Educational Talent Search and Upward Bound
Forward Service Corporation
14 questions dealing with Pre-Algebra
 10 questions from Elementary Algebra
 9 questions based on Intermediate Algebra
 9 questions from Coordinate Geometry
 14 questions from Plane Geometry
 4 Trigonometry questions
The questions assume knowledge of basic formulas and
computational skills but do not require memorization of
complex formulas or extensive computation
60 questions
60 minutes
Don’t waste time on any one problem, spend your
dftime doing as many problems as you can
The questions are arranged in order of difficulty
RELAX: answering only half of the questions right
onwill give you a score of 20 on the Math Section
Go Through The Test Twice
Take 45 minutes to go through the test
Answer the questions that you know how to do
Guess on the questions you know you’ll never get
Mark the harder questions that you’ll come back to later
Spend the last 15 minutes going over the test again
Answer the questions you skipped
Make sure you have answered every question
Spend any remaining time checking your work
ALL PROBLEMS ON THE
ACT CAN BE SOLVED
WITHOUT USING A
CALCULATOR
You may use a four-function, scientific, or
graphing calculator on the Math test
Calculators, such as TI-89 and TI-92 are NOT
permitted (see page 4 in the ACT prep booklet)
Bring a calculator that you know how to use –
bringing a more powerful calculator that you do
not know how to use isn’t going to help you
Read the problem carefully
Pay attention to what the question asks you to find
Watch for unnecessary information
Label the figures with numbers or letters
Draw a picture
There’s more than one way to solve these problems
It’s a timed test – find a quick and reliable way to
solve the problem
Do your work in your test booklet
Be careful using your calculator – it’s easy to push the
wrong button
Don’t get involved in long, complicated, or tricky
calculations
Make sure you answered the question that was asked
Each of the wrong answers represents a common
mistake that you might have made
If your answer isn’t one of the choices, reread the
question and check your work
BACKSOLVE
Take advantage of the multiple-choice format and try each
answer until you find the one that works.
WARNING: Doing it this way will take more time.
The greatest common divisor of 84, 90, and 66 (that is,
the largest exact divisor of all three numbers) is:
A. 6
84 =902٠2٠3٠7
does not go
90 into
= 2٠3٠3٠5
84 – eliminate
66 = answer
2٠3٠11 E
B. 12
36 does not go into 84 – eliminate answer D
C. 18
2٠3 18
= 6does not go into 84 – eliminate answer C
D. 36
12 does not go into 90 – eliminate answer B
E. 90
6 must be the correct answer and it checks
CORRECT ANSWER: A
BACKSOLVE (cont)
If (3  x )  1  2 , then x = ?
F. 0
G. 2
H. 4
J. 8
K. 10
xxx=
4820
Let
Let
=
Let
x==
(3
 xxx)))x)11
1
12 222
(3(3(3
(3
111
 4)
2)0)
22
122
(3(3(3
8)
2.101
2.414
(3 (32)

11 2 2
2.414
0)
2.414
5 31 1 2 2
2.236  2.414
BACKSOLVE (cont)
If (3  x )  1  2 , then x = ?
F. 0
G. 2
(3  x )  1  2
3  x  (1  2) 2
H. 4
3  x  (1  2)(1  2)
J. 8
3  x  1 2  2  2
K. 10
3 x  3 2 2
3 x  3 4 2  3 8
Correct Answer:J
GUESSTIMATE
If you can estimate the correct answer, then you should
be able to eliminate at least one or two answer choices.
What is 2% of 60?
A. 120
B. 12
C. 1.2
D. 0.12
E. 0.012
120 is greater than 60 – eliminate answer A
60 –xeliminate
0.02 =answer
1.2 B
12 is too large
CORRECT ANSWER: C
0.012 is too small – eliminate answer E
GUESSTIMATE (cont.)
In the right triangle below, how many meters long is AB ?
A
F.
G.
H.
J.
K.
12
7
5
3
?
4 meters
7
B
3 meters
12, 7, and 5 are longer than the hypotenuse –
eliminate answers F, G, and H
C
GUESSTIMATE (cont.)
In the right triangle below, how many meters long is AB ?
a b  c
2
F.
G.
H.
J.
K.
12
7
5
3
7
2
32  b 2  42
9  b  16
2
A
2
?
4 meters
b 7
2
b 7
B
CORRECT ANSWER: K
3 meters
C
EYEBALL
By looking at the figure:
thetoo
ACT,
diagrams
are “not
necessarily”
drawn
XOne
looks
bigthe
to be
5°- eliminate
answer
A
to scale, but usually they’re quite accurate, so you could
X looks too small to be either 60° or 55° - eliminate
eliminate certain answer choices by sizing things up with
answers E and D
your eyes
In the figure below, what is the value of x?
A. 5°
x
B. 30°
C. 40°
D. 55°
E. 60°
125°
85°
EYEBALL (cont.)
In the figure below, what is the value of x?
x
A. 5°
B. 30°
C. 40°
D. 55°
E. 60°
125° 55° 95° 85°
180 –125 = 55
180 - 85 = 95
180 – 55 – 95 = 30
CORRECT ANSWER: B
EYEBALL (cont.)
By looking at the figure:
Lola is making the circle graph below showing the
A
looks of
larger
than 99°–
eliminate
answer
F high
number
students
at each
grade level
in her
school.
should
the–measure
A? K
A
looksWhat
smaller
than be
240°
eliminateofanswer
F.
99°
G. 120°
H. 133°
J. 167°
K. 240°
240 freshmen
200
sophomores
A
130 seniors
150 juniors
EYEBALL (cont.)
Total number of students = 720
Total number of freshmen = 240
Total number of degrees in a circle = 360°
number of freshmen
degrees in A

number of students degrees in circle
240
A

720 360
CORRECT ANSWER:
240(360)  720 A
86400  720 A
120  A
G
PICK NUMBERS
Some problems are hard because they’re general or
abstract, so replace variables with specific numbers
If kx + k = 0, and k>1, then x = ?
A. 0
Pick a value for k that is greater than 1
B. -1
Let k = 2
Let k = 5
C. 1
kx + k = 0
kx + x = 0
D. -k
2x + 2 = 0
5x + 5 = 0
E. k
2x = -2
5x = -5
x = -1
x = -1
PICK NUMBERS (cont.)
If kx + k = 0, and k>1, then x = ?
A. 0
kx + k = 0
B. -1
k(x+1) = 0
C. 1
k= 0
OR x+1 = 0
D. -k
k= 0
OR x = -1
E. k
CORRECT ANSWER:B
PICK NUMBERS (cont.)
For all a  0, what is the slope of the line segment connecting
(a,b) and (-a,b) in the usual (x,y) coordinate plane?
F.
0
a
G.
b
b
H.
a
b
J.
a
K . 2a
( x1 , y1 ) and (x2 , y2 )
y1  y2
m
x1  x2
Let a = 1 and b = 2
(1,2) and (-1,2)
22
0
 0
1  (1) 2
Let a = -1 and b = -2
(-1,-2) and (1,-2)
2  (2) 0

0
1  1
2
PICK NUMBERS (cont.)
For all a  0, what is the slope of the line segment connecting
(a,b) and (-a,b) in the usual (x,y) coordinate plane?
F.
0
a
G.
b
b
H.
a
b
J.
a
K . 2a
( x1 , y1 ) and (x2 , y2 )
y1  y2
m
x1  x2
(a, b) and ( a, b)
bb
0
m

a  (  a ) 2a
CORRECT ANSWER: F
Question 1
In the figure below, CA is perpendicular to AB and CB is
perpendicular to BD; AB is 3 units long, AC is 4 units long,
and BD is 12 units long. How many units long is CD ?
D
C
A. 13
B. 17
4
A
12
3
C. 19
D. 24
B
E. 25
a b  c
2
2
5  12  CD
2
2
3  4  CB
2
2
9  16  CB
25  CB
2
2
169  CD
2
2
13  CD
5  CB
13
CORRECT ANSWER: A
D
5
4
A
2
25  144  CD
2
C
2
12
3
B
ABC is a 3-4-5 right triangle and BCD is a 5-12-13 right triangle
Question 2
In the figure below, line m is parallel to line n, and line t is
a transversal crossing both m and n. Which of the
following lists has 3 angles that are all equal in measure?
a
b
c
d
e
t
m
n
A. a,b,d
B. a,c,d
C. a,c,e
D. b,c,d
E. b,c,e
CORRECT ANSWER: A
Question 3
A shirt that originally cost $35 is on sale at 20% off. If
the sales tax on shirts is 5% of the purchase price, how
much would it cost to buy the shirt at its sale price?
CORRECT ANSWER: D
35 (.20) = 7
A. $ 7.35
35 – 7 = 28
28 (.05) = 1.40
B. $20.00
28 + 1.40 = 29.40
C. $26.60
D. $29.40
35 (.80) = 28
E. $29.75
28 (1.05) = 29.40
Question 4
What is the slope of the line x = 2y +3?
CORRECT ANSWER: A
A.
B.
C.
D.
1
2
1
3
2
2
E. 3
x = 2y + 3
x – 3 = 2y
x 3
y
2
x 3
 y
2 2
1
3
x  y
2
2
y = mx + b
1
m
2
3
b
2
Question 5
In the figure below, A, C, and D are collinear.
If the measure of A is 30 and the measure
ofBCD is 120, what is the measure of B?
A. 30
B
B. 60
C. 90
A
D. 120
C
D
E. 150
In the figure below, A, C, and D are collinear. If
the measure of A is 30 and the measure of
BCD is 120, what is the measure of B?
CORRECT ANSWER: C
B
90
A
30
180 – 30 – 60 = 90
60
120
C
D
180 – 120 = 60
Question 6
If the area of a circle is , what is the length
of its circumference?
A. 1
B. 2
C. 
D. 2
E. 3
Areacircle = r2
Circumference = 2r
 = r2
C = 2(1)
1 = r2
C = 2
1=r
CORRECT ANSWER: D
Question 7
What is the value of x in the solution
for the system of equations below?
2x + 5y = 20
6x – ½ y = 29
A. 4
B. 5
C. 6
D. 15
E. 20
What is the value of x in the solution
for the system of equations below?
2x + 5y = 20
6x – ½ y = 29
2x + 5y = 20
6x – ½ y = 29 multiply by 10
add equations together
solve for x
2x + 5y = 20
60x - 5y = 290
62x = 310
x=5
What is the value of x in the solution for the system of
equations below?
CORRECT ANSWER: B
2x + 5y = 20
6x – ½ y = 29
solve for y
substitute into 1st equation
solve for x
12x – y = 58
-y = 58 – 12x
y = -58 + 12x
2x + 5(-58+12x) = 20
2x + -290 + 60x = 20
62x + -290 = 20
62x = 310
x=5
Question 8
1 2 3 4 5
      ?
2 3 4 5 6
A.
B.
C.
D.
E.
1
6
17
27
13
18
7
9
5
6
1  2 3   20 
   
2  3 4   30 
1 2 4 4
   
2 3 3 6
1 8 2
  
2 9 3
1 8 6
  
2 9 9
1 2

2 9
9 4 13
 
18 18 18
CORRECT
ANSWER:
C
Question 9
For all y, 26y – (-10y) – 3y(-y+3) = ?
A. 10y
26y – (-10y) – 3y(-y+3)
B. -3y2 + 25y
26y +10y +3y2 – 9y
C. 3y2 + 7y
3y2 +27y
D. 3y2 + 25y
E. 3y2 + 27y
CORRECT ANSWER: E
Question 10
In ABC shown below, BD bisects ABC. The measure of ABC
is 100, and A measures 30. What is the measure of BDC?
C
A. 65
D
B. 70
C. 75
D. 80
E. 85
30
A
100
B
In ABC shown below, BD bisects ABC. The measure of ABC
is 100, and A measures 30. What is the measure of BDC?
C
A. 65
B. 70
C. 75
30
D
80°
100 50
50
A
B
D. 80
180 – 30 – 50 = 100
E. 85
180 – 100= 80
m BDC = 80
CORRECT ANSWER: D
Question 11
In the figure below, S is a right angle, RS is 3 units
long, and ST is 4 units long. If the measure of R is
x, then sin x = ?
T
3
A.
5
3
B.
4
4
C.
4
5
5
D.
4
x
5
E.
R
3
S
3
In the figure below, S is a right angle, RS is 3 units
long, and ST is 4 units long. If the measure of R is
x, then sin x = ?
T
opposite
sin x 
hypotenuse
a2 + b2 = c2
32
+
42
=
5
c2
4
9 + 16 = c2
25 = c2
5=c
4
sin x 
5
x
R
3
RST is a 3-4-5 right triangle
S
CORRECT
ANSWER: C
Question 12
In the figure below, the lengths of DE, EF, and FG
are given, in units. What is the area, in square
units, of triangle DEG?
A.29
B.47.5
G
C.60
D.6 149
E.120
10
D
12
E
7
F
In the figure below, the lengths of DE, EF, and FG
are given, in units. What is the area, in square units,
of triangle DEG?
G
10
D
12
E
7
AreaTriangle = ½ base x height
= ½ (12) (10)
= 60
CORRECT ANSWER: C
F
Question 13
In the figure below, A and B lie on the circle centered at O,
OA is 6 units long, and the measure of AOB is 60°. How
many units long is minor arc AB?
A
6
A. 
B. 2
C. 6
D. 12
E. 36
O
60
B
Circumference = 2r
degrees = arc
C = 26
arc
circle
C = 12
60
x

360 12
60(12) = 360 x
A
6
O
720 = 360 x
2 = x
60
B
CORRECT
ANSWER:
B
Question 14
In the right triangle below, the length of AB is 13 units and
the length of CB is 12 units. What is the tangent of A?
A.
B.
C.
D.
E.
12
5
13
12
12
13
5
12
5
13
A
13
C
12
B
opposite
tan x 
adjacent
a2 + b2 = c2
122 + b2 = 132
144 + b2 = 169
b2 = 25
12
tan x 
5
b=5
CORRECT ANSWER: A
A
13
5
C
12
B
Question 15
Points N and J have coordinates (-1, -1) and (3,3) respectively.
What is the length of line NJ ?
(the distance between ( x1 , y1 )and ( x2 , y2 ) )
Distance Formula =
A.
4
B.
8
C. 6
( x2  x1 )2  ( y2  y1 )2
(3  (1))2  (3  (1))2
42  42
32  16 2  4 2
D. 8
E. 4 2
CORRECT ANSWER: E
Question 16
What is the cost in dollars to carpet a room x yards long and
y yards wide if the carpet costs two dollars per square foot?
x yards = 3x feet
y yards = 3y feet
A. xy
B. 2xy
Arearectangle= length x width
C. 3xy
= 3x(3y) = 9xy square feet
Total Cost = area x price
D. 6xy
= 9xy (2) = 18xy
E. 18xy
CORRECT ANSWER: E
Question 17
In the figure below, the largest possible circle is cut out of a
square piece of tin. The area of the remaining piece of tin is
approximately (in square inches)
2 inches
A. 0.14
B. 0.75
C. 0.86
D. 1.0
E. 3.14
Question 17
In the figure below, the largest possible circle is cut out of a
square piece of tin. The area of the remaining piece of tin is
approximately (in square inches)
2 inches
Areacircle =  r2
A. 0.14
B. 0.75
 12 =  = 3.14
1
C. 0.86
Areasquare = s2
D. 1.0
E. 3.14
22 = 4
Areapiece = Areasquare - Areacircle
CORRECT ANSWER: C
4 – 3.14 = 0.86
Question 18
Which of the following is equal to 3.14 x 106?
A. 314
3.1400000000
B. 3140
C. 31,400
D. 314,000
E. 3,140,000
3140000.0000
CORRECT ANSWER: E
Question 19
Find the last number in the series:
8, 4, 12, 6, 18, 9,?
A. 19
B. 20
C. 22
D. 24
E. 27
84
8-4=4
or
82=4
412
4+8=12
or
4x3=12
126
12-6=6
or
122 = 6
6 18
6+12=18
or
6x3=18
189
18-9=9
or
182 = 9
9x3=27
CORRECT ANSWER: E
Question 20
Lyndsey receives grades of 91, 88, 86, and 78 on
four tests. What grade must she receive on her
fifth test to have an average test score of 85?
A. 82
Let
B. 83
(91 + 88 + 86 + 78 + x )  5 = 85
C. 84
343 + x = 425
D. 85
x = 82
E. 86
x = the fifth test score
CORRECT ANSWER: A
Question 21
One angle, A, has 3 times the measure of its
supplement, B. What is the degree measure of A?
A. 112 ½°
B. 120°
Let x = the measure of B
then 3x = the measure of A
C. 135°
x + 3x = 180°
D. 150°
4x = 180°
B = 45°
E. 157 ½°
x = 45°
A = 135°
CORRECT ANSWER: C
Question 22
A bag contains 4 red jelly beans, 5 green jelly beans,
and 3 white jelly beans. If a jelly bean is selected at
random from the bag, what is the probability that the
jelly bean selected is green?
A.
B.
C.
D.
E.
1
12
1
5
5
23
5
12
5
7
Probability of drawing a green jelly bean
number of green jelly beans 5

total number of jelly beans 12
CORRECT ANSWER: D
Question 23
For all positive values of a, b, and c with a<b and a>c,
which of the following MUST be true?
I. a+b>c
II. 2a>c
III. a+c>b
c<a<b
I.
a and b are both greater than c,
so their sum will also be – TRUE
A. I only
II.
B. II only
III. If c=2, a=4, and b=10,
then a+c<b - FALSE
C. I and II only
D. II and III only
E. I, II, and III
a>c, so 2a>c – TRUE
CORRECT ANSWER: C
Question 24
The scales on both axes of the standard (x,y) coordinate
plane below are the same. Of the following, which is
the best estimate for the slope of AB ?
A. 4
3
B.
4
1
C.
4
1
D. 4
E. -4
1
rise

slope =
run
4
A
B
1
C
CORRECT ANSWER: C
Question 25
In the figure below, D is a point on AB and E is a point
on BC such that DE AC. If DB = 4, AB = 10, BC = 20,
what is the length of EC ?
B
A. 4
B. 6
D
C. 8
E
D. 10
E. 12
A
C
CORRECT ANSWER: E
ABC and DBE are similar triangles,
so their sides are in proportion
4
10

20-x 20
4  20  10(20  x )
80  200  10x
120  10x
x  12
B
4
10
D
6
A
20
20-x
E
x
C
Things to Remember on
the Math ACT
Don’t read the directions
Bring a calculator that you know how to use
Read the question carefully
Pay attention to what the question asks you to find
Watch for unnecessary information
Draw a picture
Pace yourself (60 questions/60 minutes)
Things to Remember on
the Math ACT
Do the easy questions first, then try the hard ones
Show some work and circle your answers in your test booklet
Don’t waste too much time on one problem
Eliminate wrong answers before guessing
Answer every question
Check your work
Work for the whole 60 minutes