500 ACT Math Questions to Know by Test Day, 3rd Edition

›40    McGraw Hill 500 ACT Math Questions

MODELING 72. One gallon of paint covers about 400 square feet. How many gallons of
paint are needed for the project detailed on page 23?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

73. If the coordinates of the point X have the same sign, then X must be
located in which of the four quadrants seen in the following figure?

y

II I

0 x
III
IV

(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) III and IV only

Basic Geometry  ‹  41

74. As shown in the following figure, the angle between lines m and ℓ is
90 degrees. What must the product of their slopes equal?

y
ℓ

x

m

(A) −2
(B) −1
(C) 0
(D) 1
(E) Cannot be determined
75. Which of the following statements must be true regarding the line graphed
in the following figure?

y

x

     I. The slope of the line is positive.
   II. The x-intercept of the line is negative.
III. The y-intercept of the line is positive.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III

›42    McGraw Hill 500 ACT Math Questions

76. A rectangle has a width of 5 meters and a length of 14 meters. If a similar
rectangle has a width of 15 meters, what is its perimeter, in meters?
(A) 42
(B) 58
(C) 60
(D) 78
(E) 114

77. Lines AB and ℓ intersect at point C in the following figure. What is the
value of x in degrees?

ℓ

5x x B
AC

(A) 20
(B) 30
(C) 45
(D) 55
(E) 80

78. In the following figure, lines AB and CD intersect at a point P. If
∠ APD = 80°, what is the measure of ∠ APC in degrees?

D

A PB

C

(A) 10
(B) 100
(C) 170
(D) 200
(E) 280

Basic Geometry  ‹  43

79. Line ℓ is a transversal of the parallel lines m and n as shown in the figure.
The measures of two angles are given in terms of x and y. What is the value
of x in degrees?

y 3x m
n
75°
ℓ

(A) 25
(B) 30
(C) 35
(D) 50
(E) 75

›44    McGraw Hill 500 ACT Math Questions

80. If a line in the (x, y) coordinate plane has x-intercept (x, 0) and y-intercept
(0, y) such that x > y, which of the following could be a graph of the line?

(A) y (B)  y

55

–5
xx
–5 5 5

(C) –5 (D)  –5

y y
5 5

–5 x –5 x
5 –5 5
–5

(E) y
5

–5 x
–5 5

Basic Geometry  ‹  45

81. If lines AB and CD are perpendicular and intersect at point P, what is the
measure of ∠ APB in degrees?
(A) 30
(B) 45
(C) 60
(D) 90
(E) 180

82. If f (x) = ax + 5 and that function passes through (2, 9), then which of the
following will be parallel to f (x)?

(A) g(x) = –2x + 3
(B) h(x) = -x + 5
1
(C) g(x) = - 2 x + 4

(D) h(x) = x + 7
(E) i(x) = 2x - 3

83. The coordinates of a point P are (x, y) such that xy > 0. Given the four
quadrants of the (x, y) coordinate plane shown, which of the following
statements MUST be true?

y

II I

0 x
III
IV

(A) Point P must lie in quadrant I.
(B) Point P must lie in quadrant II.
(C) Point P must lie in quadrant III.
(D) Point P must lie in either quadrant I or quadrant II.
(E) Point P must lie in either quadrant I or quadrant III.

›46    McGraw Hill 500 ACT Math Questions

84. If the x- and y-intercepts of a line in the (x, y) coordinate plane are nonzero
and share the same sign, which of the following statements MUST be true?
(A) The slope of the line is negative.
(B) The slope of the line is positive.
(C) The slope of the line is zero.
(D) The slope of the line is undefined.
(E) The slope of the line is nonzero and defined.

85. Which of the following lines has a slope of zero?

(A) y   (B)  y

xx

(C) y   (D)  y

xx

(E) y

x

Basic Geometry  ‹  47

86. The points A (1, 0), B (1, 8), and C (5, 0) form a triangle in the (x, y)
coordinate plane. In coordinate units, what is the area of triangle ABC ?
(A) 3
(B) 12
(C) 16
(D) 32
(E) 40

87. In the following figure, lines m, n, and ℓ intersect to form angles with the
indicated measures. What is the value of x + y + z?

m
ℓn

y°
x°

z°

(A) 30
(B) 60
(C) 90
(D) 120
(E) 180
88. A circle is drawn in the (x, y) coordinate plane such that the center of the
circle is the point (0, 7). If the radius of the circle is 2, then each of the
following points lies on the circle’s circumference EXCEPT
(A) (−2, 7).
(B) (0, 9).
(C) (0, 5).
(D) (1, 7).
(E) (2, 7).

›48    McGraw Hill 500 ACT Math Questions

89. Which of the following graphs represents a line with a positive slope and a
negative x-intercept?

(A) y   (B)  y

xx

(C) y   (D)  y

xx

(E) y

x

Basic Geometry  ‹  49

90. For a line in the (x, y) coordinate plane, which of the following statements
is always true of the y-intercept?
(A) The coordinates are of the form (c, 0) for some number c.
(B) The coordinates are of the form (0, c) for some number c.
(C) If the slope is positive, the y-intercept is positive.
(D) The larger the slope, the larger the y-intercept.
(E) The origin can never be the y-intercept.

91. In the following figure, triangles ABC and DEF are similar. Which of the
following expressions represents the area of DEF in terms of x and y ?

E

B

y

AxC D 3x F

(A) xy
2

(B) 3xy
2
(C) 3xy

(D) 9xy
2

(E) 6xy

›50    McGraw Hill 500 ACT Math Questions

92. In the (x, y) coordinate plane, if the point (x, -5) lies on the graph of the
line 5y - 2x = -30, what is the value of x ?

(A) -10
(B) -8
23
(C) − 5
(D)
5
2
(E) 15

93. In the following figure, PR, MQ, and NQ are line segments such that
∠PQM = ∠NQR .

MN

PQ R

If the measure of ∠PQM is half the measure of ∠MQN, then the measure
of ∠NQR is
(A) 35°
(B) 45°
(C) 90°
(D) 100°
(E) 120°

94. If the angle between two perpendicular lines has a measure of (2x + 2)°,
then x =
(A) 44.
(B) 45.
(C) 89.
(D) 90.
(E) 91.

Basic Geometry  ‹  51

95. If lines a and b are parallel, then what is the value of x?

2x
b

146° a

(A) 17
(B) 34
(C) 73
(D) 120
(E) 146

96. In inches, what is the perimeter of a rectangle with a width of 4 inches and
a length of 10 inches?
(A) 14
(B) 18
(C) 28
(D) 40
(E) 160

97. In the following figure, the area of square ABCD is 16 square units. What
is the circumference of the circle centered at point B ?

BC

A D

(A) 2π
(B) 4π
(C) 8π
(D) 16π
(E) 32π

›52    McGraw Hill 500 ACT Math Questions

98. The area of a circle is 8π square units. What is the radius of the circle?
(A) 2 2
(B) 4
(C) 4 2
(D) 8
(E) 12

99. The length of the shortest side of a triangle is 10 meters, and the length of
one of the longer sides is 20 meters. If the two larger angles in the triangle
are equal, what is the perimeter of the triangle in meters?
(A) 40
(B) 45
(C) 50
(D) 55
(E) 87

100. The value of a square’s area is twice as large as the value of its perimeter. In
units, what is the length of one side of the square?
(A) 4
(B) 8
(C) 12
(D) 16
(E) 32

101. The length of the diagonal of a rectangle is 20 feet. If the width of the
rectangle is 12 feet, what is its area in square feet?
(A) 28
(B) 144
(C) 192
(D) 256
(E) 400

102. The sides of a triangle have lengths of 3, 3, and 5.2 meters. If one of the
smaller angles in the triangle has a measure of 30°, what is the measure of
the largest angle?
(A) 60°
(B) 90°
(C) 100°
(D) 120°
(E) 140°

Basic Geometry  ‹  53

103. Each of the smallest angles in a parallelogram is 50°. Which of the
following is the measure of each of largest angles in the parallelogram?
(A) 40°
(B) 80°
(C) 130°
(D) 260°
(E) 310°

104. The circles in the following figure are centered at point O. In square
centimeters, what is the area of the shaded region?

6 cm 2 cm
O

(A) 15π
(B) 21π
(C) 32π
(D) 35π
(E) 60π

105. Given the trapezoid in the following figure, what is the value of x ?

125° 105°

x° 75°

(A) 55
(B) 65
(C) 75
(D) 95
(E) 105

›54    McGraw Hill 500 ACT Math Questions

106. In the following figure, the points A, C, and D lie along the same line, and
ABC is a triangle with sides of the indicated length. If the measure of angle
CAB is 85°, what is the measure of angle BCD in degrees?

B

xx

A CD

(A) 85
(B) 95
(C) 105
(D) 115
(E) 120
107. In square units, what is the area of a circle with a circumference of 6π?
(A) 3π
(B) 6π
(C) 9π
(D) 12π
(E) 36π
108. In the following figure, the square ABCD is a single side of a cube with a
volume of 8 cubic inches. In inches, what is the length of the line BD ?

BC

A D

(A) 1
(B) 2
(C) 2
(D) 2 2
(E) 4

Basic Geometry  ‹  55

109. Given the right triangle in the following figure, what is the value of x ?

45°
x

5

(A) 2 2
(B) 5 2
(C) 5
(D) 2 5
(E) 10

110. In inches, what is the radius of a sphere with a volume of 36π?
(A) 1
(B) 3
(C) 6
(D) 9
(E) 12

›56    McGraw Hill 500 ACT Math Questions

111. In the following figure, AB = BC = AC, and BD bisects the angle ABC.
What is the value of x ?

B

x°

A DC

(A) 15
(B) 20
(C) 30
(D) 45
(E) 60

112. In square meters, what is the area of a rectangle with a diagonal of
25 meters and a width of 15 meters?
(A) 200
(B) 250
(C) 275
(D) 300
(E) 375

113. In the following figure, ABCD is a parallelogram such that AB has a
length of 3 centimeters and BC has a length of 7 centimeters. What is the
perimeter of ABCD?

BC

A D

(A) 10
(B) 20
(C) 30
(D) 40
(E) 50

Basic Geometry  ‹  57

114. A rectangular container measuring 4 feet wide, 8 feet long, and 3 feet tall
is filled with a 1-foot-deep layer of sand. In cubic feet, what volume of the
container remains unfilled?
(A) 18
(B) 32
(C) 42
(D) 64
(E) 72

115. Two squares of the same size overlap such that all three of the resulting
rectangles shown in the figure have the same area. If the area of the shaded
rectangle is 2, what is the area of one of the original squares?

(A) 2
(B) 4
(C) 9
(D) 16
(E) 25

MODELING 116. The following figure represents the dimensions of three rooms of a house
that are to be completely carpeted. In total, how many square feet of carpet
will be needed?

20 ft.

40 ft. 20 ft.
60 ft.

20 ft.

10 ft.

(A) 1200
(B) 1300
(C) 1500
(D) 1700
(E) 1800

›58    McGraw Hill 500 ACT Math Questions

117. Triangle ABC in the following figure is an isosceles triangle with height
BD. What is the value of a?

B
a°

40°
A DC

(A) 45
(B) 50
(C) 60
(D) 65
(E) 90
118. The interior angles of a triangle measure x°, (2x)°, and (3x)°. The value of
x MUST be
(A) 30.
(B) 35.
(C) 45.
(D) 50.
(E) 65.
119. In the following figure, the circles centered at points B and D are tangent
at the point C, and each circle has an area of 49π. What is the length of
the line segment AE ?

A B CD E

(A) 14
(B) 28
(C) 42
(D) 48
(E) 56

Basic Geometry  ‹  59

120. If one of the interior angles of a triangle measures 72°, each of the
following pairs could represent the measures of the remaining angles
EXCEPT
(A) 58°, 50°.
(B) 42°, 66°.
(C) 15°, 93°.
(D) 87°, 21°.
(E) 36°, 73°.

121. The length of a rectangle is 5 times its width. If the width is represented by
w, which of the following expressions is the perimeter of the rectangle in
terms of w?
(A) 5w
(B) 6w
(C) 10w
(D) 12w
(E) 14w

122. In the following figure, FBCE is a rectangle, while ABF and ECD are
congruent right triangles. What is the area of the quadrilateral ABCD?

B3C

2

A2 F ED

(A) 6
(B) 8
(C) 10
(D) 12
(E) 16

›60    McGraw Hill 500 ACT Math Questions

123. The perimeter of the parallelogram in the following figure is 20 feet. In
feet, what is the length of the side labeled with an x ?

4

x

(A) 4
(B) 6
(C) 8
(D) 12
(E) 16

124. A rectangle has an area of 45 square meters and a width of 9 meters. In
meters, what is the length of the rectangle?
(A) 5
(B) 7
(C) 14
(D) 27
(E) 36

MODELING 125. A modern-art piece begins as a special canvas with dimensions shown in
the following figure. Initially, the canvas will be painted with a base coat
of flat paint. If one tube of the paint can cover four square feet, how many
tubes will the artist need in order to apply the base coat?

B 12 ft. C

A 2 ft.
2 ft. D

2 ft.

(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

Basic Geometry  ‹  61

126. A circle has a diameter of length d and an area of A. If the diameter of the
circle is tripled, the area of the new circle is B. How many times larger than
A is B ?
(A) 3
(B) 6
(C) 9
(D) 12
(E) 18

127. When completely filled, a tshpeheraridciaulsboaflltohoenbcaollnotoaninws ehxeanctiltyis3c23oπmcpulebtiecly
feet of air. In feet, what is
filled?

(A) 2

(B) 46
3

(C) 4

(D) 2 6
(E) 8

128. In the following figure, the points A, B, C, and D lie along the same line.
What is the difference of x and y ?

20°
55

x° y°
AB CD

(A) 0
(B) 20
(C) 60
(D) 160
(E) 200

›62    McGraw Hill 500 ACT Math Questions

129. In square meters, what is the area of a single face of a cube that has a
volume of 125 cubic meters?
(A) 2 3
(B) 5
(C) 5 5
(D) 25
(E) 63

130. Which of the following expressions represents the area of a right triangle
with legs of lengths x and y ?

(A) xy
2

(B) x2 y2
2

(C) x+ y
2

(D) x2 − y2
2

(E) x2 + y2
2

131. In the following figure, ABCD is a rectangle, and BD is one of its
diagonals. If the area of the triangle ABD is 6 square meters, what is the
area of ABCD in square meters?

BC

A D

(A) 3
(B) 6
(C) 12
(D) 18
(E) 24

Basic Geometry  ‹  63

132. In feet, what is the perimeter of a square that has an area of 49 square feet?
(A) 14
(B) 28
(C) 56
(D) 98
(E) 196

133. In the following figure, ABCD is a rectangle, and PQR is an equilateral
triangle. Given the provided measurements are in inches, what is the area
of the rectangle ABCD in square inches?

BQC

A 1P 2 R1D

(A) 3
(B) 2 3
(C) 5
(D) 8
(E) 4 3
134. If x ≠ y, which of the following expressions represents the perimeter of the
triangle in the following figure in terms of x and y ?

x

a° a°
y

(A) x + y
(B) x + y + 2
(C) 2x + y
(D) x + 2y
(E) 2(x + y)

›64    McGraw Hill 500 ACT Math Questions

135. The parallelogram in the following figure has a perimeter of 32 feet. If x is
three times as large as y, what is the value of x in feet?

y

x

(A) 2
(B) 4
(C) 8
(D) 12
(E) 24
136. In the following figure, ABC is a triangle such that M is the midpoint of
AB and N is the midpoint of AC. If the area of triangle ABC is 12, what is
the length of side AB ?

B

M

A3 N C

(A) 2
(B) 4
(C) 6
(D) 8
(E) 10

Basic Geometry  ‹  65

137. The sides of the rhombus ABCD in the following figure all have the same
length, and the dashed line AC is a diagonal of ABCD.

B
x°

AC

D

If the measure of ∠BCD = 60°, then x =
(A) 30.
(B) 40.
(C) 60.
(D) 120.
(E) 180.
138. A circle is divided into 8 equal sections, as shown in the following figure.
What is the value of angle x in degrees?

x°

(A) 25
(B) 36
(C) 45
(D) 50
(E) 55

139. Which of the following values represents the circumference of a circle
whose radius is 1 ?
π
1
(A) 2

(B) 1
(C) 2
(D) π
(E) 2π

›66    McGraw Hill 500 ACT Math Questions

140. If the interior angles of a quadrilateral have measures x°, (2x)°, (x + 45)°,
and (x + 55)°, then x =
(A) 16.
(B) 23.
(C) 52.
(D) 65.
(E) 78.

3CHAPTER 

Integrating Essential Skills:
Average, Median, and Expressing

Numbers in Different Ways

MODELING 141. In a small graduate class of five students, Juan’s exam score was
accidentally deleted from the gradebook. The professor recalls that the
exam average was 84%. Which student earned the median exam score?

Student Exam Score

George 76%
Tamara 88%
Juan ?
Sarita 75%
Alysia 92%

(A) George
(B) Tamara
(C) Juan
(D) Sarita
(E) Alysia

142. What is the greatest common factor of 30, 90, and 130?
(A) 2
(B) 3
(C) 5
(D) 10
(E) 30

  ‹  67

›68    McGraw Hill 500 ACT Math Questions

143. What is the median of 2, 5, 3, 1, and 5 ?

(A) 2
(B) 5
(C) 3
(D) 1
(E) 5

144. Which of the following has the highest median value?
(A) 10, 13, 15, 59, 92
(B) 14, 19, 22, 23, 25
(C) 4, 3, 5, 20, 22
(D) 1, 5, 16, 38, 100
(E) −3, 0, 5, 11, 19

145. Which of the following must be true whenever a, b, and c are integers such
that a < 0 and b > c > 0?
(A) a − c > 0
(B) c − b < 0
(C) a + c > 0
(D) −a − c < 0
(E) a + b > 0

146. What is the largest integer smaller than 50 ?
(A) 4
(B) 6
(C) 7
(D) 9
(E) 12

147. The sum of 10 numbers is 250. What is the average of the 10 numbers?
(A) 25
(B) 50
(C) 110
(D) 125
(E) 240

‹Average, Median, and Expressing Numbers in Different Ways    69

148. If m and n are integers, which of the following must be an even integer?
(A) 2mn
(B) mn
(C) mn + 2
(D) mn − 2
(E) 3mn

149. If the equation x − a = y − a is true, then which of the following must be
true?

(A) x = y
(B) x > a
(C) x + y ≠ 0
(D) y < a
(E) −a + y = −a + x

150. If P is the set of multiples of 2, Q is the set of multiples of 3, and R is the
set of multiples of 7, which of the following integers will be in P and Q
but not in R ?
(A) −54
(B) −50
(C) 42
(D) 100
(E) 252

151. Using the following number line, which pairs of points satisfy the equation
1?
x− y < 2

A B CD EF

–4 –3 –2 –1 0 1 2 3 4

(A) x = A, y = B

(B) x = B, y = C
(C) x = C , y = D
(D) x = D, y = E
(E) x = E , y = F

›70    McGraw Hill 500 ACT Math Questions

152. What is the average of the numbers 5 , 1 , and x ?
4 2 2

(A) 2x + 7
12

(B) 4x + 9
12

(C) x+6
6

(D) 2x + 7
4

(E) x+9
4

153. If 3(m + n) is even, then which of the following must also be even?

(A) m + n
(B) m
(C) n
(D) 3m
(E) 3n + 1

154. The sum of seven consecutive numbers is 301. What is the value of the
largest of these seven numbers?
(A) 38
(B) 40
(C) 43
(D) 46
(E) 52

155. There are 45 students signed up for the performance band, while 30 are
signed up for the jazz band. If 19 students are signed up for both bands,
how many students are signed up for only one of the bands?
(A) 11
(B) 16
(C) 37
(D) 56
(E) 75

‹Average, Median, and Expressing Numbers in Different Ways    71

156. Which of the following inequalities represents the same set as the set
graphed on the number line shown?

–1 0 1 2 3 4 5

(A) 0 < x < 3
(B) 0 ≤ x ≤ 3
(C) −1 < x < 5
(D) −1 ≤ x < 3
(E) 0 ≤ x < 3

157. If the following numbers were ordered from smallest to largest, which
would be in the third position?

 1  −2 1
 2  3
, , 17 , π, 3

(A)  1  −2
 2 

(B) 1
3

(C) 17
(D) π
(E) 3

158. If p and q are real numbers such that ( pq)2 = 4, then which of the
following statements must also be true?
(A) pq > 0
(B) pq = 2
(C) p > 1 and q > 1
(D) If p < 0, then q > 0
(E) If p > 2, then q < 2

159. What is the median of the following set of numbers?
{1, 1, 9, 4, 6, 2}
(A) 1
(B) 2
(C) 3
(D) 4
(E) 6

›72    McGraw Hill 500 ACT Math Questions

160. What is the value of −2x − 5 − −4 + 8 when x = 7?
(A) −12
(B) −4
(C) 7
(D) 15
(E) 23

161. For positive integers x and y, x + y = 21. What is the smallest possible value
of xy?

(A) 10
(B) 20
(C) 38
(D) 54
(E) 110

162. For a real number x, the expression x is an integer. Each of the following
5
expression must also be integers EXCEPT

(A)  x  2
 5 
.

(B) 10 x .

(C) x .
25

(D) 6x .
5

(E) − x .

163. What is the value of 48 − 27 ?
(A) 0
(B) 2
(C) 3

(D) 11
(E) 21

‹Average, Median, and Expressing Numbers in Different Ways    73

164. Which of the following values of x satisfies the inequality 26 < x < 50 ?
(A) 3
(B) 5
(C) 6
(D) 9
(E) 10

165. −2 −1+ 5 =
(A) −8
(B) −6
(C) 2
(D) 6
(E) 8

166. Which of the following is a rational number?

(A) 2
2

(B) ( 2 − 2)2

(C) 100
5

(D) 3

(E) (3 + 3 )2

167. If x and y are real numbers such that 0 < x < y2, then which of the
following inequalities must be true?

(A) x < y
(B) y > 0
(C) 3x > y2
(D) x < 1
(E) x < 3 y2

168. What is the value of x + (−x)2 when x = −1?
(A) −1
(B) 0
(C) 1
(D) 2

(E) 3

›74    McGraw Hill 500 ACT Math Questions

169. What is the smallest integer larger than 26 ?
(A) 1 5
(B) 2
(C) 4
(D) 5
(E) 6

170. The average of three numbers is 18. The sum of the larger of these three
numbers is 38. What is the value of the smallest of these three numbers?
(A) 12
(B) 14
(C) 16
(D) 17
(E) 19

171. Suppose that m is an even integer and n is an odd integer. Which of the
following expressions must be an odd integer?
(A) 2(m + n)
(B) 2(m + n) + 1
(C) mn
(D) 2(mn)
(E) 3(mn)

172. Let P represent the set of numbers with a factor of 5, and let Q represent
the set of numbers that are less than 10 but larger than 0. Which of the
following numbers is in both sets?
(A) −10
(B) −5
(C) 2
(D) 5
(E) 8

173. What is the least common multiple of 12, 20, and 40?
(A) 60
(B) 120
(C) 480
(D) 800
(E) 9600

‹Average, Median, and Expressing Numbers in Different Ways    75

174. If p and q are real numbers such that p > 5 and q > 4, what is the smallest
integer larger than the product pq?
(A) 9
(B) 10
(C) 19
(D) 20
(E) 21

175. A number has three prime factors: 2, 3, and 7. Which of the following is a
possible value of the number?
(A) 12
(B) 14
(C) 49
(D) 81
(E) 84

MODELING 176. There are 53 people signed up for a race on an upcoming Saturday, and
there are 21 people signed up for a race on an upcoming Sunday. If 12
people signed up for both races, how many people are signed up for only
one race?
(A) 9
(B) 41
(C) 50
(D) 62
(E) 74

177. The value of  x + y is an integer when y = 4. Which of the following is a
y
possible value of x ?

(A) −16
(B) 2
(C) 9
(D) 15
(E) 26

178. If a set is defined as all numbers satisfying a given inequality, then which of
the following sets would NOT share any values with the set of values for x
satisfying -1 ≤ x < 5?

(A) −5 ≤ x < −1
(B) −5 < x ≤ −1
(C) 0 < x < 5
(D) 0 < x ≤ 5
(E) 0 < x ≤ 5

›76    McGraw Hill 500 ACT Math Questions

179. Let x  y = y for real numbers x and y. For what values of x is the value of
x
x  (−1) always positive?

(A) x > 0
(B) x < 0
(C) x > −1
(D) x < 1
(E) x > 1

180. If m is the number of distinct prime factors of 100, what is the value of
3m?
(A) 3
(B) 6
(C) 9
(D) 12
(E) 15

181. For any real number a, (6a5 )2 =
(A) 12a7.
(B) 36a7.
(C) 12a10.
(D) 36a10.
(E) 8a3.

182. If the average of 12 numbers is m, which of the following expressions
represents the average of these 12 numbers and 5?

(A) m + 5
(B) m + 13
(C) 13m + 5
m+5
(D) 13

(E) 12m + 5
13

‹Average, Median, and Expressing Numbers in Different Ways    77

183. Using the following number line, which of the following statements must
be true?

xy

–2 –1 0 1 2

(A) x − y < 1

(B) x − y = 1
(C) x − y > 1

(D) x− y < 1
2
1
(E) x− y = 2

184. The following table represents the number of town residents who live in
each district. If the total number of residents in the town is 650, how
many residents live in district 2?

District Number of Residents

1 147
2 ?
3
4 335
98

(A) 42
(B) 70
(C) 315
(D) 503
(E) 552

185. The average of five numbers is 12.4. The average of four of these numbers
is 11. What is the value of the fifth number?
(A) 1.4
(B) 17.0
(C) 18.0
(D) 44.0
(E) 62.0

›78    McGraw Hill 500 ACT Math Questions

186. Which of the following values of x satisfies the inequality 3 < x < 52 ?
2
9
(A) 2

(B) 20
2

(C) 25
2

(D) 29
2

(E) 31
2

187. If xy2 < 0, then which of the following statements must be true?

(A) The value of x is negative, while the value of y is positive or negative.
(B) The value of y is negative, while the value of x is positive or negative.
(C) The sign of x must be the same as the sign of y.
(D) The sign of x must be different from the sign of y.
(E) The values of x and y must be the same.

188. Which of the following is equivalent to | x | when x > 1?
(A) x x

(B) x −1
(C) −1
(D) 0
(E) 1

189. The remainder when an integer m is divided by 2 is 1. What is the
remainder when (m + 1) is divided by 2?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

‹Average, Median, and Expressing Numbers in Different Ways    79

190. The median of the following data set is 2. Which of the following is a
possible value of x ?

x, 2, 1, 6, 7

(A) 2
(B) 3
(C) 5
(D) 6
(E) 8
x y
 1   1 
191. What is the value of 3 − 2 when x = 2 and y = −1?

(A) − 17
9

(B) − 7
4

(C) − 1
9

(D) 1
8

(E) 2
3

192. If the fraction a is in simplified terms, which of the following CANNOT
25
be a factor of a ?

(A) 1
(B) 3
(C) 5
(D) 9
(E) 14

193. If the value of  4m is smaller than 1, which of the following MUST be
true ? 4n

(A) m = 1
(B) n = 1
(C) m = n
(D) m < n
(E) m > n

›80    McGraw Hill 500 ACT Math Questions

194. If x > 0, what is the median of the data set consisting of −2x, −5x, x, −8x,
and 4x ?
(A) −2x
(B) −5x
(C) x
(D) −8x
(E) 4x

195. If 3 and 2 are factors of a, which of the following must be a factor of 14a ?
(A) 4
(B) 5
(C) 8
(D) 11
(E) 19

196. The average of 5 numbers is 20. What is the sum of these same numbers?
(A) 4
(B) 5
(C) 15
(D) 25
(E) 100

MODELING 197. A professor kept track of the attendance at his Monday–Wednesday–Friday
class for one week. The average daily attendance was 32. How many
students attended his class on Friday?

Day Attendance

Monday 32
Wednesday 34
Friday ?

(A) 29
(B) 30
(C) 31
(D) 32
(E) 33

‹Average, Median, and Expressing Numbers in Different Ways    81

MODELING 198. To earn a commission for the workweek (Monday through Friday), a
salesperson must have average daily sales of $250 or greater for the week.
If a salesperson’s sales for Monday through Thursday are $98, $255, $175,
and $320, what is the LEAST whole number value of sales the salesperson
needs to have on Friday to be eligible to earn a commission?
(A) $250
(B) $299
(C) $402
(D) $1250
(E) It is not possible for the salesperson to earn a commission this week.

199. How many three-digit numbers have an odd number as a tens digit?
(A) 25
(B) 200
(C) 450
(D) 500
(E) 620

MODELING 200. The average daily rainfall for the past six days is 3.22 inches. How many
inches of rain must fall on the seventh day for the average daily rainfall
over the past week to be 3.40 inches?
(A) 2.76
(B) 3.25
(C) 3.97
(D) 4.48
(E) 4.72

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4CHAPTER 

Preparing for Higher Math:
Number and Quantity

MODELING 201. The diagram below shows an electric circuit with two batteries and
four devices called resistors. I1, I2, and I3 are symbols that represent
the values for the electric current in each branch.

I1

10 Ω 10 V
I2

20 Ω 40 V
I3

12 Ω 18 Ω

After applying energy and charge conservation, the following three
independent equations may be written:

I1 + I2 + I3 = 0
10I1 + 10 − 20I2 = 0
−18I3 + 40 − 12I3 + 20I2 − 10 = 0

  ‹  83

›84    McGraw Hill 500 ACT Math Questions

Which of the following matrix equations best models this problem?

1 1 1   I1   0 
 −20 0   I2   −10 
(A)  10   I3  =  −30 
 0 20 −30     
 

1 1 1   I1   0 
(B)  −20 0   I2   10 
 10   I3  =  30 
 0 20 −30     
 

1 1 1   I1   0 
(C)  −20 10   I2   10 
 0   I3  =  30 
 −30 20 0     
 

1 0 −30   I1   0 
(D)    I2   10 
 1 −20 20   I3  =  30 
 1 10 0     
 

1 0 −30   I1   0 
   I2   −10 
(E)  1 −20 20   I3  =  −30 
10 0     
 1  

202. Gif irve=n stw. o vectors r =  2x   and   s =  x −4  find the values of x and y
 3y   9 
  

(A) x = 2, y = 3
(B) x = 3, y = −4
(C) x = −4, y = 3
(D) x = −4, y = 9
(E) x = 4/3, y = −3

Number and Quantity  ‹  85

203. Which of the following is equivalent to 5 5 1?
−

(A) 5+ 5
4

(B) 5− 5
5

(C) 1+ 5

(D) 5+ 5
6

(E) 1− 5

204. If a, b, and c are positive integers such that x = a3cand y = b2c, then xy =
(A)  ab6c
(B) ab5c
(C) (a3b2 )c
(D) (ab)6c
(E) (ab)6c2

205. What is the product of the complex numbers (4i − 4) and (−2i + 4)?
(A)  2i
(B) 24i
(C) 24i − 24

(D) 8i2 − 16
(E) 24i − 8

206. What is the difference between 2.17 and 2.17? (Note: A bar over a set of
numbers indicates that a digit pattern is repeated.)
(A) 0
(B) 0.17
(C) 0.0017
(D) 0.0017
(E) 0.0017

›86    McGraw Hill 500 ACT Math Questions

207. Which of the following is equivalent to x − 3 + x ?
x + 5 5

(A) 2x − 3
x + 10

(B) x2 − 3x
5x + 25

(C)   x2 − 3x
5x + 5

(D) 2x −3
5( x + 5)

(E) x 2 + 10x − 15
5(x + 5)

208. If 30 = (1− n)−3, then n =
(A) −3
(B) −1
(C) 0
(D) 1
(E) 2

209. 1+ −(−5)2 =
(A) −4
(B) 6
(C) 26
(D) 1 + 5i
(E) 1 + 25i

( ) 210. If2− 1 4y 1
2 8
= , then y =

(A) −4
1
(B) − 8

(C) 7
8

(D) 3
2

(E) 21
16

Number and Quantity  ‹  87

211. If m is directly proportional to n2 and m = 2 when n = −1, then which of
the following expressions represents the value of m in terms of x when
n = x + 5?
(A) 2x + 5
(B) 2x + 10
(C) 2x2 + 10
(D) 2x2 + 10x + 25
(E) 2x2 + 20x + 50

212. Which of the following numbers is an imaginary number?
(A) −32
(B) (−5)2
(C) 5
(D) −49
(E) − 36

213. If m−3 = 64, then 8m =
(A) −192.
(B) −24.
(C) 14.
(D) 2.
(E) 32.

214. For a real number m, (2m3)4 =
(A) 2m7.
(B) 2m12.
(C) 8m7.
(D) 16m7.
(E) 16m12.

›88    McGraw Hill 500 ACT Math Questions

215. For any real numbers a and b, (10a2 )(5ab)(2ab4 ) is equivalent to
(A) 100a4b5.
(B) 100a2b4.
(C) 17a4b5.
(D) 17a2b4.
(E) 17a9b9.

216. Which of the following is equivalent to i2, where i is the imaginary
number?
(A) −1
(B) 0
(C) 1
(D) −1

(E) −12

217. Given M =  2 −1   and N =  1 0 , what is the matrix M − 4N ?
 5 3   0 1 
  

   2 −1 
(A)  5 
3 

(B) 1 0
 
 0 1 

(C)  −8 0
 
 0 12 

(D)  −2 −5 
 0 
 −1 

(E)  −2 −1 
 5 
 −1 

Number and Quantity  ‹  89

218. Given 6  x  = 3  4  , find the value of y − x.
 y   10 
   

(A) 0
(B) 3
(C) 5
(D) 6
(E) 10

219. Find the magnitude of vector r =  4  .
 −3 
 

(A)    4 
 3 


(B)   −4 
 3 


(C) 1
(D) 5

(E) 7

220. If 2−m = 1 , what is the value of m?
8
(A) −4
(B) −3
(C) 1
(D) 3
(E) 4

221. What is the value of 9a - a2 when a = 10?
(A) −90
(B) −10
(C) 70
(D) 81
(E) 90