ACT Math - Full-Length Practice Test 3

Calculator is allowed on all questions. Figures are not necessarily drawn to scale.

Question 1

If \(7x+4=46\), what is \(x\)?

A) 6
B) 5
C) 4
D) 7

Show solution

\(7x=42\), so \(x=6\).

Answer: A

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Question 2

A jacket priced at \$80 is discounted by \(25\%\). What is the sale price?

F) $20
G) $60
H) $55
J) $65

Show solution

The discount is \(0.25(80)=20\).

The sale price is \(80-20=60\).

Answer: G

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Question 3

The rectangle shown has length 15 and width 8. What is its area?

A) 23
B) 46
C) 120
D) 240

Show solution

\(A=15(8)=120\).

Answer: C

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Question 4

What is the mean of \(6,8,10,12,14\)?

F) 8
G) 9
H) 11
J) 10

Show solution

The sum is 50, and \(50/5=10\).

Answer: J

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Question 5

Which expression is equivalent to \(5(2y-3)+y\)?

A) \(11y-15\)
B) \(10y-15\)
C) \(11y-3\)
D) \(6y-15\)

Show solution

\(5(2y-3)+y=10y-15+y=11y-15\).

Answer: A

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Question 6

If \(f(x)=x^2+4\), what is \(f(5)\)?

F) 21
G) 29
H) 25
J) 45

Show solution

\(f(5)=25+4=29\).

Answer: G

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Question 7

A spinner has 8 equal sections, 3 of which are shaded. What is the probability of landing on a shaded section?

A) \(\frac18\)
B) \(\frac58\)
C) \(\frac38\)
D) \(\frac35\)

Show solution

Use \(P(\text{event})=\frac{\text{favorable outcomes}}{\text{total outcomes}}\).

Three of 8 equally likely sections are shaded, so \(P(\text{shaded})=\frac38\).

Answer: C

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Question 8

What is \(3^2\cdot3^3\)?

F) 27
G) 81
H) 729
J) 243

Show solution

\(3^2\cdot3^3=3^5=243\).

Answer: J

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Question 9

The line shown passes through \((-2,-1)\) and \((2,1)\). What is its slope?

A) \(\frac12\)
B) \(\frac14\)
C) 1
D) 2

Show solution

\(m=\frac{1-(-1)}{2-(-2)}=\frac24=\frac12\).

Answer: A

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Question 10

A circle has radius 10. What is its area?

F) \(10\pi\)
G) \(100\pi\)
H) \(50\pi\)
J) \(20\pi\)

Show solution

\(A=\pi r^2=100\pi\).

Answer: G

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Question 11

A geometric sequence begins \(3,6,12,\ldots\). What is the fourth term?

A) 15
B) 18
C) 24
D) 36

Show solution

For a geometric sequence, use \(a_n=a_1r^{n-1}\).

The common ratio is \(6/3=2\).

Substitute \(a_1=3\), \(r=2\), and \(n=4\): \(a_4=3(2)^{4-1}=3(2^3)=24\).

Answer: C

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Question 12

Which value solves \(4-x=11\)?

F) -15
G) 15
H) 7
J) -7

Show solution

\(-x=7\), so \(x=-7\).

Answer: J

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Question 13

Which best describes the scatterplot shown?

A) Positive association
B) Negative association
C) No association
D) Perfectly constant

Show solution

The points generally rise as \(x\) increases, indicating a positive association.

Answer: A

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Question 14

If \(g(x)=4x-1\), what is \(g(2)\)?

F) 5
G) 7
H) 8
J) 9

Show solution

\(g(2)=8-1=7\).

Answer: G

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Question 15

What is the volume of a rectangular prism with dimensions 3, 4, and 5?

A) 12
B) 20
C) 60
D) 47

Show solution

\(V=3(4)(5)=60\).

Answer: C

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Question 16

Solve \(4(3x-2)=2x+32\).

F) 2
G) 3
H) 5
J) 4

Show solution

\(12x-8=2x+32\), so \(10x=40\) and \(x=4\).

Answer: J

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Question 17

If \(x^2+3x-18=0\), what is the positive solution?

A) 3
B) 2
C) 6
D) 9

Show solution

\((x+6)(x-3)=0\), so the positive solution is 3.

Answer: A

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Question 18

The parabola shown has \(x\)-intercepts \(-2\) and \(2\). Which equation could represent it?

F) \(y=x^2+4\)
G) \(y=x^2-4\)
H) \(y=(x-2)^2\)
J) \(y=-x^2+4\)

Show solution

Zeros at \(-2\) and 2 give \((x+2)(x-2)=x^2-4\), and the graph opens upward.

Answer: G

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Question 19

A data set has mean 32. If every value is multiplied by 2, what is the new mean?

A) 32
B) 34
C) 64
D) 102

Show solution

Multiplying every observation by 2 multiplies the mean by 2, giving 64.

Answer: C

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Question 20

For \(x\ne-5\), which expression is equivalent to \(\frac{x^2-25}{x+5}\)?

F) \(x^2+5\)
G) \(x+5\)
H) \(x^2-5\)
J) \(x-5\)

Show solution

\(x^2-25=(x-5)(x+5)\).

The expression simplifies to \(x-5\).

Answer: J

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Question 21

Lines \(m\) and \(n\) are parallel. If the acute angle shown is \(54^\circ\), what is the measure of an obtuse angle formed by the transversal?

A) \(126^\circ\)
B) \(116^\circ\)
C) \(54^\circ\)
D) \(136^\circ\)

Show solution

An obtuse angle is supplementary to \(54^\circ\): \(180-54=126^\circ\).

Answer: A

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Question 22

If \(2^{x-1}=16\), what is \(x\)?

F) 3
G) 5
H) 4
J) 8

Show solution

\(16=2^4\), so \(x-1=4\) and \(x=5\).

Answer: G

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Question 23

In a class, 14 students study French only, 9 study both French and Spanish, 11 study Spanish only, and 6 study neither. How many students are in the class?

A) 34
B) 36
C) 40
D) 46

Show solution

\(14+9+11+6=40\).

Answer: C

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Question 24

Two similar triangles are shown. What is \(x\)?

F) 12
G) 24
H) 18
J) 15

Show solution

The scale factor is \(10/4=2.5\): \(x=6(2.5)=15\).

Answer: J

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Question 25

If \(f(x)=x+2\) and \(g(x)=3x\), what is \(g(f(4))\)?

A) 18
B) 16
C) 12
D) 24

Show solution

\(f(4)=6\), then \(g(6)=18\).

Answer: A

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Question 26

A \(90^\circ\) sector has radius 6. What is its area?

F) \(6\pi\)
G) \(9\pi\)
H) \(18\pi\)
J) \(36\pi\)

Show solution

\(\frac{90}{360}\pi(6)^2=9\pi\).

Answer: G

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Question 27

If \(P(A)=0.55\), \(P(B)=0.40\), and \(P(A\cap B)=0.20\), what is \(P(A\cup B)\)?

A) 0.35
B) 0.55
C) 0.75
D) 0.95

Show solution

\(P(A\cup B)=0.55+0.40-0.20=0.75\).

Answer: C

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Question 28

Which equation is parallel to \(y=-3x+5\)?

F) \(y=\frac13x+2\)
G) \(y=-\frac13x+4\)
H) \(y=3x+1\)
J) \(y=-3x-7\)

Show solution

Parallel lines have the same slope.

The slope must be \(-3\).

Answer: J

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Question 29

If \(h(x)=5(3^x)\), what is \(h(2)\)?

A) 45
B) 30
C) 15
D) 90

Show solution

\(h(2)=5(9)=45\).

Answer: A

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Question 30

A right triangle has legs 8 and 15. What is the hypotenuse?

F) 16
G) 17
H) 18
J) 23

Show solution

\(\sqrt{8^2+15^2}=\sqrt{289}=17\).

Answer: G

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Question 31

Given \(i=\sqrt{-1}\), what is \((2-3i)(4+i)\)?

A) \(11+10i\)
B) \(5-10i\)
C) \(11-10i\)
D) \(5+10i\)

Show solution

\(8+2i-12i-3i^2=11-10i\).

Answer: C

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Question 32

If \(\log_2(x+1)=4\), what is \(x\)?

F) 8
G) 17
H) 16
J) 15

Show solution

\(x+1=2^4=16\), so \(x=15\).

Answer: J

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Question 33

Let \(A=\begin{bmatrix}3&-2\\1&5\end{bmatrix}\) and \(B=\begin{bmatrix}4&6\\-3&2\end{bmatrix}\). What is the bottom-left entry of \(A+B\)?

A) -2
B) -4
C) 2
D) 4

Show solution

The bottom-left entry is \(1+(-3)=-2\).

Answer: A

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Question 34

A sequence is defined by \(a_1=1\) and \(a_n=3a_{n-1}-1\). What is \(a_4\)?

F) 11
G) 17
H) 14
J) 20

Show solution

Use the recurrence to build each term from the previous one.

Starting with \(a_1=1\), \(a_2=3(1)-1=2\).

Then \(a_3=3(2)-1=5\) and \(a_4=3(5)-1=14\).

Answer: G

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Question 35

For \(f(x)=\frac{x+2}{2x-6}\), which value is excluded from the domain?

A) -3
B) -2
C) 3
D) 2

Show solution

The denominator is zero when \(2x-6=0\): \(x=3\) is excluded.

Answer: C

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Question 36

The vector shown has terminal point \((4,3)\). What is its magnitude?

F) 3
G) 4
H) 7
J) 5

Show solution

The magnitude is \(\sqrt{4^2+3^2}=5\).

Answer: J

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Question 37

Which is a factor of \(x^3+2x^2-9x-18\)?

A) \(x-3\)
B) \(x+2\)
C) \(x-2\)
D) \(x+6\)

Show solution

Grouping gives \(x^2(x+2)-9(x+2)=(x+2)(x-3)(x+3)\): \(x-3\) is a factor.

Answer: A

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Question 38

In the right triangle shown, what is \(\sin\theta\)?

F) \(\frac5{13}\)
G) \(\frac{12}{13}\)
H) \(\frac5{12}\)
J) \(\frac{12}{5}\)

Show solution

Relative to \(\theta\), the opposite side is 12 and the hypotenuse is 13: \(\sin\theta=\frac{12}{13}\).

Answer: G

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Question 39

A random variable \(X\) takes values 1, 3, and 6 with probabilities 0.25, 0.50, and 0.25. What is \(E(X)\)?

A) 2.5
B) 3.0
C) 3.25
D) 4.0

Show solution

\(E(X)=1(0.25)+3(0.50)+6(0.25)=3.25\).

Answer: C

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Question 40

If \(f(x)=\begin{cases}2x+1,&x<0\\x^2+2,&x\ge0\end{cases}\), what is \(f(f(-2))\)?

F) -5
G) -3
H) 7
J) 11

Show solution

\(f(-2)=-3\).

Since \(-3<0\), \(f(-3)=2(-3)+1=-5\).

Answer: J

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Question 41

The graph shown has two branches approaching the dashed lines. What type of graph is shown?

A) Hyperbola
B) Ellipse
C) Circle
D) Parabola

Show solution

Two separated branches approaching asymptotes are characteristic of a hyperbola.

Answer: A

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Question 42

If \(2\le a\le5\) and \(3\le b\le9\), what is the greatest possible value of \(\frac{2b}{a}\)?

F) 3
G) 9
H) 6
J) 18

Show solution

Maximize the numerator and minimize the denominator: \(2(9)/2=9\).

Answer: G

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Question 43

A quadratic has zeros \(-2\) and 6 and satisfies \(f(0)=-24\). Which equation could define \(f\)?

A) \(f(x)=-2(x+2)(x-6)\)
B) \(f(x)=(x+2)(x-6)\)
C) \(f(x)=2(x+2)(x-6)\)
D) \(f(x)=(x-2)(x+6)\)

Show solution

Let \(f(x)=a(x+2)(x-6)\): then \(-12a=-24\), so \(a=2\).

Answer: C

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Question 44

A rectangle measures 20 by 10 and contains a circle of radius 4. What is the area inside the rectangle but outside the circle?

F) \(200-4\pi\)
G) \(200-8\pi\)
H) \(100-16\pi\)
J) \(200-16\pi\)

Show solution

Use the rectangle area formula \(A=lw\) to get the rectangle area, 200.

Use the circle area formula \(A=\pi r^2\) to get the circle area, \(16\pi\).

The difference is \(200-16\pi\).

Answer: J

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Question 45

A population has mean 40 and standard deviation 5. Every value is transformed by \(y=2x+7\). What are the new mean and standard deviation?

A) Mean 87, SD 10
B) Mean 80, SD 12
C) Mean 47, SD 10
D) Mean 87, SD 12

Show solution

The mean becomes \(2(40)+7=87\).

The standard deviation is multiplied by 2, giving 10.

Answer: A

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