!
Overview: About the ACT Math Test
The 60-minute, 60-question ACT Math Test contains questions from six categories of subjects taught in most
high schools up to the start of 12th grade. The categories are listed below with the number of questions from
each category:
■
Pre-Algebra (14 questions)
■
Elementary Algebra (10 questions)
■
Intermediate Algebra (9 questions)
■
Coordinate Geometry (9 questions)
■
Plane Geometry (14 questions)
■
Tr igo nom et r y ( 4 que sti ons )
Like the other tests of the ACT, the math test requires you to use your reasoning skills. Believe it or not,
this is good news, since it generally means that you do not need to remember every formula you were ever
CHAPTER
ACT Math Test
Practice
4
131
IB Math Year 1 Summer Assignment: # 1 - 80 on pages 165 - 183
On our first full day of class in August, you will complete a
scantron form with your answers to the Practice Questions # 1 - 80
on pages "165 - 183". This will be your first test grade in the class
(as well as excellent practice for the ACT!).
IB Math HL A&A Y1 Summer Assignment: p. 165, # 1 - 80
taught in algebra class. You will, however, need a strong foundation in all the subjects listed on the previous
page in order to do well on the math test. You may use a calculator, but as you will be shown in the follow-
ing lessons, many questions can be solved quickly and easily without a calculator.
Essentially, the ACT Math Test is designed to evaluate a student’s ability to reason through math prob-
lems. Students need to be able to interpret data based on information given and on their existing knowledge
of math. The questions are meant to evaluate critical thinking ability by correctly interpreting the problem,
analyzing the data, reasoning through possible conclusions, and determining the correct answer—the one
supported by the data presented in the question.
Four scores are reported for the ACT Math Test: Pre-Algebra/Elementary Algebra, Intermediate Alge-
bra/Coordinate Geometry, Plane Geometry/Trigonometry, and the total test score.
!
Pretest
As you did with the English section, take the following pretest before you begin the math review in this chap-
ter. The questions are the same type you will find on the ACT. When you are finished, check the answer key
on page 138 to assess your results.Your pretest score will help you determine in which areas you need the most
careful review and practice. For a glossary of math terms, refer to page 201 at the end of this chapter.
1. If a student got 95% of the questions on a 60-question test correct, how many questions did the stu-
dent complete correctly?
a. 57
b. 38
c. 46
d. 53
e. 95
2. What is the smallest possible product for two integers whose sum is 26?
f. 25
g. 15
h. 154
i. 144
j. 26
– ACT MATH TEST PRACTICE–
132
This "Pretest" (with answers and explanations) is not part
of the summer assignment but contains useful examples.
3. What is the value of x in the equation 2x + 1 = 4(x + 3)?
a.
!
1
6
1
!
b. 2
c.
!
1
6
1
!
d. 9
e.
!
3
5
!
4. What is the y-intercept of the line 4y + 2x = 12?
f. 12
g. 2
h. 6
i. 6
j. 3
5. The height of the parallelogram below is 4.5 cm and the area is 36 sq cm. Find the length of side QR in
centimeters.
a. 31.5 cm
b. 8 cm
c. 15.75 cm
d. 9 cm
e. 6 cm
6. Joey gave away half of his baseball card collection and sold one third of what remained. What fraction
of his original collection does he still have?
f.
!
2
3
!
g.
!
1
6
!
h.
!
1
3
!
i.
!
1
5
!
j.
!
2
5
!
PS
Q
4.5
R
– ACT MATH TEST PRACTICE–
133
7. Simplify !40
"
.
a. 2!10
"
b. 4!10
"
c. 10!4
"
d. 5!4
"
e. 2!20
"
8. What is the simplified form of (3x + 5)
2
?
f. 9x
2
+ 30x + 25
g. 9x
2
25
h. 9x
2
+ 25
i. 9x
2
30x 25
j. 39x
2
25
9. Find the measure of RST in the triangle below.
a. 69
b. 46
c. 61
d. 45
e. 23
10. The area of a trapezoid is
!
1
2
!
h(b
1
+ b
2
) where h is the altitude and b
1
and b
2
are the parallel bases. The
two parallel bases of a trapezoid are 3 cm and 5 cm and the area of the trapezoid is 28 sq cm. Find the
altitude of the trapezoid.
f. 14 cm
g. 9 cm
h. 19 cm
i. 1.9 cm
j. 7 cm
SR
T
111°
2x° x°
– ACT MATH TEST PRACTICE–
134
11. If 9m 3 = 318, then 14m = ?
a. 28
b. 504
c. 329
d. 584
e. 490
12. What is the solution of the following equation? |x + 7| 8 = 14
f. {14, 14}
g. {22, 22}
h. {15}
i. {8, 8}
j. {29, 15}
13. Which point lies on the same line as (2, 3) and (6, 1)?
a. (5, 6)
b. (2, 3)
c. (1, 8)
d. (7, 2)
e. (4, 0)
14. In the figure below, M
"
N
"
= 3 inches and P
"
M
"
= 5 inches. Find the area of triangle MNP.
f. 6 square inches
g. 15 square inches
h. 7.5 square inches
i. 12 square inches
j. 10 square inches
N
M
P
3 in
5 in
– ACT MATH TEST PRACTICE–
135
15. A
"
C
"
and BC
!
!
!
are both radii of circle C and have a length of 6 cm. The measure of ACB is 35°. Find the
area of the shaded region.
a.
!
7
2
9
!
b.
!
7
2
!
c. 36
d.
!
6
2
5
!
e. 4
16. If f(x) = 3x + 2 and g(x) = 2x 1, find f(g(x)).
f. x + 1
g. 6x 1
h. 5x + 3
i. 2x
2
4
j. 6x
2
7x 2
17. What is the value of log
4
64?
a. 3
b. 16
c. 2
d. 4
e. 644
B
C
A
6 cm
35°
– ACT MATH TEST PRACTICE–
136
18. The equation of line l is y = mx + b.Which equation is line m?
f. y = mx
g. y = x + b
h. y = 2mx + b
i. y =
!
1
2
!
mx b
j. y = mx + b
19. If Mark can mow the lawn in 40 minutes and Audrey can mow the lawn in 50 minutes, which equa-
tion can be used to determine how long it would take the two of them to mow the lawn together?
a.
!
4
x
0
!
+
!
5
x
0
!
= 1
b.
!
4
x
0
!
+
!
5
x
0
!
= 1
c.
!
1
x
!
+
!
1
x
!
= 90
d. 50x + 40x = 1
e. 90x =
!
1
x
!
20. If sin =
!
2
5
!
,find cos.
f.
!
2
5
1
!
g.
#
!
2
5
1
!
$
h.
!
5
3
!
i.
!
3
5
!
j.
#
!
2
5
1
!
$
l
y = mx + b
m
– ACT MATH TEST PRACTICE–
137
!
Pretest Answers and Explanations
1. Choice a is correct. Multiply 60 by the decimal equivalent of 95% (0.95). 60 × 0.95 = 57.
2. Choice f is correct. Look at the pattern below.
S
um Pro
duct
1 + 25 25
2 + 24 48
3 + 23 69
4 + 22 88
5 + 21 105
The products continue to get larger as the pattern progresses. The smallest possible product is 1 × 25 =
25.
3. Choice c is correct. Distribute the 4, then isolate the variable.
2x + 1 = 4(x + 3)
2x + 1 = 4x + 12
1 = 6x + 12
11 = 6x
!
1
6
1
!
= x
4. Choice j is correct. Change the equation into y = mx + b format.
4y + 2x = 12
4y = 2x + 12
y =
!
1
2
!
x + 3
The y-intercept is 3.
5. Choice b is correct. To find the area of a parallelogram, multiply the base times the height.
A = bh
Substitute in the given height and area:
36 = b(4.5)
8 = b
Then, solve for the base.
The base is 8 cm.
6. Choice h is correct. After Joey sold half of his collection, he still had half left. He sold one third of the
half that he had left (
!
1
3
!
×
!
1
2
!
=
!
1
6
!
), which is
!
1
6
!
of the original collection. In total, he gave away
!
1
2
!
and sold
!
1
6
!
,which is a total of
!
2
3
!
of the collection (
!
1
2
!
+
!
1
6
!
=
!
3
6
!
+
!
1
6
!
=
!
4
6
!
=
!
2
3
!
). Since he has gotten rid of
!
2
3
!
of the col-
lection,
!
1
3
!
remains.
7. Choice a is correct. Break up 40 into a pair of factors, one of which is a perfect square.
40 = 4 × 10.
!40
"
= !4
"
!10
"
= 2!10
"
.
– ACT MATH TEST PRACTICE–
138
8. Choice i is correct.
(3x + 5)
2
= (3x + 5)(3x + 5)
(3x + 5)(3x + 5)
(9x
2
+ 15x + 15x + 25)
(9x
2
+ 30x + 25)
9x
2
30x 25
9. Choice b is correct. Recall that the sum of the angles in a triangle is 180°.
180 = 111 + 2x + x
180 = 111 + 3x
69 = 3x
23 = x
The problem asked for the measure of RST which is 2x.Since x is 23, 2x is 46°.
10. Choice j is correct. Substitute the given values into the equation and solve for h.
A =
!
1
2
!
h(b
1
+ b
2
)
28 =
!
1
2
!
h(3 + 5)
28 =
!
1
2
!
h(8)
28 = 4h
h = 7
The altitude is 7 cm.
11. Choice e is correct. Solve the first equation for m.
9m 3 = 318
9m = 315
m = 35
Then, substitute value of m in 14m.
14(35) = 490
12. Choice j is correct.
|x + 7| 8 = 14
|x +7| = 22
|22| and |22| both equal 22. Therefore, x + 7 can be 22 or 22.
x + 7 = 22 x + 7 = 22
x = 15 x = 29
{29, 15}
– ACT MATH TEST PRACTICE–
139
13. Choice d is correct. Find the equation of the line containing (2, 3) and (6, 1). First, find the slope.
!
x
y
2
2
y
x
1
1
!
=
!
1
6
(
2
3)
!
=
!
4
4
!
= 1
Next, find the equation of the line.
y y
1
= m(x x
1
)
y 1 = 1(x 6)
y 1 = x 6
y = x 5
Substitute the ordered pairs into the equations. The pair that makes the equation true is on the line.
When (7, 2) is substituted into y = x 5, the equation is true.
5 = 7 2 is true.
14. Choice f is correct. Triangle MNP is a 3-4-5 right triangle. The height of the triangle is 4 and the base
is 3. To find the area use the formula A =
!
b
2
h
!
.
A =
!
(3)
2
(4)
!
=
!
1
2
2
!
= 6.
The area of the triangle is 6 square inches.
15. Choice d is correct. Find the total area of the circle using the formula A = r
2
.
A = (6)
2
= 36
A circle has a total of 360°. In the circle shown, 35° are NOT shaded, so 325° ARE shaded.
The fraction of the circle that is shaded is
!
3
3
2
6
5
0
!
. Multiply this fraction by the total area to find the shaded
area.
!
36
1
!
×
!
3
3
2
6
5
0
!
=
!
11
3
,7
6
0
0
0
!
=
!
65
2
!
.
16. Choice g is correct.
f(g(x)) = f(2x 1)
Replace every x in f(x) with (2x 1).
f(g(x)) = 3(2x 1) + 2
f(g(x)) = 6x 3 + 2
f(g(x)) = 6x 1
17. Choice a is correct; log
4
64 means 4
?
= 64; 4
3
= 64. Therefore, log
4
64 = 3.
18. Choice j is correct. The lines have the same y-intercept (b). Their slopes are opposites. So, the slope of
the first line is m, thus, the slope of the second line is m.
Since the y-intercept is b and the slope is m, the equation of the line is y = mx + b.
– ACT MATH TEST PRACTICE–
140
19. Choice b is correct. Use the table below to organize the information.
RATE TIME WORK DONE
Mark
!
4
1
0
!
x
!
4
x
0
!
Audrey
!
5
1
0
!
x
!
5
x
0
!
Mark’s rate is 1 job in 40 minutes. Audrey’s rate is 1 job in 50 minutes. You don’t know how long it will
take them together, so time is x.To find the work done,multiply the rate by the time.
Add the work done by Mark with the work done by Audrey to get 1 job done.
!
4
x
0
!
+
!
5
x
0
!
= 1 is the equation.
20. Choice g is correct. Use the identity sin
2
+ cos
2
= 1 to find cos.
sin
2
+ cos
2
= 1
(
!
2
5
!
)
2
+ cos
2
= 1
!
2
4
5
!
+ cos
2
= 1
cos
2
=
!
2
2
1
5
!
cos =
!
Lessons and Practice Questions
Familiarizing yourself with the ACT before taking the test is a great way to improve your score. If you are
familiar with the directions, format, types of questions, and the way the test is scored, you will be more com-
fortable and less anxious. This section contains ACT math test-taking strategies, information, and practice
questions and answers to apply what you learn.
The lessons in this chapter are intended to refresh your memory. The 80 practice questions following
these lessons contain examples of the topics covered here as well as other various topics you may see on the
official ACT Assessment. If in the course of solving the practice questions you find a topic that you are not
familiar with or have simply forgotten, you may want to consult a textbook for additional instruction.
!
Types of Math Questions
Math questions on the ACT are classified by both topic and skill level.As noted earlier,the six general topics
covered are:
Pre-Algebra
Elementary Algebra
Intermediate Algebra
!21
"
!
5
– ACT MATH TEST PRACTICE–
141
Tips
• The math questions start easy and get harder. Pace yourself accordingly.
• Study wisely. The number of questions involving various algebra topics is significantly higher than
the number of trigonometry questions. Spend more time studying algebra concepts.
• There is no penalty for wrong answers. Make sure that you answer all of the questions, even if
some answers are only a guess.
• If you are not sure of an answer, take your best guess. Try to eliminate a couple of the answer
choices.
• If you skip a question, leave that question blank on the answer sheet and return to it when you
are done. Often, a question later in the test will spark your memory about the answer to a ques-
tion that you skipped.
• Read carefully! Make sure you understand what the question is asking.
• Use your calculator wisely. Many questions are answered more quickly and easily without a cal-
culator.
• Most calculators are allowed on the test. However, there are some exceptions. Check the ACT
website (ACT.org) for specific models that are not allowed.
• Keep your work organized. Number your work on your scratch paper so that you can refer back
to it while checking your answers.
• Look for easy solutions to difficult problems. For example, the answer to a problem that can be
solved using a complicated algebraic procedure may also be found by “plugging” the answer
choices into the problem.
• Know basic formulas such as the formulas for area of triangles, rectangles, and circles. The
Pythagorean theorem and basic trigonometric functions and identities are also useful, and not that
complicated to remember.
142
Coordinate Geometry
Plane Geometry
Tr igo nom et r y
In addition to these six topics, there are three skill levels: basic, application, and analysis. Basic problems
require simple knowledge of a topic and usually only take a few steps to solve. Application problems require
knowledge of a few topics to complete the problem. Analysis problems require the use of several topics to
complete a multi-step problem.
The questions appear in order of difficulty on the test, but topics are mixed together throughout the test.
Pre-Algebra
Topics in this section include many concepts you may have learned in middle or elementary school, such as
operations on whole numbers, fractions, decimals, and integers; positive powers and square roots; absolute
value; factors and multiples; ratio, proportion, and percent; linear equations; simple probability; using charts,
tables, and graphs; and mean, median, mode, and range.
NUMBERS
■
Whole numbers Whole numbers are also known as counting numbers: 0, 1, 2, 3, 4, 5, 6,...
■
Integers Integers are both positive and negative whole numbers including zero: . . . 3, 2, 1, 0, 1,
2, 3 . . .
■
Rational numbers Rational numbers are all numbers that can be written as fractions (
!
2
3
!
), terminating
decimals (.75), and repeating decimals (.666 ...)
■
Irrational numbers Irrational numbers are numbers that cannot be expressed as terminating or
repeating decimals: or !2
"
.
O
RDER OF O
PERATIONS
Most people remember the order of operations by using a mnemonic device such as PEMDAS or Please
Excuse My Dear Aunt Sally.These stand for the order in which operations are done:
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
Multiplication and division are done in the order that they appear from left to right. Addition and sub-
traction work the same way—left to right.
Parentheses also include any grouping symbol such as brackets [ ], braces { }, or the division bar.
Examples
1. 5 + 2 × 8
2. 9 + (6 + 2 × 4) 3
2
Solutions
1. 5 + 2 × 8
5 + 16
11
– ACT MATH TEST PRACTICE–
143
2. 9 + (6 + 2 × 4) 3
2
9 + (6 + 8) 3
2
9 + 14 9
23 9
14
FRACTIONS
Addition of Fractions
To add fractions, they must have a common denominator. The common denominator is a common multi-
ple of the denominators. Usually, the least common multiple is used.
Example
!
1
3
!
+
!
2
7
!
The least common denominator for 3 and 7 is 21.
(
!
1
3
!
×
!
7
7
!
) + (
!
2
7
!
×
!
3
3
!
)Multiply the numerator and denominator ofeach fraction by the same
number so that the denominator of each fraction is 21.
!
2
2
1
!
+
!
2
6
1
!
=
!
2
8
1
!
Add the numerators and keep the denominators the same. Simplify the
answer if necessary.
Subtraction of Fractions
Use the same method for multiplying fractions, except subtract the numerators.
Multiplication of Fractions
Multiply numerators and multiply denominators. Simplify the answer if necessary.
Example
!
3
4
!
×
!
1
5
!
=
!
2
3
0
!
Division of Fractions
Ta ke t h e re ci pro ca l o f ( fli p) t he s eco nd fr a cti on a nd mu lt ip ly.
!
1
3
!
÷
!
3
4
!
=
!
1
3
!
×
!
4
3
!
=
!
4
9
!
– ACT MATH TEST PRACTICE–
144
Examples
1.
!
1
3
!
+
!
2
5
!
2.
!
1
9
0
!
!
3
4
!
3.
!
4
5
!
×
!
7
8
!
4.
!
3
4
!
÷
!
6
7
!
Solutions
1.
!
1
3
×
×
5
5
!
+
!
2
5
×
×
3
3
!
!
1
5
5
!
+
!
1
6
5
!
=
!
1
1
1
5
!
2.
!
1
9
0
×
×
2
2
!
!
3
4
×
×
5
5
!
!
1
2
8
0
!
!
1
2
5
0
!
=
!
2
3
0
!
3.
!
4
5
!
×
!
7
8
!
=
!
2
4
8
0
!
=
!
1
7
0
!
4.
!
3
4
!
×
!
7
6
!
=
!
2
2
1
4
!
=
!
7
8
!
EXPONENTS AND SQUARE ROOTS
An exponent tells you how many times to the base is used as factor. Any base to the power of zero is one.
Example
14
0
= 1
5
3
= 5 × 5 × 5 = 125
3
4
= 3 × 3 × 3 × 3 = 81
11
2
= 11 × 11 = 121
Make sure you know how to work with exponents on the calculator that you bring to the test. Most sci-
entific calculators have a y
x
or x
y
button that is used to quickly calculate powers.
When finding a square root, you are looking for the number that when multiplied by itself gives you
the number under the square root symbol.
!25
"
= 5
!64
"
= 8
!169
"
= 13
– ACT MATH TEST PRACTICE–
145
Have the perfect squares of numbers from 1 to 13 memorized since they frequently come up in all types
of math problems. The perfect squares (in order) are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169.
ABSOLUTE V
ALUE
The absolute value is the distance of a number from zero. For example, |5| is 5 because 5 is 5 spaces from
zero. Most people simply remember that the absolute value of a number is its positive form.
|39| = 39
|92| = 92
|11| = 11
|987| = 987
F
ACTORS AND M
ULTIPLES
Factors are numbers that divide evenly into another number. For example, 3 is a factor of 12 because it divides
evenly into 12 four times.
6 is a factor of 66
9 is a factor of 27
2 is a factor of 98
Multiples are numbers that result from multiplying a given number by another number. For example,
12 is a multiple of 3 because 12 is the result when 3 is multiplied by 4.
66 is a multiple of 6
27 is a multiple of 9
98 is a multiple of 2
RATIO, PROPORTION, AND PERCENT
Ratios are used to compare two numbers and can be written three ways. The ratio 7 to 8 can be written 7:8,
!
7
8
!
,or in the words “7 to 8.”
Proportions are written in the form
!
2
5
!
=
!
2
x
5
!
.Proportions are generally solved by cross-multiplying (mul-
tiply diagonally and set the cross-products equal to each other). For example,
!
2
5
!
=
!
2
x
5
!
(2)(25) = 5x
50 = 5x
10 = x
– ACT MATH TEST PRACTICE–
146
Percents are always “out of 100.” 45% means 45 out of 100. It is important to be able to write percents
as decimals. This is done by moving the decimal point two places to the left.
45% = 0.45
3% = 0.03
124% = 1.24
0.9% = 0.009
PROBABILITY
The probability of an event is P(event) = .
For example, the probability of rolling a 5 when rolling a 6-sided die is
!
1
6
!
,because there is one favor-
able outcome (rolling a 5) and there are 6 possible outcomes (rolling a 1, 2, 3, 4, 5, or 6). If an event is impos-
sible, it cannot happen, the probability is 0. If an event definitely will happen, the probability is 1.
COUNTING PRINCIPLE AND TREE DIAGRAMS
The sample space is a list of all possible outcomes. A tree diagram is a convenient way of showing the sample
space. Below is a tree diagram representing the sample space when a coin is tossed and a die is rolled.
The first column shows that there are two possible outcomes when a coin is tossed, either heads or tails.
The second column shows that once the coin is tossed, there are six possible outcomes when the die is rolled,
numbers 1 through 6. The outcomes listed indicate that the possible outcomes are: getting a heads, then
rolling a 1; getting a heads, then rolling a 2; getting a heads, then rolling a 3; etc. This method allows you to
clearly see all possible outcomes.
Another method to find the number of possible outcomes is to use the counting principle.An example
of this method is on the following page.
Coin
H
1
2
3
4
5
6
Die Outcomes
H1
H2
H3
H4
H5
H6
T
1
2
3
4
5
6
T1
T2
T3
T4
T5
T6
favorable
!
– ACT MATH TEST PRACTICE–
147
Nancy has 4 pairs of shoes, 5 pairs of pants, and 6 shirts. How many different outfits can she
make with these clothes?
Shoes Pants Shirts
4 choices 5 choices 6 choices
To find the number of possible outfits, multiply the number of choices for each item.
4 × 5 × 6 = 120
She can make 120 different outfits.
Helpful Hints about Probability
■
If an event is certain to occur, the probability is 1.
■
If an event is certain NOT to occur, the probability is 0.
■
If you know the probability of all other events occurring, you can find the probability of the remaining
event by adding the known probabilities together and subtracting that sum from 1.
M
EAN, MEDIAN
, MODE, AND
RANGE
Mean is the average. To find the mean, add up all the numbers and divide by the number of items.
Median is the middle. To find the median, place all the numbers in order from least to greatest. Count
to find the middle number in this list. Note that when there is an even number of numbers, there will be two
middle numbers. To find the median, find the average of these two numbers.
Mode is the most frequent or the number that shows up the most. If there is no number that appears
more than once, there is no mode.
The range is the difference between the highest and lowest number.
Example
Using the data 4, 6, 7, 7, 8, 9, 13, find the mean, median, mode, and range.
Mean: The sum of the numbers is 54. Since there are seven numbers, divide by 7 to find the
mean. 54 ÷ 7 = 7.71.
Median: The data is already in order from least to greatest, so simply find the middle num-
ber. 7 is the middle number.
Mode: 7 appears the most often and is the mode.
Range: 13 4 = 9.
– ACT MATH TEST PRACTICE–
148
LINEAR EQUATIONS
An equation is solved by finding a number that is equal to an unknown variable.
Simple Rules for Working with Equations
1. The equal sign separates an equation into two sides.
2. Whenever an operation is performed on one side, the same operation must be performed on the other
side.
3. Yo u r fi r s t g o a l i s t o g e t a l l o f t h e v a r i a b l e s o n o n e s i d e a n d a l l o f t h e n u m b e r s o n t h e o t h e r.
4. The final step often will be to divide each side by the coefficient, leaving the variable equal to a
number.
CROSS-M
ULTIPLYING
Yo u c a n s o l v e a n e q u a t i o n t h a t s e t s o n e f r a c t i o n e q u a l t o a n o t h e r b y cross-multiplication.Cross-
multiplication involves setting the products of opposite pairs of terms equal.
Example
!
6
x
!
=
!
x +
12
10
!
becomes 12x = 6(x) + 6(10)
12x = 6x + 60
6x 6x
!
6
6
x
!
=
!
6
6
0
!
Thus, x = 10
Checking Equations
To check an equation, substitute the number equal to the variable in the original equation.
Example
To check the equation from the previous page, substitute the number 10 for the variable x.
!
6
x
!
=
!
x +
12
10
!
!
1
6
0
!
=
!
10
1
+
2
10
!
!
1
6
0
!
=
!
2
1
0
2
!
Simplify the fraction on the right by dividing the numerator and denominator by 2.
!
1
6
0
!
=
!
1
6
0
!
Because this statement is true, you know the answer x = 10 is correct.
– ACT MATH TEST PRACTICE–
149
Special Tips for Checking Equations
1. If time permits, be sure to check all equations.
2. Be careful to answer the question that is being asked. Sometimes, this involves solving for a variable
and than performing an operation.
Example: If the question asks for the value of x 2, and you find x = 2, the answer is not 2, but
2 2. Thus, the answer is 0.
CHARTS, TABLES
, AND GRAPHS
The ACT Math Test will assess your ability to analyze graphs and tables. It is important to read each graph
or table very carefully before reading the question. This will help you to process the information that is pre-
sented. It is extremely important to read all of the information presented, paying special attention to head-
ings and units of measure. Here is an overview of the types of graphs you will encounter:
■
CIRCLE GRAPHS or PIE CHARTS
This type of graph is representative of a whole and is usually divided into percentages. Each section of the
chart represents a portion of the whole, and all of these sections added together will equal 100% of the
whole.
■
BAR GRAPHS
Bar graphs compare similar things with bars of different length, representing different values. These graphs
may contain differently shaded bars used to represent different elements. Therefore, it is important to pay
attention to both the size and shading of the graph.
Fruit Ordered by Grocer
100
80
60
40
20
0
Pounds Ordered
Week 1 Week 2 Week 3
Key
Apples
Peaches
Bananas
A
ttendance at a Baseball Game
15%
girls
24%
boys
61%
adults
– ACT MATH TEST PRACTICE–
150
■
BROKEN LINE GRAPHS
Broken-line graphs illustrate a measurable change over time. If a line is slanted up, it represents an
increase, whereas a line sloping down represents a decrease. A flat line indicates no change.
In the line graph below, Lisa’s progress riding her bike is graphed. From 0 to 2 hours, Lisa moves
steadily. Between 2 and 2
!
1
2
!
hours, Lisa stops (flat line). After her break, she continues again but at a slower
pace (line is not as steep as from 0 to 2 hours).
Elementary Algebra
Elementary algebra covers many topics typically covered in an Algebra I course. Topics include operations on
polynomials; solving quadratic equations by factoring; linear inequalities; properties of exponents and square
roots; using variables to express relationships; and substitution.
O
PERATIONS ON POLYNOMIALS
Combining Like Terms: terms with the same variable and exponent can be combined by adding the coefficients
and keeping the variable portion the same.
For example,
4x
2
+ 2x 5 + 3x
2
9x + 10 =
7x
2
7x + 5
Distributive Property: multiply all the terms inside the parentheses by the term outside the parentheses.
7(2x 1) = 14x 7
SOLVING QUADRATIC EQUATIONS BY FACTORING
Before factoring a quadratic equation to solve for the variable, you must set the equation equal to zero.
x
2
7x = 30
x
2
7x 30 = 0
Lisa’s Progress
50
40
30
20
10
0
Distance Travelled in Miles
Time in Hours
123
4
– ACT MATH TEST PRACTICE–
151
Next, factor.
(x + 3)(x 10) = 0
Set each factor equal to zero and solve.
x + 3 = 0 x 10 = 0
x = 3 x = 10
The solution set for the equation is {3, 10}.
SOLVING
INEQUALITIES
Solving inequalities is the same as solving regular equations, with one exception. The exception is that when
multiplying or dividing by a negative, you must change the inequality symbol.
For example,
3x < 9
!
3
3
x
!
<
!
9
3
!
x > 3
Notice that the inequality switched from less than to greater than after division by a negative.
When graphing inequalities on a number line, recall that < and > use open dots and and use solid
dots.
x < 2 x 2
P
ROPERTIES OF EXPONENTS
When multiplying, add exponents.
x
3
· x
5
= x
3+5
= x
8
When dividing, subtract exponents.
!
x
x
7
2
!
= x
72
= x
5
When calculating a power to a power, multiply.
(x
6
)
3
= x
6·3
= x
18
10234
10234
– ACT MATH TEST PRACTICE–
152
Any number (or variable) to the zero power is 1.
5
0
= 1 m
0
= 1 9,837,475
0
= 1
Any number (or variable) to the first power is itself.
5
1
= 5 m
1
= m 9,837,475
1
= 9,837,475
ROOTS
Recall that exponents can be used to write roots. For example, !x
"
= x
!
1
2
!
and !
3
x
"
= x
!
1
3
!
.The denominator is
the root. The numerator indicates the power. For example, (!
3
x
"
)
4
= x
!
4
3
!
and !x
5
"
= x
!
5
2
!
.The properties ofexpo-
nents outlined above apply to fractional exponents as well.
USING VARIABLES TO EXPRESS RELATIONSHIPS
The most important skill needed for word problems is being able to use variables to express relationships.
The following will assist you in this by giving you some common examples of English phrases and their math-
ematical equivalents.
■
“Increase” means add.
Example
A number increased by five = x + 5.
■
“Less than” means subtract.
Example
10 less than a number = x 10.
■
“Times” or “product” means multiply.
Example
Three times a number = 3x.
■
“Times the sum” means to multiply a number by a quantity.
Example
Five times the sum of a number and three = 5(x + 3).
■
Two variables are sometimes used together.
Example
A number y exceeds five times a number x by ten.
y = 5x + 10
■
Inequality signs are used for “at least” and “at most,” as well as “less than” and “more than.”
Examples
The product of x and 6 is greater than 2.
x × 6 > 2
– ACT MATH TEST PRACTICE–
153
When 14 is added to a number x, the sum is less than 21.
x + 14 < 21
The sum of a number x and four is at least nine.
x + 4 9
When seven is subtracted from a number x, the difference is at most four.
x 7 4
ASSIGNING V
ARIABLES IN WORD P
ROBLEMS
It may be necessary to create and assign variables in a word problem. To do this, first identify an unknown
and a known. You may not actually know the exact value of the “known,” but you will know at least some-
thing about its value.
Examples
Max is three years older than Ricky.
Unknown = Ricky’s age = x
Known = Max’s age is three years older
Therefore,
Ricky’s age = x and Max’s age = x + 3
Siobhan made twice as many cookies as Rebecca.
Unknown = number of cookies Rebecca made = x
Known = number of cookies Siobhan made = 2x
Cordelia has five more than three times the number of books that Becky has.
Unknown = the number of books Becky has = x
Known = the number of books Cordelia has = 3x + 5
SUBSTITUTION
When asked to substitute a value for a variable, replace the variable with the value.
Example
Find the value of x
2
+ 4x 1, for x = 3.
Replace each x in the expression with the number 3. Then, simplify.
= (3)
2
+ 4(3) 1
= 9 + 12 1
= 20
The answer is 20.
– ACT MATH TEST PRACTICE–
154
Intermediate Algebra
Intermediate algebra covers many topics typically covered in an Algebra II course such as the quadratic for-
mula; inequalities; absolute value equations; systems of equations; matrices; functions; quadratic inequali-
ties; radical and rational expressions; complex numbers; and sequences.
THE
QUADRATIC FORMULA
x =
!
b ±
!
2
b
a
2
4a
"
c
!
for quadratic equations in the form ax
2
+ bx + c = 0.
The quadratic formula can be used to solve any quadratic equation. It is most useful for equations that can-
not be solved by factoring.
ABSOLUTE VALUE EQUATIONS
Recall that both |5| = 5 and |5| = 5. This concept must be used when solving equations where the variable
is in the absolute value symbol.
|x + 4| = 9
x + 4 = 9
or
x + 4 = 9
x = 5 x = 13
S
YSTEMS OF EQUATIONS
When solving a system of two linear equations with two variables, you are looking for the point on the coor-
dinate plane at which the graphs of the two equations intersect. The elimination or addition method is usu-
ally the easiest way to find this point.
Solve the following system of equations:
y = x + 2
2x + y = 17
First, arrange the two equations so that they are both in the form Ax + By = C.
x + y = 2
2x + y = 17
Next, multiply one of the equations so that the coefficient of one variable (we will use y) is
the opposite of the coefficient of the same variable in the other equation.
1(x + y =2)
2x + y = 17
x y = 2
2x + y = 17
– ACT MATH TEST PRACTICE–
155
Add the equations. One of the variables should cancel out.
3x = 15
Solve for the first variable.
x = 5
Find the value of the other variable by substituting this value into either original equation to
find the other variable.
y = 5 + 2
y = 7
Since the answer is a point on the coordinate plane, write the answer as an ordered pair.
(5, 7)
C
OMPLEX NUMBERS
Any number in the form a + bi is a complex number. i = !1
"
.Operations with i are the same as with any
variable, but you must remember the following rules involving exponents.
i = i
i
2
= 1
i
3
= i
i
4
= 1
This pattern repeats every fourth exponent.
RATIONAL EXPRESSIONS
Algebraic fractions (rational expressions) are very similar to fractions in arithmetic.
Example
Write
!
5
x
!
!
1
x
0
!
as a single fraction.
Solution
Just like in arithmetic, you need to find the lowest common denominator (LCD) of 5 and 10,
which is 10. Then change each fraction into an equivalent fraction that has 10 as a denomi-
nator.
!
5
x
!
!
1
x
0
!
=
!
5
x(
(
2
2
)
)
!
!
1
x
0
!
=
!
1
2
0
x
!
!
1
x
0
!
=
!
1
x
0
!
– ACT MATH TEST PRACTICE–
156
RADICAL EXPRESSIONS
■
Radicals with the same radicand (number under the radical symbol) can be combined the same way
“like terms” are combined.
Example
2!3
"
+ 5!3
"
= 7!3
"
Think of this as similar to:
2x + 5x = 7x
■
To multiply radical expressions with the same root, multiply the radicands and simplify.
Example
!3
"
· !6
"
= !18
"
This can be simplified by breaking 18 into 9 × 2.
!18
"
= !9
"
· !2
"
= 3!2
"
■
Radicals can also be written in exponential form.
Example
!
3
x
5
"
= x
!
5
3
!
In the fractional exponent, the numerator (top) is the power and the denominator (bottom) is the root.
By representing radical expressions using exponents, you are able to use the rules of exponents to sim-
plify the expression.
INEQUALITIES
The basic solution of linear inequalities was covered in the Elementary Algebra section. Following are some
more advanced types of inequalities.
Solving Combined (or Compound) Inequalities
To solve an inequality that has the form c < ax + b < d, isolate the letter by performing the same operation
on each member of the equation.
Example
If 10 < 5y 5 < 15, find y.
Add five to each member of the inequality.
10 + 5 < 5y 5 + 5 < 15 + 5
5 < 5y < 20
– ACT MATH TEST PRACTICE–
157
Divide each term by 5, changing the direction of both inequality symbols:
!
5
5
!
<
!
5
5
y
!
<
!
20
5
!
= 1 > y > 4
The solution consists of all real numbers less than 1 and greater than 4.
Absolute Value Inequalities
|x| < a is equivalent to a < x < a and |x| > a is equivalent to x > a or x < a
Example
|x + 3| > 7
x + 3 > 7
or
x + 3 < 7
x > 4 x < 10
Thus, x > 4 or x < 10.
Quadratic Inequalities
Recall that quadratic equations are equations of the form ax
2
+ bx + c = 0.
To solve a quadratic inequality, first treat it like a quadratic equation and solve by setting the equation
equal to zero and factoring. Next, plot these two points on a number line. This divides the number line into
three regions. Choose a test number in each of the three regions and determine the sign of the equation when
it is the value of x.Determine which ofthe three regions makes the inequality true.This region is the answer.
Example
x
2
+ x < 6
Set the inequality equal to zero.
x
2
+ x 6 < 0
Factor the left side.
(x + 3)(x 2) < 0
Set each of the factors equal to zero and solve.
x + 3 = 0 x 2 = 0
x = 3 x = 2
Plot the numbers on a number line. This divides the number line into three regions.
The number line is divided into the following regions.
numbers less than 3
numbers between 3 and 2
numbers greater than 2
−4 −3 −2 −10
numbers
less than
−3
numbers
between
−3 and 2
numbers
greater
than 2
123
4
%
%
%
– ACT MATH TEST PRACTICE–
158
Use a test number in each region to see if (x + 3)(x 2) is positive or negative in that region.
numbers less than
3numbers between
3 and 2 numbers greater than 2
test # = 5 test # = 0 test # = 3
(5 + 3)(5 2) = 14 (0 + 3 )(0 2) = 6(3 + 3)(3 2) = 6
positive negative positive
The original inequality was (x + 3)(x 2) < 0. If a number is less than zero, it is negative. The
only region that is negative is between 3 and 2; 3 < x < 2 is the solution.
FUNCTIONS
Functions are often written in the form f(x) = 5x 1. You might be asked to find f(3), in which case you sub-
stitute 3 in for x. f(3) = 5(3) 1. Therefore, f(3) = 14.
MATRICES
Basics of 2
×
2 Matrices
Addition:
[]
+
[]
=
[]
Subtraction: Same as addition, except subtract the numbers rather than adding.
Scalar Multiplication: k
[]
=
[]
Multiplication of Matrices:
[][ ]
=
[]
Coordinate Geometry
This section contains problems dealing with the (x, y) coordinate plane and number lines. Included are slope,
distance, midpoint, and conics.
SLOPE
The formula for finding slope, given two points, (x
1
, y
1
) and (x
2
, y
2
) is
!
x
y
2
2
y
x
1
1
!
.
The equation of a line is often written in slope-intercept form which is y = mx + b, where m is the slope
and b is the y-intercept.
Important Information about Slope
■
A line that rises to the right has a positive slope and a line that falls to the right has a negative slope.
■
A horizontal line has a slope of 0 and a vertical line does not have a slope at all—it is undefined.
■
Parallel lines have equal slopes.
■
Perpendicular lines have slopes that are negative reciprocals.
a
11
b
11
+ a
12
b
21
a
11
b
12
+ a
12
b
22
a
21
b
11
+ a
22
b
21
a
21
b
12
+ a
22
b
22
b
11
b
12
b
21
b
22
a
11
a
12
a
21
a
22
ka
11
ka
12
ka
21
ka
22
a
11
a
12
a
21
a
22
a
11
+ b
11
a
12
+ b
12
a
21
+ b
21
a
22
+ b
22
b
11
b
12
b
21
b
22
a
11
a
12
a
21
a
22
– ACT MATH TEST PRACTICE–
159
DISTANCE
The distance between two points can be found using the following formula:
d = !(x
2
x
"
1
)
2
+ (y
"
2
y
1
)
2
"
MIDPOINT
The midpoint of two points can be found by taking the average of the x values and the average of the y values.
midpoint = (
!
x
1
+
2
x
2
!
,
!
y
1
+
2
y
2
!
)
CONICS
Circles, ellipses, parabolas, and hyperbolas are conic sections. The following are the equations for each conic
section.
Circle: (x h)
2
+ (y k)
2
= r
2
where (h, k) is the center and r is the radius.
Ellipse:
!
(x
a
2
h)
2
!
+
!
(y
b
2
k)
2
!
= 1 where (h, k) is the center. If the larger denominator
is under y, the y-axis is the major axis. If the larger
denominator is under the x-axis, the x-axis is the
major axis.
Parabola y k = a(x h)
2
or x h = a(y k)
2
The vertex is (h, k). Parabolas of the first form open
up or down. Parabolas of the second form open left
or right.
Hyperbola
!
a
x
2
2
!
!
b
y
2
2
!
= 1 or
!
a
y
2
2
!
!
b
x
2
2
!
= 1
Plane Geometry
Plane geometry covers relationships and properties of plane figures such as triangles, rectangles, circles, trape-
zoids, and parallelograms. Angle relations, line relations, proof techniques, volume and surface area, and
translations, rotations, and reflections are all covered in this section.
To begin this section, it is helpful to become familiar with the vocabulary used in geometry. The list
below defines some of the main geometrical terms:
Arc part of a circumference
Area the space inside a 2 dimensional figure
Bisect to cut in 2 equal parts
Circumference the distance around a circle
Chord a line segment that goes through a circle, with its endpoint on the circle
Diameter a chord that goes directly through the center of a circle—the longest line
you can draw in a circle
Equidistant exactly in the middle
– ACT MATH TEST PRACTICE–
160
– ACT MATH TEST PRACTICE–
161
Hypotenuse the longest leg of a right triangle, always opposite the right angle
Parallel lines in the same plane that will never intersect
Perimeter the distance around a figure
Perpendicular 2 lines that intersect to form 90-degree angles
Quadrilateral any four-sided figure
Radius a line from the center of a circle to a point on the circle (half of the
diameter)
Volume the space inside a 3-dimensional figure
BASIC FORMULAS
Perimeter the sum of all the sides of a figure
Area of a rectangle A = bh
Area of a triangle A =
!
b
2
h
!
Area of a parallelogram A = bh
Area of a circle A = r
2
Volum e o f a recta n g u l a r soli d V = lwh
B
ASIC GEOMETRIC
FACTS
The sum of the angles in a triangle is 180°.
A circle has a total of 360°.
P
YTHAGOREAN THEOREM
The Pythagorean theorem is an important tool for working with right triangles.
It states: a
2
+ b
2
= c
2
,where a and b represent the legs and c represents the hypotenuse.
This theorem allows you to find the length of any side as along as you know the measure of the other
two. So, if leg a = 1 and leg b = 2 in the triangle below, you can find the measure of leg c.
a
2
+ b
2
= c
2
1
2
+ 2
2
= c
2
1 + 4 = c
2
5 = c
2
!5
"
= c
a
b
c
PYTHAGOREAN TRIPLES
In a Pythagorean triple,the square ofthe largest number equals the sum ofthe squares ofthe other two
numbers.
Example
As demonstrated: 1
2
+ 2
2
= (!5
"
)
2
1, 2, and !5
"
are also a Pythagorean triple because:
1
2
+ 2
2
= 1 + 4 = 5 and (!5
"
)
2
= 5.
Pythagorean triples are useful for helping you identify right triangles. Some common Pythagorean
triples are:
3:4:5 8:15:17 5:12:13
M
ULTIPLES OF P
YTHAGOREAN TRIPLES
Any multiple of a Pythagorean triple is also a Pythagorean triple. Therefore, if given 3:4:5, then 9:12:15 is also
a Pythagorean triple.
Example
If given a right triangle with sides measuring 6, x, and 10, what is the value of x?
Solution
Because it is a right triangle, use the Pythagorean theorem. Therefore,
10
2
6
2
= x
2
100 36 = x
2
64 = x
2
8 = x
45-45-90 RIGHT TRIANGLES
A right triangle with two angles each measuring 45 degrees is called an isosceles right triangle.In an isosce-
les right triangle:
■
The length of the hypotenuse is !2
"
multiplied by the length of one of the legs of the triangle.
■
The length of each leg is
!
!
2
2
"
!
multiplied by the length of the hypotenuse.
45°
45°
– ACT MATH TEST PRACTICE–
162
x = y =
!
!
2
2
"
!
×
!
1
1
0
!
= 10
!
!
2
2
"
!
= 5!2
"
30-60-90 TRIANGLES
In a right triangle with the other angles measuring 30 and 60 degrees:
■
The leg opposite the 30-degree angle is half of the length of the hypotenuse. (And, therefore, the
hypotenuse is two times the length of the leg opposite the 30-degree angle.)
■
The leg opposite the 60-degree angle is !3
"
times the length of the other leg.
Example
x = 2 · 7 = 14 and y = 7!3
"
CONGRUENT
Two fig ure s are con g ru en t if t he y h av e th e sa me s ize a nd s ha pe.
TRANSLATIONS, ROTATIONS, AND REFLECTIONS
Congruent figures can be made to coincide (place one right on top of the other), by using one of the following
basic movements.
TRANSLATION (SLIDE) ROTATION REFLECTION (FLIP)
60°
30°
y
7
x
60°
30°
45
°
10
45
°
x
y
– ACT MATH TEST PRACTICE–
163
Trigonometry
Basic trigonometric ratios, graphs, identities, and equations are covered in this section.
BASIC TRIGONOMETRIC RATIOS
sin A =
!
hy
o
p
p
o
p
t
o
e
s
n
it
u
e
se
!
opposite refers to the length of the leg opposite angle A.
cos A =
!
hy
a
p
d
o
ja
te
c
n
en
u
t
se
!
adjacent refers to the length of the leg adjacent to angle A.
tan A =
!
a
o
d
p
j
p
a
o
c
s
e
i
n
te
t
!
TRIGONOMETRIC
IDENTITIES
sin
2
x + cos
2
x = 1
tan x =
!
c
si
o
n
s
x
x
!
sin 2x = 2sin x cos x
cos 2x = cos
2
x sin
2
x
tan 2x =
!
1
2
ta
ta
n
n
x
2
x
!
!
Practice Questions
Directions
After reading each question, solve each problem, and then choose the best answer from the choices given.
(Remember, the ACT Math Test is different from all the other tests in that each math question contains five
answer choices.) When you are taking the official ACT, make sure you carefully fill in the appropriate bub-
ble on the answer document.
Yo u m a y u s e a c a l c u l a t o r f o r a n y p r o b l e m , b u t m a n y p r o b l e m s a r e d o n e m o r e q u i c k l y a n d e a s i l y w i t h -
out one.
Unless directions tell you otherwise, assume the following:
■
figures may not be drawn to scale
■
geometric figures lie in a plane
■
“line” refers to a straight line
■
“average” refers to the arithmetic mean
Remember, the questions get harder as the test goes on. You may want to consider this fact as you pace
yourself.
– ACT MATH TEST PRACTICE–
164
1. How is five hundred twelve and sixteen thousandths written in decimal form?
a. 512.016
b. 512.16
c. 512,160
d. 51.216
e. 512.0016
2. 4
!
1
3
!
1
!
3
4
!
= ?
f. 2
!
1
7
2
!
g. 3
!
1
5
2
!
h. 3
!
2
3
!
i. 2
!
1
5
2
!
j. 1
!
1
8
!
3. Simplify |3 11| + 4 × 2
3
.
a. 24
b. 40
c. 96
d. 520
e. 32
4. The ratio of boys to girls in a kindergarten class is 4 to 5. If there are 18 students in the class, how
many are boys?
f. 9
g. 8
h. 10
i. 7
j. 12
5. What is the median of 0.024, 0.008, 0.1, 0.024, 0.095, and 0.3?
a. 0.119
b. 0.095
c. 0.0595
d. 0.024
e. 0.092
– ACT MATH TEST PRACTICE–
165
IB Math HL “A&A” Year 1 Summer Assignment: # 1 - 80
On our first full day of class in
August, you will complete a scantron
form with your answers to the
Practice Questions # 1 - 80 on pages
"165 - 183". This will be your first
test grade in the class (as well as
excellent practice for the ACT!).
Pages "131 - 164" of this PDF (which
is posted on the WHS website)
contains examples and brief lessons
from Algebra II.
6. Which of the following is NOT the graph of a function?
f.
g.
h.
i.
j.
– ACT MATH TEST PRACTICE–
166
7. 4.6 × 10
5
= ?
a. 4.60000
b. 0.000046
c. 4,600,000
d. 460,000
e. 0.0000046
8. What is the value of x
5
for x = 3?
f. 15
g. 243
h. 125
i.
!
5
3
!
j. 1.6
9. What is the next number in the pattern below?
0, 3, 8, 15, 24, . . .
a. 35
b. 33
c. 36
d. 41
e. 37
10. What is the prime factorization of 84?
f. 42 × 2
g. 7 × 2 × 3
h. 2
2
× 3 × 7
i. 2 × 6 × 7
j. 2
3
× 7
11. Find the slope of the line 7x = 3y 9.
a. 3
b. 9
c.
!
7
3
!
d. 3
e.
!
3
7
!
– ACT MATH TEST PRACTICE–
167
12. The perimeter of a rectangle is 20 cm. If the width is 4 cm, find the length of the rectangle.
f. 6 cm
g. 16 cm
h. 5 cm
i. 12 cm
j. 24 cm
13. Find the area of the figure below.
a. 79 square inches
b. 91 square inches
c. 70 square inches
d. 64 square inches
e. 58 square inches
14. Five cans of tomatoes cost $6.50. At this rate, how much will 9 cans of tomatoes cost?
f. $13.00
g. $11.70
h. $1.30
i. $11.90
j. $12.40
15. For all x 0,
!
3
2
x
!
+
!
1
5
!
= ?
a.
!
1
2
5x
!
b.
!
1
1
0
5
+
+
3
x
x
!
c.
!
10
1
+
5x
3x
!
d.
!
15
2
+ x
!
e.
!
5
1
x
!
10 in
7 in
7 in
3 in
– ACT MATH TEST PRACTICE–
168
16. Which inequality best represents the graph below?
f. 1.5 > x > 1
g. x 0
h. 0.5 > x > 0
i. 1.5 < x < 0
j. 1.5 x 0
17. Simplify (6x
4
y
3
)
2
.
a. 36x
6
y
5
b. 36x
2
y
c. 36x
8
y
6
d. 36x
8
y
4
e. 36xy
18. If 2x + 3y = 55 and 4x = y + 47, find x y.
f. 28
g. 16
h. 5
i. 12
j. 24
19. Which inequality represents the graph below?
a. 4x < 0
b. 20x > 5
c. x < 4
d. x 4
e. x < 4
20. Simplify !
3
16x
5
y
4
"
.
f. 2xy!
3
2x
2
y
"
g. 8x
2
y
h. 8xy!
3
2
"
i. 2xy!
3
xy
"
j. 4x
2
y
2
!
3
x
"
−40 4
−2−3 −10 1
– ACT MATH TEST PRACTICE–
169
21. The formula to convert Celsius to Fahrenheit is F =
!
5
9
!
C + 32, where F is degrees Fahrenheit, and C is
degrees Celsius. What Fahrenheit temperature is equivalent to 63° Celsius?
a. 32°
b. 95°
c. 67°
d. 83°
e. 47°
22. What are the solutions to the equation x
2
+ 8x + 15 = 0?
f. {8, 15}
g. {0}
h. {5, 3}
i. no solution
j. {2, 4}
23. If 5k = 9m 18, then m = ?
a. 5k + 18
b.
!
5
9
!
k + 2
c. 9 + 5k
d. 5k + 9
e. 9k + 18
24. What is the solution set for 5x 7 = 5(x + 2)?
f. {2}
g. {7}
h. no solution
i. all real numbers
j. all positive numbers
25. Simplify
!
4x
2
+
x +
11
3
x 3
!
for all x 3.
a. 3x
2
+ 11
b. 2x + 1
c. 4x
2
+ 12x
d. 4x
2
+ 10x 6
e. 4x 1
– ACT MATH TEST PRACTICE–
170
26. If x =
[]
and y =
[]
,find x y.
f.
[]
g.
[]
h.
[]
i.
[]
j.
[]
27. If log
3
x = 2, then x = ?
a. 6
b. 9
c.
!
2
3
!
d. 4
e.
!
1
2
!
28. Simplify
!
x
x
2
3
9
!
.
f. x 12
g. x 6
h. x + 3
i. x
2
6
j. x 3
29. The vertices of a triangle are A(1, 3), B(3, 0), and C (2, 1). Find the length of side A
"
C
"
.
a. !15
"
b. !17
"
c. 19
d. 17
e. 3!6
"
61
25
41
28
50
6 6
18
46
50
66
24
10
34
56
– ACT MATH TEST PRACTICE–
171
30. Which of the following equations has a graph that has a y-intercept of 4 and is parallel to 3y 9x = 24?
f. 12x + 4y = 16
g. 9x 3y = 15
h. 2y = 4x + 8
i. 7y = 14x + 7
j. 3x 9y = 14
31. At what point do the lines x = 9 and 3x + y = 4 intersect?
a. (3, 9)
b. (
!
5
3
!
,9)
c. (20, 9)
d. (9, 23)
e. (9, 4)
32. Which of the numbers below is the best approximation of (!37
"
)(!125
"
)?
f. 52
g. 4,600
h. 150
i. 66
j. 138
33. What is the solution set of the equation x
2
4x 4 = 2x + 23?
a. {4, 4}
b. {4, 23}
c. {1, 11.5}
d. {3, 9}
e. {5, 6}
34. If a fair coin is flipped and a die is rolled, what is the probability of getting tails and a 3?
f.
!
1
2
!
g.
!
1
1
2
!
h.
!
1
6
!
i.
!
1
4
!
j.
!
1
8
!
– ACT MATH TEST PRACTICE–
172
35. What is
!
1
2
!
% of 90?
a. 45
b. 0.045
c. 4.5
d. 0.45
e. 450
36. Between which two integers does !41
"
lie?
f. 5 and 6
g. 8 and 9
h. 4 and 5
i. 7 and 8
j. 6 and 7
37. Mike has 12 bags of shredded cheese to use to make pizzas. If he uses
!
3
4
!
of a bag of cheese for each
pizza, how many pizzas can he make?
a. 12
b. 24
c. 36
d. 9
e. 16
38. Greene ran the 100-meter dash in 9.79 seconds. What was his speed in kilometers per hour (round to
the nearest kilometer)?
f. 31 km/h
g. 37 km/h
h. 1 km/h
i. 10 km/h
j. 25 km/h
39. Larry has 4 blue socks, 6 red socks, and 10 purple socks in his drawer. Without looking, Larry ran-
domly pulled out a red sock from the drawer. If Larry does not put the red sock back in the drawer,
what is the probability that the next sock he randomly draws will be red?
a.
!
1
4
!
b.
!
1
3
0
!
c.
!
1
5
9
!
d.
!
3
7
!
e.
!
1
6
!
– ACT MATH TEST PRACTICE–
173
40. What is the product of 5 × 10
4
and 6 × 10
8
?
f. 11 × 10
4
g. 3 × 10
4
h. 1.1 × 10
5
i. 3 × 10
5
j. 5.6 × 10
4
41. What is the sine of angle B in the triangle below?
a.
!
3
4
!
b.
!
3
5
!
c.
!
4
3
!
d.
!
4
5
!
e.
!
5
4
!
42. Find tan x for the right triangle below.
f.
!
5
4
!
g.
!
3
4
!
h.
!
4
3
!
i.
!
6
3
!
j.
!
5
3
!
4
5
3
x°
A
B
8
6
C
– ACT MATH TEST PRACTICE–
174
43. The surface area of a box is found by taking the sum of the areas of each of the faces of the box. Find
the surface area of a box with dimensions 6 inches by 8 inches by 10 inches.
a. 480 sq in
b. 138 sq in
c. 346 sq in
d. 376 sq in
e. 938 sq in
44. Find the area of the shaded region. Recall that the area of a circle is r
2
,where r is the radius of the
circle.
f. 65
g. 6
h. 25
i. 5
j. 33
45. The area of square WXYZ is 100 square centimeters. Find the length of diagonal WY in centimeters.
a. 10!2
"
cm
b. 20 cm
c. 10 cm
d. 2!5
"
cm
e. 10!5
"
cm
46. Find the hypotenuse of the triangle below.
f. !13
"
g. !5
"
h. !65
"
i. !97
"
j. 13
4
9
3
4
3
– ACT MATH TEST PRACTICE–
175
47. A circular lid to a jar has a radius of 3
!
1
2
!
inches. Find the area of the lid.
a.
!
1
4
2
9
!
sq in
b.
!
4
1
9
2
!
sq in
c.
!
4
4
9
!
sq in
d.
!
7
2
!
sq in
e.
!
4
4
9
!
sq in
48. What is the value of x when y is equal to 15 for the equation y = 4x
2
1?
f. 2
g. 16
h. 64
i. !5
"
j. 0
49. The senior class at Roosevelt High has 540 students. Kristen won the election for class president with
60% of the vote. Of that 60%, 75% were female. Assuming that the entire senior class voted, how many
females voted for Kristen?
a. 195
b. 405
c. 324
d. 227
e. 243
50. If cos =
!
1
6
7
!
and tan =
!
5
6
!
,then sin = ?
f.
!
1
5
7
!
g.
!
6
5
!
h.
!
1
5
7
!
i.
!
5
6
!
j.
!
1
2
!
51. The formula for the volume of a rectangular solid is V = lwh.Ifeach dimension is tripled,how many
times the original volume will the new volume be?
a. 3
b. 9
c.
!
1
3
!
d. 27
e. 81
– ACT MATH TEST PRACTICE–
176
52. In a right triangle, the two non-right angles measure 7x and 8x.What is the measure ofthe smaller
angle?
f. 15°
g. 60°
h. 30°
i. 48°
j. 42°
53. What is the length of the missing leg in the right triangle below?
a. !181
"
b. 1
c. !19
"
d. 4
e. !21
"
54. The length of a rectangle is twice the width. If the perimeter of the rectangle is 72 feet, what is the
length of the rectangle?
f. 12 feet
g. 6 feet
h. 36 feet
i. 48 feet
j. 24 feet
55. The area of a triangle is 80 square inches. Find the height if the base is 5 inches more than the height.
a.
!
1+
!
2
629
"
!
b.
!
9±
2
!5
"
!
c. 4 ± !85
"
d. 5 !665
"
e.
!
5+
2
!
665
"
!
10
x
9
– ACT MATH TEST PRACTICE–
177
56. Three of the vertices of a square are (2, 3), (5, 3), and (2, 4). What is the length of a side of the
square?
f. 5
g. 4
h. 3
i. 7
j. 8
57. Which of the following lines is perpendicular to y = 3x + 1?
a. 6x + 5 = 2y
b. 4 + y = 3x
c. 9y = 3 + 2x
d. 2x + y = 4
e. 3y + x = 5
58. Which statement best describes the lines 2x + 3y = 12 and 60 + 15y = 10x?
f. the same line
g. parallel
h. skew
i. perpendicular
j. intersect at one point
59. What is the midpoint of X
"
Y
"
if X(4, 2) and Y(3, 8)?
a. (7, 6)
b. (0.5, 3)
c. (1, 6)
d. (7, 10)
e. (2, 1.5)
60.
!
3
4
x
!
+
!
x
5
1
!
= ?
f.
!
x
1
+
5x
3
!
g.
!
x
8
+
x
3
!
h.
!
3
x
x
+
+
3
5
!
i.
!
3x
2
1
3
5
x
x
+ 20
!
j.
!
x
2
+
1
4
5
x
x
1
!
– ACT MATH TEST PRACTICE–
178
61. Simplify (
!
2
1
x
2
!
)
3
.
a. 6x
6
b. 8x
6
c.
!
6
1
x
6
!
d.
!
8
3
x
5
!
e.
!
8
1
x
5
!
62. If 4x = 3y + 15 and 2y x = 0, find x.
f. 6
g. 3
h. 2
i. 1
j. 5
63. Simplify 36
!
2
3
!
.
a. 6
b. 216
c. 12
d.
!
2
1
16
!
e.
!
2
1
16
!
64. If x
3
= 50, the value of x is between which two integers?
f. 3 and 4
g. 7 and 8
h. 3 and 4
i. 2 and 3
j. 7 and 8
65. Find the value of x.
a. 25°
b. 136°
c. 112°
d. 68°
e. 34°
112°
x°
x°
– ACT MATH TEST PRACTICE–
179
66. Line l is parallel to line m.Find the measure ofangle x.
f. 99°
g. 39°
h. 21°
i. 121°
j. 106°
67. Find the radius of the circle with center (4, 2) that is tangent to the y-axis.
a. 2
b. 6
c. 1
d. 4
e. 10
68. Find the area, in square units, of the circle represented by the equation (x 5)
2
+ (y 2)
2
= 36.
f. 6
g. 36
h. 25
i. 2
j. 4
69. mABC = 120° and mCDE = 110°. Find the measure of BCD.
a. 70°
b. 50°
c. 60°
d. 150°
e. 40°
AB
C
D
E
120°
110°
x°
21°
120°
l
m
– ACT MATH TEST PRACTICE–
180
70. The ratio of the side lengths of a right triangle is 1:1:!2
"
.Find the sine ofthe smallest angle.
f.
!
1
2
!
g.
!
!
2
2
"
!
h. !2
"
i. 1
j. 2
71. What is the minimum value of 9cos x?
a. 9
b. 0
c. 90
d. 2
e. 9
72. A triangle with angles measuring 30°, 60°, and 90° has a smallest side length of 7. Find the length of
the hypotenuse.
f. 14
g. 7!3
"
h. 2
i. 12
j. 18
73. The Abrams’ put a cement walkway around their rectangular pool. The pool’s dimensions are 12 feet
by 24 feet and the width of the walkway is 5 feet in all places. Find the area of the walkway.
a. 748 square feet
b. 288 square feet
c. 460 square feet
d. 205 square feet
e. 493 square feet
74. Tr ian g le XYZ is an equilateral triangle. Y
"
W
"
is an altitude of the triangle. If Y
"
X
"
is 14 inches, what is the
length of the altitude?
f. 7!3
"
inches
g. 7 inches
h. 7!2
"
inches
i. 6!3
"
inches
j. 12 inches
– ACT MATH TEST PRACTICE–
181
75. What is the sum of the solutions to the equation 2x
2
= 2x + 12?
a. 4
b. 7
c. 1
d. 9
e. 1
76. Find the value of sin A if angle A is acute and cos A =
!
1
9
0
!
.
f.
!
!
10
11
"
!
g.
!
5
4
!
h.
!
1
9
0
!
i.
!
1
1
0
9
0
!
j.
!
!
10
19
"
!
77. Find the value of x.
a. 2
b. 1
c. !7
"
d. !10
"
e. 2!5
"
x
45°
√
¯¯¯
5
– ACT MATH TEST PRACTICE–
182
78. Which equation corresponds to the graph below?
f.
!
2
x
5
2
!
+
!
y
9
2
!
= 1
g. 25x
2
+ 9y
2
= 1
h.
!
2
x
5
2
!
!
y
9
2
!
= 1
i.
!
2
y
5
2
!
+
!
x
9
2
!
= 1
j. 5x
2
+ 3y
2
= 3
79. What is the inequality that corresponds to the graph below?
a. y > 3x + 2
b. y 3x + 2
c. y 3x + 2
d. y < 3x + 2
e. y < 3x + 2
80. What is the domain of the function f(x) =
!
x
2
4
+
x
3
x
5
4
!
?
f. {x | x 0}
g. Ø
h. All real numbers
i. {x | x 3}
j. {x | x 4 and x 1}
(0,3)
(5,0)
(0,3)
(5,0)
– ACT MATH TEST PRACTICE–
183
