Additional Practice Test 1: ACT Math Study Guide

This form follows a broadly balanced current-enhanced mix and rises in difficulty overall. It begins with quick core points, introduces more representation changes through the middle, and ends with several higher-demand items that require choosing a relationship before calculating. Local variation remains possible, so question position should guide pacing only broadly, not determine how much time any one item deserves.

Time: 50 minutes. Questions: 45. Calculator permitted throughout.

Work in order unless a question becomes locally expensive. Choose the best answer and move on when the route is not developing. Do not leave any question unanswered. A useful pacing checkpoint is roughly 15 questions complete by 14–16 minutes and roughly 30 complete by 31–34 minutes, preserving 16–19 minutes for the final 15.

▸ Questions

Question 1

A store reduces the price of a $72 jacket by 25%. What is the sale price?

A. $18
B. $47
C. $54
D. $60

Question 2

If 3x − 7 = 20, what is x?

F. 7
G. 9
H. 11
J. 13

Question 3

The mean of 6, 9, 11, and 14 is:

A. 10
B. 9
C. 11
D. 40

Question 4

For f(x) = 2x² − 3, what is f(2)?

F. 5
G. 1
H. 8
J. 13

Question 5

A right triangle has legs 5 and 12. What is the length of the hypotenuse?

$Figure from the book$

A. 15
B. 13
C. 17
D. 25

Question 6

Which expression is equivalent to x³ · x⁵?

F. x¹⁵
G. x⁸
H. 2x⁸
J. x²

Question 7

A car travels 168 miles in 3 hours at a constant rate. At that rate, how far will it travel in 5 hours?

A. 224
B. 260
C. 336
D. 280

Question 8

Solve the inequality −2x + 5 > 13.

F. x < −4
G. x > −4
H. x < 4
J. x > 4

Question 9

The point (4, −3) is reflected across the x-axis. What are the coordinates of its image?

A. (−4, −3)
B. (4, 3)
C. (−4, 3)
D. (3, 4)

Question 10

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

F. 10
G. 14
H. 18
J. 24

Question 11

A bag contains 4 red, 3 blue, and 5 green marbles. One marble is chosen at random. What is the probability it is blue?

A. 1/3
B. 3/5
C. 1/4
D. 5/12

Question 12

What is the value of i¹⁸?

F. 1
G. i
H. −i
J. −1

Question 13

The graph of y = 2x + 1 is shown. Which statement about the graph is true?

$Figure from the book$

A. Its y-intercept is 1.
B. Its slope is 1/2.
C. It is decreasing.
D. It passes through (1, 1).

Question 14

A rectangular garden is 3 feet longer than twice its width. If the width is w feet, which expression gives its area?

F. 2w+3
G. 2w(w+3)
H. w(2w+3)
J. w²+3

Question 15

The circle (x−2)²+(y+1)²=25 has radius:

A. 2
B. 25
C. √24
D. 5

Question 16

A value increases from 80 to 92. What is the percent increase?

F. 12%
G. 13%
H. 15%
J. 20%

Question 17

Which value is irrational?

A. 0.125
B. √18
C. 7/9
D. √49

Question 18

A survey randomly assigns 200 volunteers to two study groups and gives only one group a new study method. If the groups differ significantly afterward, which conclusion is best supported?

F. The study method caused the difference in these volunteers.
G. The study method is effective for every student.
H. The groups were observational only.
J. No causal conclusion is possible from random assignment.

Question 19

Solve the system: x+y=9 and 2x−y=6. What is x?

A. 5
B. 3
C. 4
D. 6

Question 20

The graph of a quadratic has x-intercepts at −2 and 5. Which expression could represent the quadratic?

F. (x−2)(x+5)
G. (x+2)(x+5)
H. (x+2)(x−5)
J. (x−2)(x−5)

Question 21

A scale drawing uses 1 inch to represent 6 feet. A rectangular room measures 2.5 inches by 3 inches on the drawing. What is the actual area of the room?

A. 45 ft²
B. 270 ft²
C. 90 ft²
D. 540 ft²

Question 22

If sin(θ)=3/5 for an acute angle θ, what is cos(θ)?

F. 2/5
G. 3/4
H. 5/3
J. 4/5

Question 23

For what value of k does x²−6x+k=0 have exactly one real solution?

A. 9
B. 3
C. 6
D. 36

Question 24

An arithmetic sequence begins 7, 12, 17, 22, … . What is the 20th term?

F. 102
G. 97
H. 107
J. 112

Question 25

A box plot has minimum 12, Q1=18, median=24, Q3=31, maximum=45. What is the interquartile range?

$Figure from the book$

A. 19
B. 27
C. 33
D. 13

Question 26

Simplify (3+2i)(1−i).

F. 1−i
G. 5+i
H. 1+5i
J. 5−i

Question 27

A phone plan charges $25 per month plus $0.08 per text message. Another plan charges $40 per month plus $0.03 per text. At how many texts do the plans cost the same?

A. 300
B. 200
C. 250
D. 500

Question 28

For f(x)=(x²−9)/(x−3), x≠3. What is the y-value of the hole in the graph?

F. 0
G. 6
H. 3
J. 9

Question 29

Two similar triangles have corresponding side lengths 6 and 15. If the area of the smaller triangle is 24, what is the area of the larger triangle?

$Figure from the book$

A. 60
B. 150
C. 96
D. 120

Question 30

A recipe uses 2.5 cups of flour for 8 servings. At the same rate, how many cups are needed for 20 servings?

F. 5.0
G. 7.5
H. 6.25
J. 8.0

31. The table shows values of a linear function. What is f(10)?

x	f(x)
1	11
4	17
7	23

A) 23

B) 27

C) 31

D) 29

Question 32

A circle has radius 8. What is the length of an arc subtended by a central angle of 135 degrees?

F. 3π
G. 8π
H. 12π
J. 6π

Question 33

If log₂(x)=5, what is x?

A. 10
B. 25
C. 32
D. 64

Question 34

The expression (x²−16)/(x−4) is undefined at x=4. For x≠4, which expression is equivalent?

F. x−4
G. x+4
H. x²+4
J. 1

Question 35

A normal distribution has mean 70 and standard deviation 5. A score of 80 has z-score:

A. 1
B. 5
C. 2
D. 10

Question 36

A cylinder has radius 3 and height 10. What is its volume?

F. 90π
G. 30π
H. 60π
J. 180π

Question 37

A worker earns $18 per hour for the first 40 hours and 1.5 times that rate for overtime. If the worker works 46 hours, what is the total pay?

A. $882
B. $828
C. $846
D. $936

Question 38

The solutions to |2x−5|=9 are:

F. x=7 only
G. x=7 or x=−2
H. x=−2 only
J. x=2 or x=−7

Question 39

A geometric sequence has first term 3 and common ratio 2. What is the sum of the first 6 terms?

A. 63
B. 96
C. 189
D. 192

Question 40

In triangle ABC, A=40°, B=65°, and side a=10. Which expression gives side b?

F. 10 sin40/sin65
G. 10 cos65/cos40
H. 10 tan65/tan40
J. 10 sin65/sin40

Question 41

A poll of 625 voters estimates 52% support for a proposal with a margin of error of ±4 percentage points. Which population value is NOT consistent with the reported interval?

A. 49%
B. 52%
C. 55%
D. 57%

Question 42

The polynomial p(x)=x³−4x²−x+4 has x=1 as a zero. Which factorization is correct?

F. (x−1)²(x−4)
G. (x+1)(x−4)(x−1)
H. (x−1)(x−2)²
J. (x+1)(x−2)(x+2)

Question 43

The graph of y=−2(x−3)²+18 represents the height of an object, in feet, after x seconds. What is the maximum height?

$Figure from the book$

A. 3
B. 21
C. 18
D. 36

Question 44

A rectangular tank is 4 ft by 3 ft by 2.5 ft and is 80% full. How many cubic feet of water are in the tank?

F. 24
G. 18
H. 30
J. 37.5

Question 45

A line tangent to the circle x²+y²=20 at the point (2,4) has what slope?

A. 1/2
B. −1/2
C. 2
D. −2

▸ Answers and Explanations

Show the answers and explanations

1: C. A 25% discount means paying 75% of the original price: 72(0.75)=$54. The requested object is the final sale price, not the discount amount.

2: G. Add 7 to both sides: 3x=27, so x=9.

3: A. The sum is 40 and there are 4 values, so the mean is 40/4=10.

4: F. Substitute x=2: 2(2²)−3=8−3=5.

5: B. By the Pythagorean theorem: c=√(5²+12²)=√169=13.

6: G. For the same base, add exponents when multiplying: x^(3+5)=x⁸.

7: D. The unit rate is 168/3=56 miles per hour. In 5 hours the car travels 56(5)=280 miles.

8: F. Subtract 5 from both sides: −2x>8. Dividing by −2 reverses the inequality: x<−4.

9: B. Reflecting across the x-axis keeps x the same and changes the sign of y: (4,−3)→(4,3).

10: H. Evaluate inside first: g(2)=2+4=6. Then h(6)=3(6)=18.

11: C. There are 12 marbles total and 3 are blue, so P(blue)=3/12=1/4.

12: J. Powers of i repeat every 4. Since 18 leaves remainder 2 when divided by 4, i¹⁸=i²=−1.

13: A. In slope-intercept form y=mx+b, the y-intercept is b=1. The slope is 2, so the graph increases, and at x=1 the y-value is 3, not 1.

14: H. Length=2w+3, so area=width×length=w(2w+3). This is a setup item; do not calculate a number that was never asked for.

15: D. Circle form is (x−h)²+(y−k)²=r². Here r²=25, so r=5.

16: H. The increase is 92−80=12. Dividing by the original value: 12/80=0.15=15%.

17: B. √18=3√2, which is irrational since 18 is not a perfect square. The other three values are all rational.

18: F. Random assignment supports a causal conclusion for this study's sample. It does not automatically justify generalizing to every student, since the volunteers were not described as a random sample of all students.

19: A. Adding the two equations eliminates y: 3x=15, so x=5.

20: H. A zero at −2 gives the factor (x+2); a zero at 5 gives the factor (x−5).

21: B. The actual dimensions are 2.5(6)=15 ft and 3(6)=18 ft. Area=15×18=270 ft². A common error is applying the linear scale factor only once to an area calculation.

22: J. Using sin²θ+cos²θ=1: cos²θ=1−9/25=16/25, so cosθ=4/5 (positive for an acute angle).

23: A. Exactly one real solution means the discriminant equals 0: (−6)²−4(1)(k)=0, so 36−4k=0, giving k=9.

24: F. aₙ=a₁+(n−1)d: a₂₀=7+19(5)=7+95=102.

25: D. IQR=Q3−Q1=31−18=13.

26: J. Distribute: (3+2i)(1−i)=3−3i+2i−2i²=3−i+2=5−i.

27: A. Set the costs equal: 25+0.08t=40+0.03t. Then 0.05t=15, so t=300.

28: G. Factor: (x−3)(x+3)/(x−3)=x+3 for x≠3. The missing point occurs at x=3, giving y=3+3=6.

29: B. The linear scale factor is 15/6=2.5. Area scales by the square of that factor: 24(2.5²)=24(6.25)=150.

30: H. Scale by 20/8=2.5: 2.5 cups × 2.5=6.25 cups.

31: D. From the table, y increases by 6 when x increases by 3, so the slope is 2. Using the point (1,11): f(x)=2x+9. Thus f(10)=2(10)+9=29.

32: J. Arc length=(135/360)(2π·8)=(3/8)(16π)=6π.

33: C. log₂(x)=5 means 2⁵=x, so x=32.

34: G. Factor the numerator: (x−4)(x+4). Cancel x−4 for x≠4, leaving x+4.

35: C. z=(80−70)/5=10/5=2.

36: F. V=πr²h=π(3)²(10)=90π.

37: A. Regular pay=40(18)=720. Overtime rate=1.5(18)=27, and overtime pay=6(27)=162. Total=720+162=882.

38: G. Two cases: 2x−5=9 gives x=7; 2x−5=−9 gives x=−2.

39: C. The terms are 3,6,12,24,48,96. Their sum is 189, which also follows from Sₙ=a(1−rⁿ)/(1−r).

40: J. By the Law of Sines, a/sin A=b/sin B. Solving for b gives b=10 sin(65°)/sin(40°).

41: D. The plausible interval is 52%−4%=48% to 52%+4%=56%. Since 57% lies outside that interval, it is the value not consistent with the reported margin of error.

42: G. Factor by grouping: x²(x−4)−1(x−4)=(x²−1)(x−4)=(x−1)(x+1)(x−4).

43: C. The function is in vertex form. The vertex is (3,18), and since the leading coefficient is negative, this vertex is a maximum. Maximum height=18 feet.

44: F. Full volume=4×3×2.5=30 ft³. At 80% full: 0.8(30)=24 ft³.

45: B. The radius from the origin to (2,4) has slope 4/2=2. A tangent line is perpendicular to the radius at the point of tangency, so its slope is the negative reciprocal, −1/2.

Additional Practice Test 1

▸ Questions

▸ Answers and Explanations

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