ACT MATH PREPARATION

ACT MATH
PREPARATION
QUESTIONS
…
DR……

Athanasios Drougas
http://mathhmagic.blogspot.gr/
1
ACT MATH PREPARATION QUESTIONS
1.The rectangular prism below has a length of 8,a width of 3,
and a height of 4.That means its volume is equal to:
Α.69
B.96
C.86
D.106
E.68
2. Andrew and Bob are sharing a pack of gum. There are 9 sticks of gum in the pack,and
Andrew takes 3 of them. Since Andrew has 3 of 9 sticks in the pack of gum he has ….of
gum.
Α.9/3
B.6/9
C.0.3
D.1/3
E.3.333
3. if a=9,b=5,and c=-6 what does (a+b-c)(b+c) equal ?
Α.-20
B.-8
C.19
D.20
E.20
4. The sum of two numbers is S. Suppose 3 is added to each number and then each of the
resulting number is doubled. What is the sum of the final two numbers?
A.2S+3
B.3S+2
C.3S+6
D.2S+6
E.2S+12
5. A bolt making machine makes 60 bolts per minute. A second bolt making machine makes
80 bolts per minute. The second machine starts making bolts 2 minutes after the first machine
starts . Both machines stops making bolts 8 minutes after the first machine starts. Together,
the 2 machines made how many bolts?
Α.480
B.600
C.680
D.720
E.960
6.The first term is -1 in the geometric sequence -1,3,-9,27,….What is the seventh term of
geometric sequence
Α.-243
B.-30
C.81
D.-729
E.189
8
3
4

Athanasios Drougas
2
7.Bob ,Alice ,and John shared a pizza .Bob ate 1/3 of the pizza , Alice ate ½ of the pizza and
John ate the rest. What is the ratio of John’s share to Alice ‘ s share to Bob ‘s share;
Α.6:2:3
Β.3:6:2
C.2:3:6
D.2:6:3
E.3:2:6
8.For triangle FGH ,shown below, which of the following is an expression of x in terms of
y?
Α.y+3
Β. 2
3y +
C. 2
3y −
D. 2
9y −
E. 2
9y +
9.The inequality 5(x+2) < 6(x-5) is equivalent for which of the following inequalities
Α. 30x >
Β. 40x >
C. 20x >
D. 20x > −
E. 30x > −
10.A bag contains 10 red marbles,5 yellow marbles ,and 12 blue marbles .How many
marbles must be added to the 27 marbles already in the bag so that the probability of
randomly drawing a yellow marble is
14
25
?
Α. 22
Β. 23
C. 11
D. 10
E. 45
11.The coordinates of the endpoints of CD ,in the standard (x,y) coordinate plane are (-2,4)
and (-6,18) .What is the y-coordinate of the midpoint of CD .
Α.10
Β. 11
C. -4
D. -3
E. 9
H
GF
y cm
x cm
3 cm

Athanasios Drougas
3
12.Peter’s bicycle has a wheel circumference of 96 cm (or 0.96 m). It is a three-speed bicycle
with a low, a middle and a high gear. The gear ratios of Peter’s bicycle are:
Low 3:1 Middle 6:5 High 1:2
How many pedal turns would Peter take to travel 960 m in middle gear?
A gear ratio of 3:1 means 3 complete pedal turns yields 1 complete wheel turn.
Α.1200 pedal turns
B.1300 pedal turns
C. 300 pedal turns
D. 9600 pedal turns
E. 1000 pedal turns
13.The coordinates of the endpoints of CD ,in the standard (x,y) coordinate plane are (2,14)
and (6,-18) .What is the x-coordinate of the midpoint of CD .
Α.16
Β.-4
C.4
D.-2
E. -16
14.What is the surface area ,in square inches,of an 10-inch cube?
Α.400
Β. 600
C. 4000
D. 6000
E. 800
15. if a=2017,b=2016 what does 3(a-2)+2(a+3)-5(a-2b) equal ?
Α.20170
B.20160
C.0
D.20017
E.20016
16.If 2w+z is equal to 50% of 2w ,what is the value of
w
z
?
Α.
1
3
Β.
1
9
C.
1
4
D.
1
5
E.
1
2

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17.What is the surface area ,in square inches, of an 10-inch cube
Α.400
Β. 600
C. 4000
D. 6000
E. 800
and a height of 4.That means its surface area is equal to:
Α.156
B.96
C.120
D.256
E.154
19.Space station Mir remained in orbit for 15 years and circled Earth some 86500 times
during its time in space.The longest stay of one cosmonaut in the Mir was around 680 days.
Approximately how many times did this cosmonaut fly around Earth?
A. 110
B. 1100
C. 11000
D. 110000
E. 1100000
20.The table below shows the temperatures, in degrees Fahrenheit ,in a city in over a one
week period .If m represents the median temperature, f represents the temperature that
occurs more often, an a represents the average (arithmetic mean) of the seven temperatures,
which of the following is the correct order of m,f and a?
A. a m f< <
B. a f m< <
C. m a f< <
D. m f a< <
E. a m f= <
21.A number is increased by 45% and the resulting number is then decreased by 40%.The
final number what percent is of the original number ?
Α.85%
B. 87%
C. 89%
D. 92%
E.94%
8
3
4
Mon Tue Wed Thu Fri Sat Sun
66 78 75 69 78 77 70

Athanasios Drougas
5
22.Bob buys three sandwiches for two dollars each. He also buys a few bottles of water .Each
bottle of water costs seventy-five cents. Bob pays with fifteen dollars and gets six dollars
and seventy-five cents in change. How many bottles of water did he buy?
Α.3
B.4
C.2
D.5
E.6
23.We write all integers from 1 to 100.How many times appears 3?
Α.9
B.10
C.11
D.12
E.3
24.A bag contains 10 red marbles,5 yellow marbles ,and 12 blue marbles.How many marbles
must be added to the 27 marbles already in the bag so that the probability of randomly
drawing a yellow marble is
14
25
?
Α. 22
Β. 23
C. 11
D. 10
E. 45
25.The equations below are linear equations of a system ,a,b,c are positive integers
ax=c+1 -by
ax+bc=c
Which of the following describes the graph of at least 1 such system of equations in the
standard(x,y) coordinate plane?
I. two parallel lines
II.two intersecting lines
III. a single line
Α.I only
B.II only
C.III only
D.I or II only
E.I,II or III
26.Which of the following mathematical expressions is equivalent to the verbal expression
”A number x squared is 16 more than the product of 6 and x”?
Α.x2
=16+6x
B.2x=16+6x
C.2x=16x+6x
D. x2
=16+x6
E. x2
=16-6x

Athanasios Drougas
6
27.The determinant of a matrix
a b
c d
 
 
 
equals ad-bc.What must be the value of x for the
matrix
1
5 10
x x − 
 
 
to have a determinant of 55?
Α. 15
B. 12
C. 10
D. 14
E. 11
28. For 3 consecutive integers ,which of the following statements is false ?
Α.Half the sum of the smaller and the larger integer is equal to the middle integer.
B. The difference between the third number and the second is equal to the difference between
the second number and the first.
The average for 3 numbers is the number in the middle.
C. At least one of the three integers is odd
D. At least one of the three integers is even.
E. At least one of the three integers is prime.
29. A function f(x) is defined as f(x)=-256x-3
.What is f(-4)?
Α.3
B.-2
C.2
D.-4
E. 4
30.What is the sum of all roots of (2x+3)(x-4)+(2x+3)(x-6)=0?
A.7/2
B.4
C.5
D.7
E.13
31.A function f(x) is defined as
10
5
1
( )
1
x
f x
x
−
=
−
. 1x ≠ .What is 5
( 13)f ?
Α.14
B.12
C.11
D. 10
E. 13
32. If 2 63x
= ,the which of the following must be true?
Α. 4 5x< <
B. 3 4x< <
C. 5 6x< <
D. 2 3x< <
E. 6 x<

Athanasios Drougas
7
33.What is the least common multiple of 30,70,130?
Α. 140
B. 26000
C. 21000
D. 910
E. 273000
34.A building with 100 flours has 3 elevators. The elevator A stops only every 3th
floor ,
the elevator B stops only every 5th
floor ,and the elevator C stops only every 7th
floor .
Which floor of the building is possible for the three elevators (A,B,C) to meet?
Α. 15th
floor
B. 35th
floor
C. 21th
floor
D. 91th
floor
E. There is no such a floor
35. A rectangle has an area of 40 square feet and a perimeter of 26 feet .What is the shortest
of the side lengths ,in feet, of the rectangle?
Α.8
B.5
C.2
D.4
E.3
36. The mean of three numbers is 10 more than the least of the numbers and 15 less than the
greatest .The median of the three numbers is 5.What is their sum?
A.5
B.20
C.25
D.30
E.36
37. A rectangle has an area of 12 square feet and a perimeter of 14 feet .What is the
length ,in feet, of the diagonal ?
Α.5
B.6
C.3
D.4
E.7
38.The expression (4x-7y)(3x+7y) is equivalent to:
F. 8x2
-49y2
G. 16x2
+49y2
H. 49y2
-16x2
J. 16x2
-49y2
K. 16x2
-14y2

Athanasios Drougas
8
39. Discount tickets to a Baseball tournament sell for $ 3.00 each .Bob spent $40.00 on
discount tickets , $30.00 less than if he had bought the tickets at the regular price. What was
the regular ticket price?
Α. $ 5.00
B. $ 4.50
C. $ 5.50
D. $ 10.00
E. $ 23.33
40. In triangle ABC , the sum of the measures of A∠ and B∠ is 57o
.What is the measure of
C∠ ?
Α. 57o
B. 67o
C. 77o
D. 133o
E. 123o
41. If A is a two- digit positive integer ,then
( 10) ( 11) ( 12) .... ( 97) ( 98) ( 99) ?A A A x x x− × − × − × × − × − × − =
Α. 100
B. -100
C. 10
D. 0
E. cannot be determined from the given information
42.For what value of b would the following system of equations have an infinite number of
solutions?
-3x+5y=10
9x-15y=2b
Α. 10
B. -10
C. 3
D. 15
E. -15
43.A bag contains 10 red marbles,5 yellow marbles ,and 12 blue marbles.How many marbles
must be added to the 27 marbles already in the bag so that the probability of randomly
drawing a yellow marble is
14
25
?
Α. 22
Β. 23
C. 11
D. 10
E. 45

Athanasios Drougas
9
44.The determinant of a matrix
a b
c d
 
 
 
equals ad-bc.What must be the value of x for the
matrix
14x
x x
 
 
 
to have a determinant of -49?
Α. -7
B. -6
C. 7
D.
49
13
−
E.
49
13
−
45.The solution set of which the following equations is the set of all real numbers that are 6
units from -4?
Α. 4 1 6x − =
B. 6 4x − =
C. 6 4x + =
D. 4 6x + =
E. 4 6x − =
and a height of 4.That means,its surface area not visible from
your point of view is equal to:
Α.120
B.60
C.72
D36
E.196
47.A function f is defined as follows :
For x>0, 4 3 2
( ) 23 23f x x x x= + − −
For x <0, 4 3 2
( ) 23 23f x x x x= − + + −
What is the value of f(-1)
Α. 46
B. -46
C. 0
D. -1
E. -2
8
3
4

Athanasios Drougas
10
48.The number line graph below is the graph of which of the following inequalities?
Α. 2 x− ≤ and 5 x≤
B. 2 x− ≤ and 5 x≥
C. 2 x− ≤ or 5 x≤
D. 2 x− ≥ or 5 x≤
E. 2 x− ≥ or 5 x≥
49. If two sides of a triangle have lengths 7 and 8 , the perimeter of the triangle could be
which of the following?
A.11
B.15
C.24
D.31
E.30
50. The solution set of which the following equations is the set of all real numbers that are
less than 3 units from 7?
Α. 3 7x − >
B. 3 7x + >
C. 3 7x + <
D. 3 7x − <
E. 7 3x − <
51.What is the sum of the first 5 terms of the arithmetic sequence in which the 7th
term is
10 and the 8th
term is 12?
Α.14
B. 16
C. 20
D. 18
E. 22
-2 5

Athanasios Drougas
11
52.Triangles ABC∆ and PQR∆ are shown below. The given side lengths are in centimeters.
The area of ABC∆ is 40 square centimeters. What is the area of PQR∆ ?
Α.20
B.10
C.40
D.30
E. 60
53. In a poll 33 people were in favor pf voting John Doe ,15 people were against him, and
two persons had no opinion. What percent of those polled were in favor of voting John Doe?
Α.
33
100
B.
66
100
C.
30
100
D.
12
100
E.
70
100
54.Which of the following mathematical expressions is equivalent to the verbal expression
”A number x plus his squared product with 25 is 60 less than the product of 343 and twice
x”?
Α.x+625x2
=686x-60
B. x+625x2
=686x+60
C. x-25x2
=343x+60
D. x+25x2
=343x-60
E. x+25x2
=686x-60
1200
o
R
P
Q
y
x
B
A
C
x
y
60o

Athanasios Drougas
12
55.For the population that grows at a constant rate of r% per year ,the formula
( ) 1
100
t
r
P t Po= +
 
 
 
models the population t years after an initial population of Po people
is counted.
The population of city of San Antonio was 1,151,979 in 2010.Assume the population
grows at a constant rate of 5% per year .According to this formula ,which of the following
is an expression for the population of San Antonio in the year 2016?
Α. 1,151,979 (6)6
B. 1,151,979 (1.5)6
C. 1,151,979 (1.05)6
D. (1,151,979 x 1.5)6
E. (1,151,979 x 1.05)6
56.Triangle ABC∆ is shown in the figure below .The measure of A∠ is 40o
,AB=18cm and
AC=12 cm. Which of the following is the length , in centimeters ,of BC ?
(Note: For a triangle with sides of length a,b and c opposite angles A∠ , B∠ , C∠ ,
respectively the law of sines states
sin sin sinA B C
a b c
∠ ∠ ∠
= = and the law of cosines states
2 2 2
2 cosc a b ab C= + − ∠ )
Α. 0
18cos70
B. 0
12sin 70
C. 2 2
12 18+
D. 2 2
18 12+
E. 2 2
18 12 2(12)(18)sin 70o
+ −
57.In the equation 2
2 0x mx n+ + = , m,n are integers .The only possible value for x is
-2.What is the value of m?
Α. 2
B. -2
C. 4
D. -4
E. 1
A
C
B
12 cm
18 cm
70o

Athanasios Drougas
13
58.In the equation 2
2 0x mx+ + = , 0m ≠ .If a,b are the roots of the equation .The
expression
2 2 3a b abm
m
+ +
is equal to:
Α.6
B.4
C.-2
D.2
E. 3
59. The length of a rectangle is 5 times the length of a smaller rectangle .The two
rectangles have the same width .The area of the smaller rectangle is A square units .The area
of the larger rectangle is kA square units .Which of the following is the value of k?
Α.
1
5
B.
1
25
C. 5
D.25
E.1
60.The dimensions of the right triangle shown below are given in feet( k positive integer).
What is sinθ?
Α.
5
13
B.
12
13
C.
5
12
D.
12
5
E.
13
12
θ
5κ
13κ
12κ

Athanasios Drougas
14
61.The hair color of 190 people were determined for a study as shown in the figure below.
If one person from the study is randomly selected, what is the probability that this person has
either red hair or blond hair.
Α.
89
190
B.
12
190
C.
112
190
D.
25
190
E.
77
190
62. Given that a=bc5
,a=2017,c=10,what is b?
Α. 2.017
B. 0.2017
C. 0.02017
D. 20.17
E. 4
201.7 10x −
63. If f(x)=(5x-6)2
then f(f(1))=?
Α.-1
B.1
C. 0
D. 10
E. 100
55
33
77
20
10
30
40
50
60
70
80
black brow
n
red blond Hair color
Number
of people

Athanasios Drougas
15
64. If f(x)=(5x-6)2
then f(f(1))=?
Α.-1
B.1
C. 0
D. 10
E. 100
65.Bob the scientist studying motion of his parrot Jim ,so Bob observed Jim flying at a
constant rate along a straight line .The table below gives the distance ,d feet, Jim the parrot
was from a reference point at 2-second intervals from t=0 second to t=6 seconds .
Which of the following equations represents this relationship between d and t?
Α. 4t+29
B. t+25
C. 4t+25
D. t-25
E. 5t+24
66.In scientific notation 2350,000,000-450,000,000=?
Α.1.9 x 10-9
B. 1.9 x 107
C. 1.9 x 108
D. 1.9 x 1015
E. 1.9 x 109
67. If f(x)=(6x-16)3
then f(1)=?
Α.10
B.100
C. 1
D. 1000
E. 10000
68.In scientific notation 6250,000,000/250=?
Α.2.5 x 10-7
B. 2.5 x 107
C. 2.5 x 108
D. 2.5 x 10-9
E. 2.5 x 10-8
69.We write all integers from 1 to 100.How many integers of them have at least a 5?
Α.9
B.10
C.11
D.12
E.8
t
d
0
25
2
33
4
41
6
49

Athanasios Drougas
16
70.In triangle ABC , the measure of A∠ is 90o
and the measure of B∠ is twice the measure
of C∠ .What is the measure of C∠ ?
Α. 60o
B. 30o
C. 45o
D. 120o
E. 90o
71.In which of the following are
2 6 3
, ,
3 7 2
,
6
11
and arranged in ascending order ?
Α.
2 6 6 3
3 7 11 2
< < <
B.
2 6 6 3
3 11 7 2
< < <
C.
3 6 6 2
2 11 7 3
< < <
D.
3 2 6 6
2 3 11 7
< < <
E.
6 6 3 2
11 7 2 3
< < <
72. 3 2 2 3
3 2 6a ab a b⋅ ⋅ is equivalent to:
Α. 6 5
11a b
B. 5 6
12a b
C. 6 5
12a b
D. 5 5
11a b
E. 6 6
11a b
73. If 3( 4) 4 11 12x x x− + = − , the x=?
Α. 1
B. 3
C. every real number
D. 0
E. 2
74. If 20 2a t= − then
5
2
?
2
a t+ 
= 
 
Α. 100
B. 1000
C. 10000
D. 500
E. 5000
75.Which of the following numbers is prime?
Α. 28713
B. 25726
C.15675
D.78250
E. 2244

Athanasios Drougas
17
76.For all pairs of positive integers a,b, which of the following numbers is even?
Α. ab
B. 2 2
a a b b+ + +
C. 2 2
2a ab b+ +
D. 2 2
a b ab+
E. a b−
77.What is the y-coordinate of the point in the standard (x,y) coordinate plane at which the
three lines y-3=0,y=25x+25,
50
0
150
y
x
+
− = ?
Α. 4
B. 5
C. 50
D. 0
E. 3
78.If x and y are positive numbers such that the greatest common factor of 4
x y and 2 3
x y is
147,which of the following is a possible value for x?
Α. 147
B. 21
C. 7
D. 3
E. 9
79.The sum of two numbers is 12,and the product of the same two numbers is 4.The sum of
the reciprocals of the two numbers is equal to:
Α. 4 2
B. 23
C. 4
D. 3
E. 2
(Reciprocal of number x( 0x ≠ ) is the
1
x
. )
80.If 2
19a b ab a∇ = + + for all nonnegative numbers ,what is the value of 6 10∇ ?
Α. 25
B. 35
C. 45
D. 615
E. 635
81.A cat chases a mouse , who has a 60–m head start. For every 7 m the mouse runs ,the cat
runs 9 m. How far must the cat run to catch the mouse?
Α. 72
B. 73
C. 74
D. 84
E. 124

Athanasios Drougas
18
82.If 7 hats cost a dollars and b hats would cost how many dollars?
Α.
7a
b
B. 7ab
C.
7
ab
D.
7
a
b
E.
7
ab
83.For x such that
2
x
π
π < < , the expression
2 2
1 sin 1 cos
cos sin
x x
x x
− −
+ is equivalent to:
Α. 0
B. 1
C. 2
D.
1
tan x
E. tan x
84.For x such that 0
2
x
π
< < , the expression
2 2
1 sin 1 cos
cos sin
x x
x x
− −
− is equivalent to:
Α. 0
B. 1
C. 2
D.
1
tan x
E. tan x
85.For x such that 0
2
x
π
< < , the expression
( ) ( )
2 2
sin cos sin cos
2
x x x x− + +
is equivalent to:
Α. 2
B. 0
C. 1
D. cos x
E. cos x−

Athanasios Drougas
19
86.Consider the functions ( )f x x= and ( ) 2g x x b= + .In the standard (x,y) coordinate
plane , ( ( ))y f g x= passes through (3,5).What is the value of b?
Α.19
B. 20
C. 21
D. 22
E. -22
87.What is the real value of x in the equation 2 2 3log 96 log 3 log x− =
Α.32
B.27
C.9
D.125
E.243
88.What is the real value of x in the equation 2 2
15
2
log 5 2log 3
log
log 45
x
+
=
Α.5
B.45
C.3
D.15
E.9
89.What is the average of the following numbers
1+w, 2-2w, 3+4w,16, 3+3w , 7-6w
Α.8+2w
B.4+2w
C.4
D.8
E. 32
90.What are the real solutions to the equation
2
4 5 0?x x+ − =
Α. 5±
B. 1±
C.1 and 5
D. -1 and -5
E. 5± and 1±
91.What are the real solutions to the equation
2
9 6 1 0?x x− + =
Α. 3±
B.
1
3
±
C.1 and
1
3
D. -1 and
1
3
−
E. 3± and
1
3
±

Athanasios Drougas
20
92.The parallel sides of the isosceles trapezoid shown below are 10 feet long and 16 feet long,
respectively. What is the distance, in feet , between these 2 sides?
Α.3
B.4
C.5
D.10
E.16
93.Normally, a penguin couple produces two eggs every year. Usually the chick from the
larger of the two eggs is the only one that survives. With rockhopper penguins, the first egg
weighs approximately 78 g and the second egg weighs approximately 110 g.
By approximately how many percent is the second egg heavier than the first egg?
A. 29%
B. 32%
C. 41%
D. 71%
D. 61%
respectively. What is the distance, in feet, between these 2 sides?
Α.13
B.14
C.12
D.20
E.16
respectively. What is the area of triangle ABE, in square feet?
Α.240
B.120
C.260
D.160
E.100
5 ft 5 ft
16 ft
10 ft
13 ft 13 ft
20 ft
10 ft
13 ft 13 ft
20 ft
10 ftA B
C
D E

Athanasios Drougas
21
96.For all x in the domain of the function 3
2 1
4
x
x x
−
−
, this function is equivalent to:
Α. 2
1
2x x−
B. 3 2
1
2x x−
C. 2
2 1
4 1 4 1x x
−
− −
D. 2
1 1
2x x
+
E. 2
1
2x x+
97.The number line graph below is the graph of which of the following inequalities?
Α. 21 x≤ and 10 x≤
B. 21 x≥ and 10 x≤
C. 21 x≥ or 10 x≤
D. 21 x≥ or 10 x≤
E. 21 x≥ or 10 x≥
98. For the pentagram ABCD shown below , ?A B C D F∠ + ∠ + ∠ + ∠ + ∠ =
Α.360o
B.180o
C.270o
D.540o
E.120o
A
B
C
D
E
10 21

Athanasios Drougas
22
99. 3 3 3 3 321 2 13
13 7 3
a a b a b⋅ ⋅ is equivalent to:
Α. 27 9
2a b
B. 27 9
3a b
C. 9 6
2a b
D. 9 6
3a b
E. 9 9
2a b
100.In the diagram below, the congruent circles are tangent to the larger square and each
other as shown and their centers are vertices of the smaller square. The area of the smaller
square is 4.The area of the larger square is equal to:
Α.8
B.16
C.8π
D.16π
E.4π
101.Jack rode his bike from home to the river, which is 4 km away. It took him 9 minutes. He
rode home using a shorter route of 3 km. This only took him 6 minutes.
Jack's average speed, in km/h, for the trip to the river and back is equal to:
A. 60 km/h
B. 50 km/h
C. 40 km/h
D. 30 km/h
E. 46.6 km/h
102. Given the six numbers
1916, 2817,3718,4619,5520,6421
The percent of their average to their sum is equal to:
Α.20%
B.25%
C.30%
D.35%
E.15%
103. If a=1-i then (a(1+i)-1)20
=?
Α.0
B.1
C. -1
D.i
E.-i

Athanasios Drougas
23
B C
E
A
104. If 6 4x + < which of the following statements are true?
I. 2x < −
II. 2x > −
III. 10x > −
Α.I only
B.II only
C.I and II only
D.II and III only
E. I and III only
105. If 23 3x − < − which of the following statements are true?
I. 20x <
II. 23x >
III. 26x > −
Α.I only
B.II only
C.I and II only
D.II and III only
E. none the three statements is true
106.In the figure below, EC=10 centimeters
AB=5 centimeters and the inscribed circle
has a radii of 2 centimeters then AC is
equal to:
Α.12
B.13
C.14
D.10
E.9
107.Which of the following is true for all consecutive integers m and n such m <n?
Α. m is odd
B. n is odd
C. n-m is even
D. n2
-m2
is odd
E. n2
+m2
is even
108.Two numbers are reciprocals if their product is equal to 1.If x and y are reciprocals and
x>1 ,the y must be:
Α. less than -1
B. between 0 and -1
C. between 0 and 1
D. equal to 0
E. greater than 1

Athanasios Drougas
24
109.Which of the following is true for all consecutive integers m and n such m <n?
Α. m is odd
B. n is odd
C. mn is even
D. mn+1 is even
E. n2
+1 is odd
110.Mount Fuji is only open to the public for climbing from 1 July to 27 August each year.
About 200,000 people climb Mount Fuji during this time. On average, about how many
people climb Mount Fuji each day?
A. 340
B. 710
C. 3,400
D. 7,100
E. 7,400
111.The Gotemba walking trail up Mount Fuji is about 9 km long. Walkers need to return
from the 18km walk by 8pm. Toshi estimates that he can walk up the mountain at 1.5km/h on
average, and down at twice that speed. These speeds take into account meal breaks and rest
times. Using Toshi's estimated speeds, what is the latest time he can begin his walk so that he
can return by 8pm?
A. 8:30 am
B. 9:30 am
C. 9:00 am
D.10:00 am
E.11:00 am
112.Toshi wore a pedometer to count his steps on his walk along the Gotemba trail. His
pedometer showed that he walked 22,500 steps on the way up. Estimate Toshi's average step
length for his walk up the 9km Gotemba trail. Give your answer in centimeters.
A. 30 cm
B. 40 cm
C. 25 cm
D. 35 cm
E. 45 cm
113.If 3
3 3
log
log log
b
a b
a
b a
=
+
⊙ for all nonnegative numbers ,what is the value of 1 2017⊙ ?
Α. 1
B. 0
C. 2017
D.2018
E. 3

Athanasios Drougas
25
114.A revolving door includes three wings which rotate within a circular-shaped space. The
inside diameter of this space is 2 meters (200 centimeters). The three door wings divide the
space into three equal sectors. The plan below shows the door wings in three different
positions viewed from the top.
The door makes 4 complete rotations in a minute. There is room for a maximum of two
people in each of the three door sectors.
What is the maximum number of people that can enter the building through the door in 30
minutes?
A. 60
B. 180
C. 300
D. 240
E. 720
115.What is the slope of the line through (-2,4) and (-3,9) in the standard(x,y) coordinate
plane?
A.5
B.-5
C.6
D.
7
5
E.
7
5
−
116.The figure below consists of congruent squares and has total area 100 square feet. What
is the outside perimeter of the figure?
A.80
B.40
C.50
D.20
E.60

Athanasios Drougas
26
117.According to the measurements given in the figure below, which of the following
expressions give the distance, in miles from the dog to the cat.
A.30cos52o
B.30tan52o
C.30sin52o
D. 0
30
sin52
E. 0
30
cos52
118.The sum of the real numbers x and y is 36.Their difference is 5.What is the value of xy?
A.3
B.6
C.9
D.36
E.41
119.Each side of the smaller square in the figure below is x inches long, and each side of the
larger square is y inches longer than a side of the smaller square. The area of the larger square
is how many square inches greater than the area of the smaller square?
A. y2
B. xy
C. 4y(x + y)2
D. 2xy + y2
E. 2xy+ x2
x
x
y
y
30 miles 52o
cat
dog
Dog house

Athanasios Drougas
27
120.On the number line above ,there are 9 equal intervals between 0 and 1.What is the
value of x?
A.
1
9
B.
1
81
C.
1
27
D.
1
3
E. 1
121.A cube with edges
1
2
inch long is shown below. What is the length, in inches, of a
diagonal that runs from one corner of the cube to the opposite corner?
A.
1
4
B.
3
2
C.
3
4
D.
2
2
E.
3
2
1
2
inch
1
2
inch
1
2
inch
0 1
x

Athanasios Drougas
28
122.In the 8–sided figure below, adjacent sizes meet at right angles and the lengths given
are in meters.What is the perimeter of the figure, in meters?
A.90
B.120
C.180
D.200
E. 400
123. When
1 1
1
5 3
a a+ = ,what is the value of k?
A.
1
15
B.
8
15
C.
15
8
D. 8
E. 15
124.Hexagon ABCDFG shown below was drown on a grid with unit squares.Each vertex is at
intersection of 2 grid lines .What is the area of hexagon?
A. 62
B. 63
C.64
D.65
E.66
40
38
12

Athanasios Drougas
29
125.In the figure above, what is the median of the slopes of OA,OB ,OC ,OD and OE ?
Α.
1
4
B.
1
2
C.1
D. 2
E.
2
4
126. The hair color of 180 people were determined for a study as shown in the figure below.
If one person from the study is randomly selected, what is the probability that this person has
either red or blond or brow hair.
Α.
77
180
B.
80
180
C.
103
180
D.
23
180
E.
134
180
0
B C D EA
2
1 2 3 4 5
1
6 7 8
3
4
5
6
7
x
y
77
20
10
30
40
50
60
70
80
black brow
n
red blond Hair color
Number
of people

Athanasios Drougas
30
127.The median of a set of 11 consecutive integers 23.What is the greatest of these 11
integers?
Α.26
B.27
C.28
D.29
E.30
128.Let the function f defined by f(x)=x-1.If 3f(k)=3,what is the value of f(5k)?
Α.2
B.3
C.4
D.5
E.6
129.In the figure above, what is the median of the slopes of GA,GB ,GC ,GD and GE ?
Α.5
B.4
C.3
D.2
E.1
130.Two identical jars A and B are filled with equal numbers of marbles .The marbles are
colored black or white. The ratio of red to white marbles is 7:1 in Jar A and 9:1 in Jar B. If
there are 90 white marbles all together, how many black marbles are in Jar B?
F. 180
G. 300
H. 360
J. 400
K. 420
G
2
1 2 3 4 5
1
6 7 8
3
4
5
6
7
x
y
D
C
B
A
E
0

Athanasios Drougas
31
131.A cube is illustrated in the diagram below. In this cube what is the measure of ABC∠ ?
F. 90o
G. 60o
H. 30o
J. 120o
K. 45o
132.If A is the set of all odd integers ,B is the set of positive integers and C is the set of all
integers more than 8 which of the following integer will be in all three sets?
F. -10
G. 10
H. 7
J. 12
K. 9
133.If 7 19w+ = then ?w =
F. 9
G. 90
H. 100
J. 81
K. -90
134.Given that the number 1948a8 is divisible by 18,what are the possible values for a?
F.3
G.4
H.5
J.6
K.7
135.The range of values for a in the triangle shown below is equal to:
Α. 11a <
B. 6 11a< <
C.5 11a< <
D. 5 12a< <
E. 6 12a< <
B
A
C
3 8
α

Athanasios Drougas
32
136. 17 17 17 17
2 2 2 2 ?+ + + =
F. 18
2
G. 19
2
H. 20
2
J. 21
2
K. 1028
137. OEC is a quarter circle. ABCO is a rectangle with sides AB = 4 and OA = 3. What is the
length of the arc EBC ?
A. 5π
B. 10π
C. 2.5π
D. 16
E. 14
138. The smallest positive integer value for x which
6
1x −
yields an positive integer is equal
to:
A.0
B.1
C. 2
D.4
E. 6
139. In the figure below the ratio a:b is 1:3.What is the diagonal in terms of b?
A. 12b
B.
10
3
b
C. 10b
D.
2
3
b
E.
3
b
O
A
E
D
C
B
α
b

Athanasios Drougas
33
140. In the figure below ,what is the ratio of ABC∠ to DBC∠ ?
A.2:7
B.2:6
C.7:2
D.6:7
E. 3:7
141. In the figure below ,what is value of x?
A.110o
B.100o
C.30o
D.40o
E.50o
142.If 3 3
( ) 5 5f x x x= − − ,what is the value of 3 3
( 1) 5 5f x x− = − −
A.10
B.0
C.-10
D.25
E.-25
143.If 2 2
5x y+ = and ( )
2
9y x− = what is the value of xy ?
A.1
B.2
C.-2
D.3
E.4
3α+20ο
2α-40ο
Α
Β
C
D
70o
x0
A
C B

Athanasios Drougas
34
144.If
2
1
25a
a
 
+ = 
 
what is the value of 2
2
1
a
a
+ ?
A. 5
B.27
C. 23
D. 7
E. 3
145.If
1
( )
1
f x
x
x
=
+
what is
1
f
x
 
 
 
?
A. 1
B. ( )f x
C. ( )f x−
D.
1
( )f x
E.
1
( )f x
−
146.What is the range of the function 2
( ) 9f x x= −
A. All real numbers
B. All real numbers greater than or equal to 0.
C. All real numbers greater than or equal to 3.
D.All real numbers greater than or equal to 3 or less than or equal to -3..
E. All real numbers between -3 and 3 ,inclusive.
147.What is the domain of the function 2
( ) 16f x x= − −
A. All real numbers
B. The empty set.
C. All real numbers greater than or equal to 4.
D. All real numbers greater than or equal to 4 or less than or equal to -4.
E. All real numbers between -4 and 4 ,inclusive.
148.What is the domain of the function
4
3
( 4)
( )
( 4)
x
f x
x
−
=
−
?
A. All real numbers
B. { : 4 4}x x− < <
C. { : 4}x x ≠
D. { : 4}x x ≠ −
E. { : 4 4}x x or x< − >

Athanasios Drougas
35
149.The figure below shows the graph of an equation y=αx2
+bx+c .Which of the answer
choices CANNOT be true?
A. a b>
B. a c<
C. a b=
D. b c<
E. a c>
150. Which of the following function has a range of ( ) 2f x ≥
A. ( ) 2f x x= +
B. ( ) 2f x x= −
C. ( ) 2 2f x x= − +
D. ( ) 2 2f x x= − −
E. ( ) 2 2f x x= + −
151. Of the 133 employees of a factory ,9 are women. What is the ratio of workers who are
women to those who are men?
A. 1:11
B. 1:12
C. 1:13
D. 1:14
E. 1:15
y
x

Athanasios Drougas
36
152.In the figure above ,each of the four angles in the parallelogram is as shown.What is
the value of c?
A. 40o
B. 60o
C. 120o
D. 80o
E. 50o
Solution
153.For the real numbers x and y ,when is the equation 0x y x y+ + − = true?
A. only x=0
B. only y=0
C. always
D. x=0 or y=0
E. x=0 and y=0
0
2 100a −
α
Α Β
C
D
α-20ο
b c

Athanasios Drougas
37
154.As shown below, rectangle ABCD
is divided into 1 large square (labeled
K) x inches on a side , 10 rectangles
(Labeled R) each x inches by y inches
and 6 small squares(labeled M ) each
side 2y inches .What is the total area,
of ABCD?
A.x2
+10xy+4y2
B. x2
+10xy+6y2
C. 2x+10xy+6y2
D. x2
+10xy+24y2
E. 2x+10xy+6y2
155.What are the Quadrants of the standard (x,y) coordinate plane below that contain
points on the graph of the equation 6x+3y=3
A.I and III only
B. I and II and III only
C. II and III only
D. II and III and IV only
E. I and II and IV only
156.For 2
1i = − ,( )
2
1 ?i− =
A. 0
B. 2
C. 2i
D. 2i−
E. i−
K
E
E
E
E
E
E
B
CD
A
M M
M M
M M
E
E E E
III
III IV
Quadrants of the
standard (x,y)
coordinate plane

Athanasios Drougas
38
157. The graph of y=-2x2
+3 passes through (2, )
5
a
in the standard (x,y) coordinate plane .
What is the value of a?
A.5
B.-5
C.25
D.-25
E.-20
158. If 22x
=44x-3
,what is the value of x?
A.2
B.1
C.3
D.4
E.-2
159. What is the average (arithmetic mean) of the numbers
3 2
,
2 3
,and
5
6
A.
1
6
B.1
C.
8
3
D.
5
6
E.2
160. If (x-α)(x-b)=x2
-7x-13,what is the value of
a b
ab
+
?
A.
8
13
B.
13
7
C.
7
13
D. 1
E. it cannot be determined from the given information

Athanasios Drougas
39
161. If (x-α)(x-b)=x2
-7x-13,what is the value of
a b
ab
+
?
A.
8
13
B.
13
7
C.
7
13
D. 1
E. it cannot be determined from the given information
162.Ann and Bob pick two digits numbers .If you multiply Ann ‘s number with 4 and
subtract Bob’s number the result is 100.If you multiply Bob ‘s number with 6 then add Ann‘s
number the sum is 200.What is the sum of their two numbers ?
A.50
B.100
C.150
D.200
E.250
163.In the figure below ,Square ABCD is inscribed with circle P. If the area of the circle is
12π,what is the area of the square?
A. 2 3
B.24
C.48
D. 24 3
E. 16 3
164.In the figure below the rectangle ABCD consists of two congruent squares(EBCF,AEFD)
.If diagonal BD= 5 .The diagonal BF is equal to:
A. 3
B. 2
C.4
D. 1
E.
5
2
D C
BA
.P
F
E
D C
BA

Athanasios Drougas
40
165.What is the fraction of the square is shaded?
A.
1
3
B.
2
5
C.
5
12
D.
3
7
E.
1
2
166.Figures A1,A2 ,A3 are squares.The perimeter of A3 is 8 and the perimeter of A2 is
20.The perimeter of A1 is :
A.25
B.28
C.32
D.24
E.26
167.If X is 50 percent of 4Y, then Y what percent of X?
A.50%
B.75%
C.100%
D.200%
E.250%
168.For a linear function f, f(0)=2 and f(2)=-4 .If k=f(6) what is the value of k?
A.-13
B.-14
C.-15
D.-16
E.-17
A1
A2
A3

Athanasios Drougas
41
169.If 10% of 10% of a certain number is 2, this certain number is:
A.20
B.100
C.120
D.200
E.300
170.Bob can eat 20 sandwiches every 3 days. How many sandwiches can Bob eat in 12 days
?
A.160
B.120
C.80
D.60
E. 40
170.The average of eleven 11’s is :
A.121
B.11
C.111
D.1
E.131
171.If f(2x)= 4x+1,what is the value of α for which 11
2
a
f
 
= − 
 
?
A.0
B.-1
C.-2
D.-3
E.-4
178.If
log
1000
log
a
b
= the numerical value of
log
log
a
b
b
 
 
  is:
A.900
B.899
C.999
D.1999
E.1000

Athanasios Drougas
42
179.If ( )
2
1x yi i+ = + ,where x,y are real , what is the value of x+y
A.x=1 and y=2
B. x=1 and y=-2
C. x=0 and y=-2
D. x=0 and y=2
E. x=1 and y=0
180.If a and b , are the roots of 2
2 5 4 0x x− + = , what is the value of
2 2log loga b+ ?
A.0
B.1
C.2
D.3
E.4
180.For positive integers a,b,and c let
a
b
c
be defined by 2
2
a
b abc b c
c
= − − .What is the
value of
2
3
4
?
A.0
B.1
C. 2
D. 3
E.4
181. For positive integers a,b, let a#b be defined by 2
a ##b (2 )(2 )a b a b b= − + + .What
is the value of4##2017?
A.0
B. 4
C.8
D.12
E.16

Athanasios Drougas
43
182.If
7
10
0.20
is expressed as a fraction in lowest terms ,what number will be in the
denominator?
A.1
B.2
C.4
D.5
E.10
183.If 180 270o o
θ< < and tanθ=0.75 what is the value of sinθ+cosθ?
A.3
B. -3
C. 1.4
D.-1.4
E.0.71
184.What is the value of 12 x− if 15x = ?
A.-3
B. 27
C. 3
D.-27
E.-9
185. Which of the following is equal to 15 3 ?
A. 5 3
B. 3 5
C. 9 5
D. 3 3
E. 5
186. For the integers a and b such that ab=10, which of the following is NOT possible value
of x?
A. 5−
B. 4−
C. 2−
D. 5
E. 4
187. How many pairs of prime numbers are there whose sum is 825?
A. 0
B. 1
C. 2
D. 3
E. 4

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