Review Guide Math

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ACT Math Concepts to Know

Just Numbers
Terms
? Concept
Undefined

Imaginary number

Integers
Rational/ Irrational number

Adding and subtracting negative
numbers

Multiplying and dividing with
negative numbers

Explanation
An expression is undefined when the
denominator equals zero.


To take the square root of a negative
number, first take the square root of
the number as if it were positive, then
add “i”
 i2 = -1
Whole numbers, including negative
numbers and zero
Rational – Can be expressed as a decimal or
fraction
Irrational – Cannot be expressed as a
decimal or fraction
Adding a positive and a negative –
1) Ignore the signs
2) Subtract the smaller number from the
larger number
3) Put on the sign from the larger number
Subtracting negative numbers –
1) Change the subtraction into addition
1) Ignore the signs and do the problem
without signs
2) If there is an odd number of negative
signs, add a negative sign to the final
answer

1

Example/Visual
𝑎+𝑏

If f(x) = 𝑥−𝑐 , for what value of x would this
function be undefined?
Answer: c, because c - c = 0
Solve: √3𝑥 = 6𝑖
1) Square both sides  3x = -6
2) Divide both sides by 3  x = -2

-2, 0, 3, 7
Rational - .45, ¾
Irrational - √2, √3, 𝜋
-38 + 25
1) 38 - 25 = 13
2) -38 is larger than 25, so the answer is -13

13 - (-23) =
13 + (+23) = 36
-3 x -5 x -2
1) 3 x 5 x 2 = 30
2) Odd number of negative signs  -30

PEMDAS/ Order of operations

1) Parentheses
2) Exponents
3) Multiplication and Division
4) Addition and Subtraction

Absolute value

A number inside an absolute value sign
becomes positive
1) Do what's inside the absolute value sign
first
2) Make the result positive
Subtract the two numbers and then add 1

The number of integers from one
number to another

We need to add 1 to include the first
number

|2 - 5| =
|-3| = 3

How many integers are there from 12 to 25?
25 - 12 = 13
13 + 1 = 14

Divisibility
? Concept
Factors and Multiples

Prime number
Prime Factorization

Explanation
Factor – A number that divides into an
integer with no remainder
Multiple – A number that the integer
divides into with no remainder
The only factors are 1 and the number.
Note: 1 is NOT a prime number
Keep breaking a number into factors until
all the factors are prime

2

Example/Visual
Factors of 20:
1, 2, 4, 5, 10, 20
Multiples of 6:
6, 12, 18, 24, 30, 36…
2, 3, 5, 7, 13…
Prime factorization of 48:
48 =
12 x 4 =
(3 x 4) x (2 x 2) =
3x2x2x2x2

Least Common Multiple

Greatest Common Factor

Even or Odd?

Is a number divisible by 2, 3, 4, 5,
9, and 10?

Remainder

The lowest number that is a multiple of
both numbers
1) Find multiples of the larger number until
you get to one that is also a multiple of the
smaller number.
The highest number that is a factor of both
numbers
1) Figure it out mentally
OR
1) Break each number into its prime factors
2) Multiple the prime factors they have in
common
To find out if an answer will be even or odd,
just plug in simple numbers like 1 and 2

Divisible by…
2 – If the last digit is even
3 – If the sum of the digits is divisible by 3
4 – If the last two digits are divisible by 4
5 – If the last digit is 5 or 0
9 – If the sum of the digits is divisible by 9
10 – If the last digit is 0
The whole number left over after division

The LCM of 12 and 15:
15, 30, 45, 60
60 is also a multiple of 12
The LCM is 60
The GCM of 16 and 24
Prime factorization of 16:
2x2x2x2
Prime factorization of 24:
2x2x2x3
16 and 24 have 2 x 2 x 2 in common
The GCM is 2 x 2 x 2 = 8
Will 2x3 + 1 be odd or even?
x=12+1=3
x = 2  16 + 1 = 17
2x3 + 1 will always be odd
36  the last digit, 6, is even
357  3 + 5 + 7 = 15 (15 is divisible by 3)
524  24 is divisible by 4
55  the last digit is 5
396  3 + 9 + 6 = 18 (18 is divisible by 9)
730  the last digit is 0
14/4 = 3 Remainder 2

Fractions and Decimals
? Concept
Reducing fractions

Adding and subtracting fractions

Explanation
Cancel out all the factors that the
numerator (top number) and the
denominator (bottom number) have in
common
1) Find a common denominator
2) Add or subtract the numerators

3

Example/Visual
15
3×5
𝟑
=
=
20
4×5
𝟒
1 2
5
8
𝟏𝟑
+ =
+
=
4 5
20 20
𝟐𝟎

Multiplying fractions
Dividing fractions
Changing a mixed number to an
improper fraction

Changing an improper fraction to
a mixed number

Reciprocal
Which fraction is greater?

Converting fractions to decimals
Finding a particular digit in a
repeating decimal

Percent formula

1) Multiply the numerators
2) Multiply the denominators
1) Flip the second fraction
2) Multiple the two fractions
1) Multiply the whole number by the
denominator, then add the numerator
2) Put the number from #1 over the same
denominator
1) Divide the denominator into the
numerator to get a whole number and a
remainder
2) The whole number remains a whole
number, and the remainder is the
numerator. The denominator stays the
same
Flip the numerator and the denominator
1) Convert both fractions so they have a
common denominator
OR
2) Convert both fractions to decimals
Divide the top number by the bottom
number
1) Which digit are you trying to find?
2) How many digits are repeating?
3) Find the multiple of #2 that is closest to
the digit you are trying to find, and then
count up or down to find the digit you are
looking for
1) Change the percent into a decimal
(divide it by 100)
2) Change “of” into a multiplication sign,
and Change “is/are” into an equal sign
3) Complete the problem

4

3 5
15
× =
8 7
56
2 7
2 9
𝟏𝟖
÷ = ÷ =
5 9
5 7
𝟑𝟓
(3 × 7) + 3
3
𝟐𝟒
3 =
=
7
7
𝟕
23
5
1) 23 ÷ 5 = 4 R 3
𝟑
2) 𝟒 𝟓

2
𝟓
→
5
𝟐
5
9
10
or  or
8

16

16

9
16

𝟓
𝟖

 is greater

5 ÷ 8 = .625
9 ÷ 16 = .5625
9 ÷ 16 = .5625
1) To find the 101th digit of
6

6
11

2) = .54545454 (2 digits are repeating)
11
3) The digit of every multiple of 2 is 4. The
100th digit is 4. The 101th digit is 5.
70% of 50 is what?
1) 70% ÷ 100 = .7
2) .7 x 50 = ?
3) .7 x 50 = 35

Percent increase and decrease

Multiple increases and decreases

Percent increase –
1) Convert the percent to a decimal, add 1
2) Multiply
Percent decrease –
1) Convert the percent to a decimal
2) Multiply
3) Check if you need to subtract your
answer from the original
1) Start with 100 and then apply the
increases and decreases

$120 increased by 25%
1) 25% = .25
.25 + 1 = 1.25
1.25 x $120 = $150
$120 decreased by 25%
1) 25% = .25
2) .25 x $120 = $30
3) $120 - $30 = $90
A price is increased by 20% and then the new
price is decreased by 30%. What is the net
change?
20% increase  100 x 1.20 = 120
30% decrease  120 x .30 = 36
120 - 36 = 84
Net change  100 - 84 = 16
16% decrease

Ratios, Proportions, and Rates
? Concept
Setting up a ratio

Solving a proportion

Solving rate problems

Explanation
1) Put the number after “of” on top
2) Put the number after “to” on bottom
3) Reduce
1) Cross multiply

Rate x Time = Distance
Average Rate = Total Distance / Total Time

5

Example/Visual
What is the ratio of 3 cats to 5 dogs?
Answer  3:5
2
𝑥
=
5
15
5x = 30
x=6
A car travels 294 miles in 6 hours. What is the
rate at which it is traveling?
Rate x 6 = 294
Rate = 49mph

Averages
? Concept
Average formula

Explanation
Average = Sum of terms / Number of terms

Average of evenly spaced
numbers

Just average the smallest and largest
numbers

Using the average to find the sum

Sum = Average x Number of terms
Then, subtract the numbers you already
have to find the answer

Counting the possibilities

Multiply the number of choices for the first
thing by the number of choices for the
second thing

Probability

Number of items / Total number of items =
Probability/Percentage

Example/Visual
What is the average of 25, 39, and 42?
25 + 39 + 44
108
→
→ 𝟑𝟔
3
3
What is the average of all the even numbers
from 12 to 36?
12 + 36
48
→
→ 𝟐𝟒
2
2
Jim’s average score after four tests is 88. What
score on the fifth test would bring Martin’s
average up to exactly 90?
Answer:
90 x 5 = 450
450 - 88 - 88 - 88 - 88 = 98
John has 5 different shirts and 7 different pairs
of pants. How many different combinations of
shirts and pants can he have?
Answer  5 x 7 = 35 different combinations
In a group of 30 students, 12 are male. What
percentage of the group is male?
Answer  12 / 30 = .4  40% are male

Roots
? Concept
Simplifying square roots

Adding and subtracting roots

Explanation
1) Factor out the perfect squares
2) Put the square root of the perfect
square(s) in front of the radical

Example/Visual
√27
√9 × 3
𝟑√𝟑
2√5 + 6√5 = 8√5

If the number under the radical is the
same, you can add or subtract them

6

Multiplying and dividing roots

Matrices – Adding and
subtracting

You can multiply two different roots by
first multiplying the numbers under the
roots
You can divide two different roots by first
dividing the numbers under the roots
Simply add or subtract the spaces that
correspond to each other

7

√3 × √5 = √15
√15 ÷ √5 = √3

Algebra
Algebraic Expressions
? Concept
Multiplying and Dividing Powers

Example/Visual
𝑥3 × 𝑥4 = 𝑥7

Raising powers to powers

Explanation
To multiply powers with the same base:
add the exponents
To divide powers with the same base,
subtract the exponents
Multiply the exponents

Evaluating an algebraic expression

Plug in the values for the unknown

Adding and subtracting algebraic
expressions
Multiplying monomials (one term
by one term)
Multiplying binomials (two terms
by two terms) using FOIL

Add and subtract like terms

If f(x) = x3 - x2 + x, what is the value of f(-2)?
Answer:
(-2)3 - (-2)2 + (-2)
(-8) - (4) + (-2) = -14
2x + 3x = 5x
x2 + 3x2 = 4x2
3𝑥 2 × 5𝑥 3 = 15𝑥 5

Multiply the coefficients and the variables
separately
In this order, multiply the:
1) First terms
2) Outside terms
3) Inside terms
4) Last terms

𝑥5 ÷ 𝑥3 = 𝑥2
𝑥 3 × 𝑥 4 = 𝑥 12

(2x + 2)(x – 2) =
(2x)(x) + (2x)(-2) + (2)(x) + (2)(-2) =
2x2 + (-4x) + 2x + (-4) =
2x2 - 2x -4

Factoring Algebraic Expressions
? Concept
Factoring out a common divisor
Factoring the difference of
squares

Explanation
If all the terms have a common factor, it
can be factored out
The ACT likes to test this.
x2 - (number)2 = (x - number)(x + number)

8

Example/Visual
2x2 - 8x =
2x(x - 4)
x2 - 9 =
(x - 3)(x + 3)

Factoring the square of a binomial
(a + b)2 or (a - b)2

Factoring other algebraic
expressions

Simplifying an algebraic fraction

The ACT likes to test this
If the last number is a perfect square, check
if the algebraic expression is a square of a
binomial
Think about what binomials you could use
FOIL on to result with the algebraic
expression
1) What first terms could get you the
squared term?
2) What last terms could get you the
number?
3) What combinations of first and last
terms could get you the middle term?
1) Factor the numerator and denominator
2) Cancel out factors that are in both the
numerator and denominator

x2 + 8x + 16 =
(x + 4)2

6x2 - 16x + 8 =
1) 3 x 2 or 6 x 1
2) 4 x 2 or 8 x 1
3) (3 x 4) + (2 x 2) = 16
(3x2 - 2)(2x2 - 4)

𝑥+3
+ 5𝑥 + 6
(𝑥 + 3)
(𝑥 + 3)(𝑥 + 2)
𝟏
𝒙+𝟐
𝑥2

Solving Equations
? Concept
Solving a linear equation

Solving “in terms of”

Explanation
1) Add and subtract terms to get the x
terms on one side
2) Divide (or multiply) to solve for x
To solve for one variable in terms of
another, do the same thing above for
the variable you are solving for. The
other side will have the variable you are
solving in terms of.

9

Example/Visual
2x + 6 = 5x
* Subtract 2x from both sides
6 = 3x
2=x
Solve for x in terms of y: 2x - y - 4 = y
* add y to both sides
2x – 4 = 2y
* divide both sides by 2
x–2=y

Translating from English into
algebra

1) Break the word problem into parts
2) Write out the algebraic expression for
the different parts and then put them
together according to what the problem
is asking for

Solving a quadratic equation

1) Factor the algebraic expression
2) Find out what values will make the
expression equal zero
OR
1) Use the quadratic formula
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎

Solving a system of equations

The toll for driving a segment of a certain freeway is
$1.20 plus 20 cents for each mile traveled. John paid
a $25.00 toll for driving a segment of the freeway.
How many miles did he travel?
1.20 + .20(m) = 25
.20(m) = 23.80
m = 119 miles
x2 + 5x + 6 = 0
(x + 2)(x + 3) = 0
x = -2 or -3
−5 ± √52 − 4(1)(6)
2(1)
−5 ± √1
𝑥=
2
−4
−6
𝑥=
𝑜𝑟
2
2
𝒙 = −𝟐 𝐨𝐫 − 𝟑
𝑥=

Combine the equations in a way that one
of the variables cancels out

Elimination
Solving an equation that has
absolute value signs

There will be two different answers: one
that results in a positive number and one
that results in a negative number

10

Substitution

Solving an inequality

1) Solve for the variable as if it is a linear
equation
2) If you multiply or divide by a negative
number, flip the inequality sign

Graphic inequalities on a
number line

Use a solid circle if the point is included
and an open circle of the point is not
included

Graphing inequalities on a grid

1) Choose a coordinate to plug into the
equation
2) If the coordinate makes the inequality
true, shade in that side of the line/curve.
If not, shade in the other side.

11

Coordinate geometry
? Concept
Finding the distance between two
points

Explanation
1) Use the Pythagorean Theorem
(x1 - x2)2 + (y1 - y2)2 = c2
OR
2) Use special right triangles

Example/Visual

Finding the midpoint

Using two points to find the slope

Slope = Change in y / Change in x

12

A line perpendicular to the slope

Take the reciprocal of the slope and change
the sign

Using an equation to find the
slope

Use the slope-intercept form
y = mx + b
m is the slope, b is the y-intercept

Using an equation to find the
intercept

To find the y-intercept: Plug in 0 for x and
solve for y
To find the x-intercept: Plug in 0 for y and
solve for x
(x - h)2 + (y - k)2 = r2
r = radius
(h,k) is the center of the circle

Equation for a circle

13

Equation for a parabola

y = ax2 + bx + c

Equation for an ellipse

x2/a2 + y2/b2 = 1
a = x-intercepts
b = y-intercepts

Factored: (x + 1)(x - 3)

14

Plane Geometry
Lines and Angles
? Concept
Intersecting lines

Parallel lines and transversal (a
line that crosses through parallel
lines)

Explanation
When two lines intersect:
 Adjacent angles, or angles next to each
other, are supplementary and add up
to 180 degrees
 Vertical angles, or angles across from
each other, are equal
Forms four equal acute angles and four
equal obtuse angles

Triangles – General
? Concept
Interior angles of a
triangle

Explanation
The three angles add up to 180
degrees

Example/Visual

15

Example/Visual
∠a + ∠b = 180°
∠b = ∠d

Exterior angle of a
triangle




An exterior angle equals the
sum of the two angles it is not
next to
The three exterior angles of a
triangle add up to 360 degrees
∠a + ∠b + ∠c = 360°

Similar triangles

Have the same shape, but different
size
 Corresponding angles are
equal
 Corresponding sides are
proportional

Area of a triangle




Area = ½(Base)(Height)
For non-right triangles, the
height is inside the triangle

Right Triangles
? Concept
Pythagorean Theorem

Explanation
(leg1)2 + (leg2)2 = (hypotenuse)2

Example/Visual

16

∠a = ∠C + ∠D

Special right triangles

3-4-5 (and multiples)
5-12-13 (and multiples)
30-60-90 – 1:√𝟑:2
45-45-90 – 1:1: √𝟐

Other Polygons
? Concept
Rectangle

Parallelogram

Explanation
 Has 4 right angles
 Opposite sides are equal
 Diagonals are equal
 Area = Length x Width






Example/Visual



Has 2 pairs of parallel sides
Opposite sides are equal
Opposite angles are equal
Angles next to each other add up to
180 degrees
Area = Base x Height

Square




Rectangle with 4 equal dies
Area = (Side)2

Trapezoid




One pair of parallel sides
One pair of non-parallel sides



Area =

base1 + base2
2

× ℎ𝑒𝑖𝑔ℎ𝑡

17

Interior angles of a polygon


(n−2)x 180

o

= Interior angle
n is the number of sides

n

for a polygon with 8 sides

Circles
? Concept
Circumference (outside) of a circle

Explanation
Circumference = 2r

Area of a circle

Area = r2

Length of an arc (piece of the
circumference)

Arc = n/360 * (2r)
n = the arc’s central angle

Example/Visual

18

Area of a sector (piece of the area
of a circle)

Sector = n/360 x (r2)
n = the sector’s central angle

Solids
? Concept
Surface area of a rectangular
solid

Explanation
Surface area = 2(lw + wh + lh)

Volume of a rectangular solid

Volume = lwh

Volume of a cylinder

Volume of a cylinder = r2H

Volume of a cone

Volume of a cone = 1/3r2H

Example/Visual

19

Volume of a sphere

Volume of a sphere = 4/3r3

Surface area of a sphere

Surface area of a sphere = 4r2

20

Trigonometry
? Concept
Sine, Cosine, and Tangent
of acute angles

Cotangent, Secant, and
Cosecant of acute angles

Trigonometric functions of
other angles

Simplifying trigonometric
expressions

Explanation
SOH-CAH-TOA
Sine = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite/Adjacent

Example/Visual

These are the reciprocals of Sine, Cosine,
and Tangent
Cotangent = 1/Tangent =
Adjacent/Opposite
Secant = 1/Cosine = Hypotenuse/Adjacent
Cosecant = 1/Sine = Hypotenuse/Opposite
For angles greater than 90 degrees
1) Draw a circle with radius 1 centered on
the coordinate grid
2) Rotate the appropriate number of
degrees counterclockwise
3) Draw an acute triangle depending on
where the angle lands
4) Find the answer
sin2x + cos2x = 1
Also, use the trigonometric properties to
simplify

21

Graphic trigonometric
functions

x-axis  angle
y-axis  value of the trigonometric
function
Use special angles like 0, 30, 45, 60, 90,
etc. to plot key points

Radians

π = 180°

2π = 360°
π = 180°
π/2 = 90°

22



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