 14 and 3. But,
this cannot be possible because the range of the
six numbers would be 14 – 3 = 11 and not 12 as
stated. Therefore, x must be either the greatest or
the least of the six numbers. If x is the greatest
of the six numbers, then 3 is the least, and
x – 3 = 12. It follows that x = 15. On the other
hand, if x is the least of the six numbers, then 14
is the greatest, and 14 – x = 12. It follows that
x = 2. Thus, there are only two possible values of
x, namely 15 and 2, and so the difference between
the greatest and least possible values of x is
15 – 2 = 13.

(B)

(C)

11%

(D)

12.5%

(E)

14.8%

The correct answer is D.

Arithmetic Percents

If the typical dosage is 2 cubic centimeters per
15 pounds of body weight, then the typical
dosage for a person who weighs 120 pounds
⎛ 120 ⎞
is 2⎜
= 2(8) = 16 cubic centimeters. The
⎝ 15 ⎟⎠
prescribed dosage of 18 cubic centimeters is,
⎛ ⎛ 18 − 16 ⎞
⎞
therefore, ⎜ ⎜
× 100 ⎟ % or 12.5% greater
⎟
⎝ ⎝ 16 ⎠
⎠
than the typical dosage.

(A)

(B)

144

Company P had 15 percent more employees in
December than it had in January. If Company P had
460 employees in December, how many employees
did it have in January?

(C)

162

(A)

391

(D)

216

(B)

400

(E)

324

(C)

410

(D)

423

(E)

445

What number is 108 more than two-thirds of itself?

87.

209

GMAT® Ofﬁcial Guide 2018

Arithmetic Percents

evaporated by the original amount and multiply
by 100 to convert the decimal to a percent. Thus,
0.2
× 100 = 0.02 × 100 or 2%.
10
The correct answer is D.

It is given that 460 is 115% of the number of
employees in January. Therefore, the number of
employees in January was
460 460 ⎛ 100 ⎞ ⎛ 460 ⎞
=
⎜
⎟ =⎜
⎟ (100 ) = ( 4 )(100 ) = 400.
1.15 1.15 ⎝ 100 ⎠ ⎝ 115 ⎠

90.

If m and p are positive integers and m2 + p2 < 100,
what is the greatest possible value of mp ?

The correct answer is B.
88.

The function f is defined by f ( x ) = x − 10 for all
positive numbers x. If u = f(t) for some positive
numbers t and u, what is t in terms of u ?
u + 10

(A)
(B)

(

u + 10

)

(C)

u 2 + 10

(D)

(u + 10)2

(E)

(u2 + 10)2

Algebra Functions

The question can be answered by solving
u = t – 10 for t in terms of u. Adding 10 to both
sides of this equation gives u + 10 = t . Squaring
both sides of the last equation gives (u + 10)2 = t,
which gives t in terms of u.
The correct answer is D.
89.

A glass was filled with 10 ounces of water, and
0.01 ounce of the water evaporated each day during a
20‐day period. What percent of the original amount of
water evaporated during this period?

(B)

(C)

(D)

(E)

Arithmetic Operations with integers

Trying various integer values for m and
corresponding values of p that satisfy
m2 + p2 < 100 might be the quickest way to solve
this problem. First, m < 10 and p < 10; otherwise,
m2 + p2 < 100 is not true.
If m = 9, then for m2 + p2 < 100 to be true,
p < 100 − 81 = 19 , so p ≤ 4, and the greatest
possible value for mp is (9)(4) = 36.
Similarly, if m = 8, then p < 100 − 64 = 36, so
p ≤ 5, and the greatest possible value for mp is
(8)(5) = 40.
If m = 7, then p < 100 − 49 = 51, so p ≤ 7, and
the greatest possible value for mp is (7)(7) = 49.
If m = 6, then p < 100 − 36 = 64, so p ≤ 7, and
the greatest possible value for mp is (6)(7) = 42.

(A)

0.002%

(B)

0.02%

(C)

0.2%

Thus, the greatest possible value for mp is 49.

(D)

The correct answer is D.

(E)

20%

Arithmetic Percents

Since 0.01 ounce of water evaporated each day for
20 days, a total of 20(0.01) = 0.2 ounce evaporated.
Then, to find the percent of the original amount of
water that evaporated, divide the amount that

210

(A)

If m ≤ 5 and p ≤ 9, it follows that mp ≤ 45.

5.5 Problem Solving Answer Explanations

91.

From this it follows that Equation III will hold
b2
only if 2 = 1, which can be false. For example,
a
if x = a = c = 1 and y = b = d = 2 (a choice of
x c
d b
values for which = and = are true), then
y d
c a
b2
2 1
≠ 1 and Equation III is = , which is false.
1 2
a2

x c
d b
= and = , which of the following must be
y d
c a
true?

If =

y b
=
x a
x y
=
a b
y x
=
a b

I.
II.
III.

The correct answer is C.

(A)

I only

(B)

II only

(C)

I and II only

If k is an integer and (0.0025)(0.025)(0.00025) × 10k
is an integer, what is the least possible value of k ?

(D)

I and III only

(A)

–12

(E)

I, II, and III

(B)

–6

(C)

(D)

(E)

92.

Algebra Ratio and proportion

Equation I is true:
x
y
y
x
d
c
y
x

=
=
=
=

c
d
d
c
b
a
b
a

given

Arithmetic Properties of numbers

Let N = (0.0025)(0.025)(0.00025) × 10k.
Rewriting each of the decimals as an
integer times a power of 10 gives
N = (25 × 10−4)(25 × 10−3)(25 × 10−5) × 10k =
(25)3 × 10k − 12. Since the units digit of (25)3 is
5, it follows that if k = 11, then the tenths digit
of N would be 5, and thus N would not be an
integer; and if k = 12, then N would be
(25)3 × 100 = (25)3, which is an integer. Therefore,
the least value of k such that N is an integer is 12.

take reciprocals
given
use last two equations

Equation II is true:
y b
= Equation I (shown true)
x a
y=

bx
multiply both sides by x
a

y x
= divide both sides by b
b a
Equation III is false, since otherwise it would
follow that:
bx
y=
from above
a

The correct answer is E.
93.

If a(a + 2) = 24 and b(b + 2) = 24, where a ≠ b, then
a+b=
(A)

–48

(B)

–2

(C)

(D)

(E)

y bx
=
divide both sides by a
a a2
x bx
=
use Equation III (assumed true)
b a2
x=

b 2x
multiply both sides by b
a2
211

GMAT® Ofﬁcial Guide 2018

Algebra Second-degree equations

a(a + 2) = 24

given

a2 + 2a = 24

use distributive property

a2 + 2a – 24 = 0

subtract 24 from both sides

(a + 6)(a – 4) = 0 factor

A
95.

So, a + 6 = 0, which means that a = –6, or
a – 4 = 0, which means a = 4. The equation with
the variable b has the same solutions, and so
b = –6 or b = 4.
Since a ≠ b, then a = –6 and b = 4, which means
a + b = −6 + 4 = −2, or a = 4 and b = −6, which
means that a + b = 4 + (−6) = −2
The correct answer is B.
94.

0.06N + 3,200

(B)

0.1N + 7,200

(C)

0.4N + 7,200

(D)

0.1N + 8,000

(E)

0.06N + 8,000

Algebra Percents

If N represents the total number of votes cast and
40% of the votes cast were cast by independent
voters, then 60% of the votes cast, or 0.6N votes,
were cast by voters registered with a political
party. Ms. Robbins received 10% of these, and so
Ms. Robbins received (0.10)(0.6N) = 0.06N votes
cast by voters registered with a political party.
Thus, Ms. Robbins received 0.06N votes cast
by voters registered with a political party and
8,000 votes cast by independent voters, so she
received 0.06N + 8,000 votes in all.
The correct answer is E.

212

In the figure shown, the triangle is inscribed in the
semicircle. If the length of line segment AB is 8 and the
length of line segment BC is 6, what is the length of
arc ABC ?
(A)

15π

(B)

12π

(C)

10π

(D)

7π

(E)

5π

Geometry Circles, triangles

Because ΔABC is inscribed in a semicircle,
∠ABC is a right angle. Applying the Pythagorean
theorem gives (AB)2 + (BC)2 = (AC)2. Then
substituting the given lengths, 82 + 62 = (AC)2,
and so (AC)2 = 100 and AC = 10. Thus, the
diameter of the circle is 10, the circumference of
the entire circle is 10π, and the length of arc ABC
is half the circumference of the circle, or 5π.

In a recent election, Ms. Robbins received 8,000 votes
cast by independent voters, that is, voters not
registered with a specific political party. She also
received 10 percent of the votes cast by those voters
registered with a political party. If N is the total number
of votes cast in the election and 40 percent of the
votes cast were cast by independent voters, which of
the following represents the number of votes that Ms.
Robbins received?
(A)

The correct answer is E.
96.

A manufacturer makes and sells 2 products, P and Q.
The revenue from the sale of each unit of P is $20.00
and the revenue from the sale of each unit of Q is
$17.00. Last year the manufacturer sold twice as
many units of Q as P. What was the manufacturer’s
average (arithmetic mean) revenue per unit sold of
these 2 products last year?
(A)

$28.50

(B)

$27.00

(C)

$19.00

(D)

$18.50

(E)

$18.00

Arithmetic Statistics

Let x represent the number of units of Product P
the manufacturer sold last year. Then 2x
represents the number of units of Product Q
the manufacturer sold last year, and x + 2x = 3x
represents the total number of units of Products P
and Q the manufacturer sold last year. The total

5.5 Problem Solving Answer Explanations

revenue from the sale of Products P and Q was
$(20x) + $(17(2x)) = $(54x), so the average
$(54 x )
revenue per unit sold was
= $18.
3x

98.

The correct answer is E.
97.

On a certain day, orangeade was made by mixing a
certain amount of orange juice with an equal amount
of water. On the next day, orangeade was made by
mixing the same amount of orange juice with twice
the amount of water. On both days, all the orangeade
that was made was sold. If the revenue from selling
the orangeade was the same for both days and if the
orangeade was sold at $0.60 per glass on the first
day, what was the price per glass on the second day?
(A)

$0.15

(B)

$0.20

(C)

$0.30

(D)

$0.40

(E)

$0.45

A worker carries jugs of liquid soap from a production
line to a packing area, carrying 4 jugs per trip. If the
jugs are packed into cartons that hold 7 jugs each,
how many jugs are needed to fill the last partially filled
carton after the worker has made 17 trips?
(A)

(B)

(C)

(D)

(E)

Arithmetic Remainders

Carrying 4 jugs per trip, the worker carries a
total of 4(17) = 68 jugs in 17 trips. At 7 jugs per
carton, these jugs will completely fill 9 cartons
with 5 jugs left over since (9)(7) + 5 = 68. To fill
the 10th carton, 7 – 5 = 2 jugs are needed.
The correct answer is B.
w

Arithmetic Applied problems

The ratio of the amount of orangeade made and
sold on the first day to amount of orangeade
made and sold on the second day is 2:3, because
the orangeade on the first day was 1 part orange
juice and 1 part water, while on the second day it
was 1 part orange juice and 2 parts water. Thus,
the ratio of the number of glasses of orangeade
made and sold on the first day to the number of
glasses of orangeade made and sold on the second
day is 2:3. Since the revenues for each day were
equal and 2 glasses were sold on the first day for
every 3 glasses that were sold on the second day,
2($0.60) = 3p, where p represents the price per
glass at which the orangeade was sold on the
⎛ 2⎞
second day. Therefore, p = ⎜ ⎟ ($0.60 ) = $0.40.
⎝ 3⎠
The correct answer is D.

99.

The figure shown above represents a modern painting
that consists of four differently colored rectangles,
each of which has length ℓ and width w. If the area of
the painting is 4,800 square inches, what is the width,
in inches, of each of the four rectangles?
(A)

(B)

(C)

(D)

(E)

Geometry Area

From the figure, ℓ = 3w, and the area of the
painting is ℓ(w + ℓ). Substituting 3w for ℓ gives
3w(w + 3w) = 3w(4w) = 12w2. It is given that
the area is 4,800 square inches, so 12w2 = 4,800,
w2 = 400, and w = 20.
The correct answer is B.
213

GMAT® Ofﬁcial Guide 2018

100. A certain fruit stand sold apples for $0.70 each and
bananas for $0.50 each. If a customer purchased
both apples and bananas from the stand for a total of
$6.30, what total number of apples and bananas did
the customer purchase?
(A)

(B)

(C)

(D)

(E)

Algebra First-degree equations; Operations
with integers

If each apple sold for $0.70, each banana sold for
$0.50, and the total purchase price was $6.30, then
0.70x + 0.50y = 6.30, where x and y are positive
integers representing the number of apples and
bananas, respectively, the customer purchased.
0.70x + 0.50y = 6.30
0.50y = 6.30 – 0.70x
0.50y = 0.70(9 – x)
y=

7
(9 − x )
5

Since y must be an integer, 9 – x must be
divisible by 5. Furthermore, both x and y must be
positive integers. For x = 1, 2, 3, 4, 5, 6, 7, 8, the
corresponding values of 9 – x are 8, 7, 6, 5, 4, 3, 2,
and 1. Only one of these, 5, is divisible by 5.
7
Therefore, x = 4 and y = ( 9 − 4 ) = 7 and the
5
total number of apples and bananas the customer
purchased is 4 + 7 = 11.
The correct answer is B.
101. In the xy‐plane, what is the slope of the line with
equation 3x + 7y = 9 ?

214

(A)

−

7
3

(B)

−

3
7

(C)

3
7

(D)

(E)

Algebra Coordinate geometry

Since the given equation of the line is equivalent
3
9
to 7y = –3x + 9, or y = − x + , the slope of the
7
7
3
line is − . Alternatively, choose 2 points lying on
7
the line and then use the slope formula for these
2 points. For example, substitute x = 0 in
9⎞
⎛
7y = –3x + 9 and solve for y to get ⎜ 0, ⎟ ,
⎝
7⎠
substitute y = 0 in 7y = –3x + 9 and solve for x
to get (3,0), then use the slope formula to get
9
9
−0
3
7
7
=
=− .
0 − 3 −3
7
The correct answer is B.
102. Working simultaneously and independently at an
identical constant rate, 4 machines of a certain type
can produce a total of x units of product P in 6 days.
How many of these machines, working simultaneously
and independently at this constant rate, can produce a
total of 3x units of product P in 4 days?
(A)

(B)

(C)

(D)

(E)

Algebra Applied problems

Define a machine day as 1 machine working for
1 day. Then, 4 machines each working 6 days is
equivalent to (4)(6) = 24 machine days. Thus,
x units of product P were produced in 24 machine
days, and 3x units of product P will require
(3)(24) = 72 machine days, which is equivalent
72
= 18 machines working independently and
to
4
simultaneously for 4 days.
The correct answer is B.

5.5 Problem Solving Answer Explanations

103. At a certain school, the ratio of the number of second
graders to the number of fourth graders is 8 to 5, and
the ratio of the number of first graders to the number
of second graders is 3 to 4. If the ratio of the number
of third graders to the number of fourth graders is 3
to 2, what is the ratio of the number of first graders to
the number of third graders?
(A)

16 to 15

(B)

9 to 5

(C)

5 to 16

(D)

5 to 4

(E)

4 to 5

Arithmetic Ratio and proportion

If F, S, T, and R represent the number of first,
second, third, and fourth graders, respectively,
S 8
F 3
then the given ratios are: (i) = , ( ii ) = ,
R 5
S 4
T 3
F
and ( iii ) = . The desired ratio is . From (i),
R 2
T
8
3
S = R , and from (ii), F = S . Combining these
5
4
3
3⎛8 ⎞ 6
results, F = S = ⎜ R ⎟ = R . From (iii),
4
4⎝5 ⎠ 5
6
F 5R 6 2 4
3
T = R . Then
=
= ⋅ = . So, the
T 3R 5 3 5
2
2
ratio of the number of first graders to the number
of third graders is 4 to 5.
The correct answer is E.

Arithmetic Operations with integers

If Δ represents addition, subtraction,
multiplication, or division, then 6 Δ 3 is equal
to either 6 + 3 = 9, or 6 – 3 = 3, or 6 × 3 = 18,
or 6 ÷ 3 = 2. Since it is given that 6 Δ 3 ≤ 3, Δ
represents either subtraction or division.
Statement I is true for subtraction since 2 – 2 = 0
but not true for division since 2 ÷ 2 = 1.
Statement II is not true for subtraction since
2 – 2 = 0 but is true for division since 2 ÷ 2 = 1.
Statement III is true for subtraction since
4 – 2 = 2 and is true for division since 4 ÷ 2 = 2.
Therefore, only Statement III must be true.
The correct answer is C.
105. The average distance between the Sun and a certain
planet is approximately 2.3 × 1014 inches. Which of the
following is closest to the average distance between
the Sun and the planet, in kilometers? (1 kilometer is
approximately 3.9 × 104 inches.)
(A)

7.1 × 108

(B)

5.9 × 109

(C)

1.6 × 1010

(D)

1.6 × 1011

(E)

5.9 × 1011

Arithmetic Measurement conversion

Convert to kilometers and then estimate.
⎛

⎞

= 2.3 × 10
( 2.3 × 10 in ) ⎜⎝ 3.91×km
⎟
10 in ⎠ 3.9 × 10
14

104. The symbol Δ denotes one of the four arithmetic
operations: addition, subtraction, multiplication, or
division. If 6 Δ 3 ≤ 3, which of the following must be
true?
I.

2Δ2=0

II.

2Δ2=1

III.

4Δ2=2

(A)

I only

(B)

II only

(C)

III only

(D)

I and II only

(E)

I, II, and III

= 2.3 × 1014 − 4 km
3.9
≈ 2 × 1010
4
= 0.5 × 1010
= 5 × 10 9
The correct answer is B.

215

GMAT® Ofﬁcial Guide 2018

1
106. If mn ≠ 0 and 25 percent of n equals 37 percent of
2
12n
?
m, what is the value of
m
(A)

(B)

32
3

(C)

(D)

(E)

9
8

The correct answer is E.
108. Last year Joe grew 1 inch and Sally grew 200 percent
more than Joe grew. How many inches did Sally grow
last year?

Algebra Percents; First-degree equations

It is given that (25%)n = (37.5%)m, or
12n
0.25n = 0.375m. The value of
can be
m n
found by first finding the value of and then
m
multiplying the result by 12. Doing this gives
12n ⎛ 0.375 ⎞
=⎜
(12) = 18. Alternatively, the
⎝ 0.25 ⎟⎠
m
numbers involved allow for a series of simple
equation transformations to be carried out, such
as the following:
0.25n
25n
50n
2n
12n
12n
m

=
=
=
=
=

0.375m
37.5m
75m
3m
18m

= 18

given
multiply both sides by 100
multiply both sides by 2
divide both sides by 25
multiply both sides by 6
divide both sides by m

The correct answer is A.
107. In the coordinate plane, a circle has center (2,−3) and
passes through the point (5,0). What is the area of the
circle?
(A)

3π

(B)

3 2π

(C)

3 3π

(D)

9π

(E)

18π

Geometry Coordinate geometry; Circles; Area

The area of a circle is given by πr2, where r is the
radius of the circle. The value of r2 is the square
of the distance from the center to a point
216

of the circle. Using the distance formula,
r2 = (2 − 5)2 + (−3 − 0)2 = 9 + 9 = 18. Therefore,
the area of the circle is 18π.

(A)

(B)

(C)

(D)

(E)

Arithmetic Percents

Joe grew 1 inch last year and Sally grew 200 percent
more than Joe grew, so Sally grew 1 inch plus
200 percent of 1 inch or 1 + 2(1) = 3 inches.
The correct answer is D.
109. The cost C, in dollars, to remove p percent of a certain
pollutant from a pond is estimated by using the formula
100,000p
C=
. According to this estimate, how much
100 − p
more would it cost to remove 90 percent of the
pollutant from the pond than it would cost to remove
80 percent of the pollutant?
(A)

$500,000

(B)

$100,000

(C)

$50,000

(D)

$10,000

(E)

$5,000

Algebra; Arithmetic Simplifying algebraic
expressions; Operations on rational numbers

Removing 90% of the pollutant from the pond
(100, 000 )( 90 ) = 9, 000, 000 =
would cost
100 − 90
10
900,000 dollars, and removing 80% of the
(100, 000 )(80 ) =
pollutant would cost
100 − 80
8, 000, 000
= 400, 000 dollars. The difference is,
20
then, $900,000 − $400,000 = $500,000.
The correct answer is A.

5.5 Problem Solving Answer Explanations

110. If xy ≠ 0 and x2y2 – xy = 6, which of the following could
be y in terms of x ?
I.
II.

Algebra Simplifying algebraic expressions

1
( 5, 280 ), s = 40 into the
2
given formula and calculate the value for N.
Substitute L = 2, d =

1
2x
−

2
x

III.

3
x

(A)

I only

(B)

II only

(C)

I and II

(D)

I and III

(E)

II and III

⎛ 1⎞
20 ( 2 ) ⎜ ⎟ ( 5, 280 )
⎝ 2⎠
N=
600 + 40 2
20 ( 5, 280 )
=
600 + 1, 600
=

2 ( 528 )
22
528
=
11
= 48
=

Algebra Second-degree equations

x2y2 – xy = 6 given
x2y2 – xy – 6 = 0 subtract 6 from both sides
(xy + 2)(xy – 3) = 0 factor
2
So, xy + 2 = 0, which means xy = –2 and y = − ,
x
or xy – 3 = 0, which means that xy = 3 and
3
y = . Thus, y in terms of x could be given by the
x
expressions in II or III.
The correct answer is E.
111. At a certain instant in time, the number of cars, N,
traveling on a portion of a certain highway can be
estimated by the formula
N=

20Ld

155

(B)

(C)

(D)

(E)

The correct answer is D.
112.

4.8 × 109 is closest in value to
(A)

2,200

(B)

70,000

(C)

220,000

(D)

7,000,000

(E)

22,000,000

Arithmetic Operations on radical expressions

4.8 × 10 9 = 48 × 108

600 + s 2

where L is the number of lanes in the same direction, d
is the length of the portion of the highway, in feet, and
s is the average speed of the cars, in miles per hour.
Based on the formula, what is the estimated number
1
of cars traveling on a ‐ mile portion of the highway
2
if the highway has 2 lanes in the same direction and
the average speed of the cars is 40 miles per hour?
(5,280 feet = 1 mile)
(A)

20 ( 5, 280 )
2, 200

substitute 48 × 108
for 4.8 × 10 9

= 48 × 108
≈ 49 × 108

ab = a × b
49 ≈ 48

and then
49 × 108 = 7 × 10 4

49 = 7,

(10 )

4 2

108 =

= 10 4

= 70, 000
The correct answer is B.

217

GMAT® Ofﬁcial Guide 2018

113. Three printing presses, R, S, and T, working together
at their respective constant rates, can do a certain
printing job in 4 hours. S and T, working together at
their respective constant rates, can do the same job
in 5 hours. How many hours would it take R, working
alone at its constant rate, to do the same job?
(A)

(B)

(C)

(D)

(E)

Algebra Applied problems

Let r be the portion of the job that printing
press R, working alone, completes in 1 hour;
and let s and t be the corresponding portions,
respectively, for printing press S and printing
press T. From the given information, it follows
1
1
that r + s + t = and s + t = . Subtracting
4
5
1 1
1
these two equations gives r = − = . It
4 5 20
follows that printing press R, working alone, will
1
complete
of the job in 1 hour, and therefore
20
printing press R, working alone, will complete the
job in 20 hours.
The correct answer is E.
114. For a party, three solid cheese balls with diameters of
2 inches, 4 inches, and 6 inches, respectively, were
combined to form a single cheese ball. What was the
approximate diameter, in inches, of the new cheese
4
ball? (The volume of a sphere is π r 3 , where r is the
3
radius.)
(A)

(B)

(C)

(D)

3 8

(E)

23 36

Since the diameters of the cheese balls are given
as 2 inches, 4 inches, and 6 inches, the radii of
the cheese balls are 1 inch, 2 inches, and 3 inches,
4
respectively. Using V = π r 3, the combined
3

)

Thus, if R represents the radius of the new cheese
ball, then the volume of the new cheese ball
4
4
is π R 3 = π ( 36 ) and R3 = 36, from which
3
3
it follows that R = 3 36 inches. Therefore,
the diameter of the new cheese ball is
2R = 2 3 36 inches.
The correct answer is E.
115. The sum of all the integers k such that −26 < k < 24 is
(A)

(B)

−2

(C)

−25

(D)

−49

(E)

−51

Arithmetic Operations on integers

In the sum of all integers k such that
−26 < k < 24, the positive integers from 1 through
23 can be paired with the negative integers from
−1 through −23. The sum of these pairs is 0
because a + (−a) = 0 for all integers a.
Therefore, the sum of all integers k such that
−26 < k < 24 is −25 + (−24) + (23)(0) = −49.
The correct answer is D.
R

0
116. The number line shown contains three points R, S, and
T, whose coordinates have absolute values r, s, and t,
respectively. Which of the following equals the average
(arithmetic mean) of the coordinates of the points R, S,
and T ?

Geometry Volume

218

(

4
volume of the 3 cheese balls is π 13 + 23 + 33
3
4
or π (36) cubic inches.
3

(A)

(B)

s+t−r

(C)

r −s−t
3

(D)

r +s+t
3

(E)

s+t −r
3

5.5 Problem Solving Answer Explanations

Arithmetic Absolute value; Number line

Algebra Sets

Because point R is to the left of 0 on the number
line, the coordinate of R is negative. It is given
that r is the absolute value of the coordinate of R
and so the coordinate of R is −r. Because points
S and T are to the right of 0 on the number line,
their coordinates are positive. It is given that s
and t are the absolute values of the coordinates of
S and T, and so the coordinates of S and T are s
and t. The arithmetic mean of the coordinates of
s+t −r
.
R, S, and T is
3

Since x students are taking music, y students are
taking art, and z students are taking both music and
art, the number of students taking only music is
x − z, and the number of students taking only art is
y − z, as illustrated by the following Venn diagram.

music

x–z

The correct answer is E.
117. Mark and Ann together were allocated n boxes of
cookies to sell for a club project. Mark sold 10 boxes
less than n and Ann sold 2 boxes less than n. If Mark
and Ann have each sold at least one box of cookies,
but together they have sold less than n boxes, what is
the value of n ?
(A)

(B)

(C)

(D)

(E)

Algebra Inequalities

Mark sold n − 10 boxes and Ann sold n − 2 boxes.
Because each person sold at least one box, it
follows that n − 10 ≥ 1 and n − 2 ≥ 1, which
implies that n ≥ 11. On the other hand, together
they sold less than n boxes, so (n − 10) + (n − 2) < n,
which implies that n < 12. Therefore, n is an
integer such that n ≥ 11 and n < 12, which
implies that n = 11.
The correct answer is A.
118. A certain high school has 5,000 students. Of these
students, x are taking music, y are taking art, and z
are taking both music and art. How many students are
taking neither music nor art?
(A)

5,000 − z

(B)

5,000 − x − y

(C)

5,000 − x + z

(D)

5,000 − x − y − z

(E)

5,000 − x − y + z

art

y–z

Therefore, the number of students taking
neither music nor art is
5,000 − [(x − z) + z + (y − z)] = 5,000 − x − y + z.
The correct answer is E.
119. Yesterday’s closing prices of 2,420 different stocks
listed on a certain stock exchange were all different
from today’s closing prices. The number of stocks
that closed at a higher price today than yesterday was
20 percent greater than the number that closed at a
lower price. How many of the stocks closed at a higher
price today than yesterday?
(A)

484

(B)

726

(C)

1,100

(D)

1,320

(E)

1,694

Arithmetic Percents

Let n be the number of stocks that closed at a
lower price today than yesterday. Then 1.2n is the
number of stocks that closed at a higher price
today than yesterday, and 1.2n is the value asked
for. Because the total number of stocks is 2,420, it
follows that n + 1.2n = 2,420, or 2.2n = 2,420.
2, 420
= 1,100, and hence
2.2
1.2n = (1.2)(1,100) = 1,320.

Therefore, n =

The correct answer is D.
219

GMAT® Ofﬁcial Guide 2018

120. Each person who attended a company meeting was
either a stockholder in the company, an employee
of the company, or both. If 62 percent of those
who attended the meeting were stockholders and
47 percent were employees, what percent were
stockholders who were not employees?
(A)

34%

(B)

38%

(C)

45%

(D)

53%

(E)

62%

Amount Budgeted

Amount Spent

$110,000

$117,000

40,000

42,000

2,500

2,340

Payroll
Taxes
Insurance

122. The table shows the amount budgeted and the
amount spent for each of three accounts in a certain
company. For which of these accounts did the amount
spent differ from the amount budgeted by more than
6 percent of the amount budgeted?

Arithmetic Sets

(A)

Payroll only

Let M represent the number of meeting
attendees. Then, since 62% of M or 0.62M were
stockholders and 47% of M or 0.47M were
employees, it follows that 0.62M + 0.47M = 1.09M
were either stockholders, employees, or both. Since
1.09M exceeds M, the excess 1.09M − M = 0.09M
must be the number of attendees who were both
stockholders and employees, leaving the rest
0.62M − 0.09M = 0.53M, or 53%, of the meeting
attendees to be stockholders but not employees.

(B)

Taxes only

(C)

Insurance only

(D)

Payroll and Insurance

(E)

Taxes and Insurance

The correct answer is D.
121. A gym class can be divided into 8 teams with an equal
number of players on each team or into 12 teams with
an equal number of players on each team. What is the
lowest possible number of students in the class?
(A)

(B)

(C)

(D)

(E)

Arithmetic Properties of numbers

The lowest value that can be divided evenly
by 8 and 12 is their least common multiple
(LCM). Since 8 = 23 and 12 = 22(3), the LCM is
23(3) = 24.
The correct answer is B.

220

Accounts

Arithmetic Percents

For Payroll, 6% of the budgeted amount
is (0.06)($110,000) = $6,600. Since
$117,000 – $110,000 = $7,000 > $6,600, the
amount spent differed from the amount budgeted
by more than 6%.
For Taxes, 6% of the budgeted amount
is (0.06)($40,000) = $2,400. Since
$42,000 – $40,000 = $2,000 < $2,400, the
amount spent did not differ from the amount
budgeted by more than 6%.
For Insurance, 6% of the budgeted
amount is (0.06)($2,500) = $150. Since
$2,500 – $2,340 = $160 > $150, the amount
spent differed from the amount budgeted by more
than 6%.
Thus, the amount spent differed from the amount
budgeted by more than 6% for Payroll and
Insurance.
The correct answer is D.

5.5 Problem Solving Answer Explanations

Then rotating this figure clockwise about point P
through an angle of 120° will produce the figure
shown below.

123. In the figure above, triangle ABC is equilateral, and point
P is equidistant from vertices A, B, and C. If triangle ABC
is rotated clockwise about point P, what is the minimum
number of degrees the triangle must be rotated so that
point B will be in the position where point A is now?
(A)

(B)

120

(C)

180

(D)

240

(E)

270

P
B

In this figure, point B is in the position where
point A was in the original figure. The triangle
was rotated clockwise about point P through
120° + 120° = 240°.

Geometry Angles

The correct answer is D.

P
A

Since ΔABC is equilateral, the measure of
∠ACB is 60°. Therefore, the measure of
∠BCD is 180° − 60° = 120°. Rotating the figure
clockwise about point P through an angle of 120°
will produce the figure shown below.

(A)

(B)

(C)

(D)

(E)

Arithmetic Operations on rational numbers

2
of the members must vote in favor of
3
1
a resolution, then no more than of the members
3
can be voting against it. On this 40‐member
1
1
committee, (40) = 13 , which means that no
3
3
more than 13 members can vote against the
resolution and still have it pass.
If at least

P
C

2
of the 40 members of a committee must
3
vote in favor of a resolution for it to pass. What is the
greatest number of members who could vote against
the resolution and still have it pass?

124. At least

The correct answer is E.
D
221

GMAT® Ofﬁcial Guide 2018

126. If n = 20! + 17, then n is divisible by which of the following?

number of computers assembled

20
18

II.

III.

(A)

None

(B)

I only

(C)

II only

(D)

I and II

(E)

II and III

12
10
8
6
4
2
0

Arithmetic Properties of numbers

day
125. The graph shows the number of computers assembled
during each of 6 consecutive days. From what day to
the next day was the percent change in the number of
computers assembled the greatest in magnitude?
(A)

From Day 1 to Day 2

(B)

From Day 2 to Day 3

(C)

From Day 3 to Day 4

(D)

From Day 4 to Day 5

(E)

From Day 5 to Day 6

Arithmetic Percents

The following table shows the percent change
from each day to the next and the magnitude of
the percent change.
Time period

Because 20! is the product of all integers from 1
through 20, it follows that 20! is divisible by each
integer from 1 through 20. In particular, 20! is
divisible by each of the integers 15, 17, and 19.
Since 20! and 17 are both divisible by 17, their
sum is divisible by 17, and hence the correct
answer will include II. If n were divisible by 15,
then n − 20! would be divisible by 15. But,
n − 20! = 17 and 17 is not divisible by 15.
Therefore, the correct answer does not include I.
If n were divisible by 19, then n − 20! would be
divisible by 19. But, n − 20! = 17 and 17 is not
divisible by 19. Therefore, the correct answer does
not include III.
The correct answer is C.

Percent change

Magnitude of percent change

From Day 1 to Day 2

⎛ 12 − 20
⎞
⎛ 8
⎞
× 100 ⎟ % = ⎜ −
× 100 ⎟ % = −40%
⎜⎝
⎠
⎝ 20
⎠
20

From Day 2 to Day 3

⎛ 18 − 12
⎞
⎛ 6
⎞
× 100 ⎟ % = ⎜ × 100 ⎟ % = 50%
⎜⎝
⎠
⎝
⎠
12
12

From Day 3 to Day 4

⎛ 10 − 18
⎞
⎛ 8
⎞
× 100 ⎟ % = ⎜ − × 100 ⎟ % ≈ −44%
⎜⎝
⎠
⎝ 18
⎠
18

From Day 4 to Day 5

⎛ 16 − 10
⎞
⎛ 6
⎞
× 100 ⎟ % = ⎜ × 100 ⎟ % = 60%
⎜⎝
⎠
⎝
⎠
10
10

From Day 5 to Day 6

⎛ 8 − 16
⎞
⎛ 8
⎞
× 100 ⎟ % = ⎜ − × 100 ⎟ % = −50%
⎜⎝
⎠
⎝ 16
⎠
16

The correct answer is D.
222

5.5 Problem Solving Answer Explanations

127. The product of two negative numbers is 160. If the
lesser of the two numbers is 4 less than twice the
greater, what is the greater number?
(A)

–20

(B)

–16

(C)

–10

(D)

–8

(E)

–4

Algebra Second-degree equations

Let x and y be the two numbers, where x is the
lesser of the two numbers and y is the number
desired. From the given information it follows
that xy = 160 and x = 2y – 4, from which it
follows that (2y – 4)y = 160. Dividing both sides
of the last equation by 2 gives (y – 2)y = 80. Thus,
80 is to be written as a product of two negative
numbers, one that is 2 less than the other. Trying
simple factorizations of 80 quickly leads to the
value of y: (–40)(–2) = 80, (–20)(–4) = 80,
(–10)(–8) = 80. Therefore, y = –8. Note that
because –8 is one of the answer choices, it is not
necessary to ensure there are no other negative
solutions to the equation (y – 2)y = 80.
Alternatively, (y – 2)y = 80 can be written as
y2 – 2y – 80 = 0. Factoring the left side gives
(y + 8)(y – 10) = 0, and y = –8 is the only negative
solution.
The correct answer is D.
128. According to a certain estimate, the depth N(t), in
centimeters, of the water in a certain tank at t hours past
2:00 in the morning is given by N(t) = −20(t − 5)2 + 500
for 0 ≤ t ≤ 10. According to this estimate, at what time
in the morning does the depth of the water in the tank
reach its maximum?
(A)