MATHEMATICS TEST - ACT Practice Test 1 - 1,296 ACT Practice Questions

1,296 ACT Practice Questions, 3rd Edition (2013)

ACT Practice Test 1

2. MATHEMATICS TEST

60 Minutes—60 Questions

DIRECTIONS: Solve each problem, choose the correct answer, and then darken the corresponding oval on your answer document.

Do not linger over problems that take too much time. Solve as many as you can; then return to the others in the time you have left for this test.

You are permitted to use a calculator on this test. You may use your calculator for any problems you choose, but some of the problems may best be done without using a calculator.

Note: Unless otherwise stated, all of the following should be assumed:

1. Illustrative figures are NOT necessarily drawn to scale.

2. Geometric figures lie in a plane.

3. The word line indicates a straight line.

4. The word average indicates arithmetic mean.

1. In the hiking trail shown below, X marks the trail’s halfway point. If measures 24 kilometers and is the length of , what is the total length, in kilometers, of the trail?

A. 144

B. 104

C. 96

D. 72

E. 48

2. What is the value of x when + 7 = 6?

G. −

H. −1

J. −

K. −5

3. Cyclist A averages 80 pedal revolutions per minute, and Cyclist B averages 61 pedal revolutions per minute. At these rates, how many more minutes does Cyclist B need than Cyclist A to make 9,760 pedal revolutions?

A. 19

B. 38

C. 122

D. 141

E. 160

4. The perimeter of a square is 36 inches. What is the area of the square, in square inches?

F. 6

G. 9

H. 18

J. 36

K. 81

5. For the rectangle shown in the standard (x,y) coordinate plane below, what are the coordinates of the unlabeled vertex?

A. ( 4,5)

B. ( 4,7)

D. ( 6,7)

E. (10,4)

6. Carla has 5 times as many notebooks as her brother does. If they have 42 notebooks between them, how many notebooks does Carla have?

F. 30

G. 33

H. 35

J. 37

K. 47

7. If G is in the interior of right angle DEF, then which of the following could be the measure of GEF?

A. 85°

B. 95°

C. 105°

D. 115°

E. 125°

8. Susie has three T-shirts: one red, one blue, and one black. She also has three pairs of shorts: one red, one blue, and one black. How many different combinations are there for Susie to wear exactly one T-shirt and one pair of shorts?

F. 3

G. 6

H. 8

J. 9

K. 27

9. 20% of 20 is equal to 50% of what number?

A. 2

B. 4

C. 8

D. 10

E. 200

10. There are 45 musicians in an orchestra, and all play two instruments. Of these musicians, 36 play the piano, and 22 play the violin. What is the maximum possible orchestra members who play both the piano and the violin?

F. 9

G. 13

H. 22

J. 23

K. 36

11. What is the largest value of m for which there exists a real value of n such that m² = 196 – n²?

A. 14

B. 98

C. 182

D. 196

E. 392

12. Phil earned $800 at his summer job and saved all of his earnings. He wants to buy a deluxe drum kit that is regularly priced at $925 but is on sale for off. The drum kit is subject to 5% sales tax after all discounts are applied. If Phil buys the kit on sale and gives the sales clerk his entire summer earnings, how much change should he receive?

F. $23

G. $37

H. $40

J. $77

K. None; Phil still owes $171.25.

13. Which of the following numbers is an imaginary number?

C. −

D. −

E. −

14. Which of the following correctly factors the expression 25x⁴ − 16y⁸?

F. (25 − 16)(x² − y⁴)(x² + y⁴)

G. (5x² − 4y⁴)(5x² + 4y⁴)

H. (25x² − y⁴)(x² + 16y⁴)

J. (5x⁴ − 4y⁸)(5x⁴ + 4y⁸)

K. (5x⁴ − 8y⁸)(5x⁴ + 2y⁸)

15. The figure below shows a portion of a tile floor from which the shaded polygon will be cut in order to make a repair. Each square tile has sides that measure 1 foot. Every vertex of the shaded polygon is at the intersection of 2 tiles. What is the area, in square feet, of the shaded polygon?

A. 9.5

B. 10.0

C. 10.5

D. 11.0

E. 11.5

16. The percent P of a population that has completed 4 years of college is given by the function P(t) = −0.001t² + 0.4t where t represents time, in years. What percent of the population have completed four years of college after 20 years, to the nearest tenth?

F. 0.1

G. 7.6

H. 8.0

J. 8.4

K. 160.0

17. At Fatima’s Fruits, a bag of eight grapefruits costs $4.40. At Ernie’s Edibles, a bag of three grapefruits costs $1.86. How much cheaper, per grapefruit, is the cost at Fatima’s Fruits than at Ernie’s Edibles?

A. $0.07

B. $0.35

C. $0.59

D. $1.17

E. $2.54

18. Which of the following is equivalent to (x⁴ − 4)(x⁴ + 4)?

F. 2x⁴

G. x⁸ − 16

H. x⁸ + 16

J. x¹⁶ − 16

K. x⁸ − 8x⁴ − 16

19. Wade is making a tile mosaic. He begins the project by laying tile at a speed of 50 pieces per hour for 3.5 hours. He is then interrupted from his work for 60 minutes. He resumes working and lays tile at a speed of 35 pieces per hour, until he has laid 280 pieces of tile total. How many hours did Wade spend working on the mosaic after he was interrupted?

A. 2.5

B. 3

C. 3.5

D. 4

E. 4.5

20. Point C (1,2) and point D (7,−10) lie in the standard coordinate plane. What are the coordinates of the midpoint of ?

F. (1, 8)

G. (3,−6)

H. (4,−4)

J. (4,−6)

K. (7,−4)

21. Michael is planning to put fencing along the edge of his rectangular backyard, which is 22 yards by 16 yards. One long side of the backyard is along his house, so he will need to fence only 3 sides. How many yards of fencing will Michael need?

A. 38

B. 54

C. 60

D. 76

E. 352

22. What is the y-intercept of the line given by the equation 7x – 3y = 21?

F. −7

G. −

J. 7

K. 21

23. On April 8th, a flower at Blooming Acres Florist was 15.0 centimeters tall. On April 16th, the flower was 17.4 centimeters tall. If the flower grew at a constant rate, on what day was the flower 16.5 centimeters tall?

A. April 11th

B. April 12th

C. April 13th

D. April 14th

E. April 15th

24. Which of the following expressions is equivalent to the expression given below?

(2x³ – x − 1) − 3(x⁴ + 2x³ − 2x² – x + 3)

F. x¹⁴ − 3

G. −3x¹⁴

H. −3x⁴ + 8x³ − 6x² − 4x + 8

J. −3x⁴ + 4x³ − 2x² − 2x − 3

K. −3x⁴ − 4x³ + 6x² + 2x − 10

25. The playground equipment shown below has a ladder that is 6 feet tall and a diagonal slide that is 7 feet long. If the ladder makes a right angle with the ground, approximately how many feet is the base of the slide from the base of the ladder?

A. 2

B. 4

C. 6

D. 8

E. 10

26. In a data set of 5 points, the mean, median, and mode are each equal to 8. Which of the following could be the data set?

F. {5, 7, 8, 8, 12}

G. {7, 7, 8, 8, 12}

H. {7, 8, 8, 8, 12}

J. {7, 8, 8, 10, 12}

K. {7, 8, 8, 12, 12}

27. In a certain sequence of numbers, each term after the 1st term is the result of adding 2 to the previous term and multiplying that sum by 3. If the 4th term in the sequence is 186, what is the 2nd term?

A. 2

B. 4

C. 18

D. 60

E. 174

28. Which of the following values of x does NOT satisfy the inequality |x−3|≥12?

F. −15

G. −12

H. −9

J. 9

K. 15

29. For all real numbers s, t, u, and v, such that s + t + u = 29 and s < v, which of the following statements is true?

A. s + t + v < 29

B. t + u + v > 29

C. s + t + v = 29

D. s + u + v = 29

E. s + t + v > 29

30. In the figure below, rectangle ABCD shares with CDE, diagonal of the rectangle extends in a straight line beyond D to E to create , and the measure of CDE is 155°. What is the measure of CBD?

F. 25

G. 55

H. 65

J. 90

K. 155

31. If a, b, and c are positive prime numbers, in the equation a – b = c, either b or c must represent which number?

A. 13

B. 11

C. 7

D. 5

E. 2

32. Pierre competes in a triathlon, along a course as shown in the figure below. He begins swimming at starting point S and swims straight across the lake, gets on his bicycle at station A, bikes to station B, and then runs to finishing line F. The judges use a stopwatch to record his elapsed times of t_A, t_B, and t_F hours from point S to points A, B, and F, respectively. If the distance, in miles, between points S and A along the racecourse is denoted by SA, then what is Pierre’s average speed for this race, in miles per hour?

33. The triangle shown below has a hypotenuse with a length of 13 feet. The measure of A is 20° and the measure of B is 70°. Which of the following is closest to the length, in feet, of ?

(Note: sin 70° ≈ 0.9397
cos 70° ≈ 0.3420
tan 70° ≈ 2.747)

A. 4.4

B. 5.0

C. 12.0

D. 12.2

E. 35.7

34. What is the value of when x = −3 and y = −4?

F. −

G. −

35. As shown in the figure below, with angles as marked, a ramp is being designed that will have a vertical height of 4 feet. Which of the following is closest to the horizontal length of the ramp, in feet?

A. 5

B. 6

C. 7

D. 8

E. 9

36. In the diagram below, ABC is isosceles and BCD is equilateral. = and the measure of ABC is half the measure of BAC. What is the measure of ABD?

F. 36°

G. 60°

H. 72°

J. 96°

K. 150°

Use the following information to answer questions 37–39.

The coordinates of the vertices of MON are shown in the standard (x,y) coordinate plane below. Rectangle MPQR is shown shaded. Point P lies on , point Q lies on , and point R lies on .

37. What is the slope of MON?

A. −2

B. −

C. 0

E. 2

38. Which of the following is closest to the perimeter, in coordinate units, of MON?

F. 12.0

G. 16.9

H. 18.0

J. 20.9

K. 92.0

39. What is the value of cos (MON)?

C. 2

40. In a Spanish class there are m students, of which n did NOT pass the last exam. Which of the following is a general expression for the fraction of the class that did receive a passing grade?

41. The solution set of 5x + 9 ≥ 2(3x + 4) + 7 is shown by which of the following number line graphs?

42. An artist wants to cover the entire outside of a rectangular box with mosaic tiles. The dimensions of the box shown below are given in centimeters. If each tile is exactly one square centimeter, and the artist lays the tiles with no space between them, how many tiles will he need?

F. 75

G. 96

H. 108

J. 126

K. 150

43. In the figure shown below, and are parallel and = . If ABC is 130° and BAF is 22°, what is the measure of AEF?

A. 50°

B. 118°

C. 152°

D. 158°

E. 164°

44. Given the figure below, what is the area of the trapezoid, in square inches?

F. 18

G. 30

H. 42

J. 50

K. 52

45. What is the solution set of

A. {4}

B. {8}

C. {−4, 8}

D. {−8, 4}

E. {−2, ±2}

46. As shown in the figure below, a skateboard ramp leading from the top of a boulder is 10 feet long and forms a 32° angle with the level ground. Which of the following expressions represents the height, in feet, of the boulder?

F. 10 tan 32°

J. 10 sin 32°

K. 10 cos 32°

47. The 4 integers j, j, k, and n have an average of 0. Which of the following equations must be true?

A. k = n

B. k = –j

C. k + n = −2j

D. k + n = 0

E. k + n = j

48. If f(x) = and the composite function f(g(x)) = , which of the following could be g(x)?

H. 2x² − 25

J. 4x² − 5

K. 16x⁴ − 5

Use the following information to answer questions 49−51.

In the qualifying rounds for a race, Rusty and Dale drive their cars around a 6,000-foot oval track. Rusty and Dale each drive 8 laps in the qualifying rounds in lanes of identical length.

49. On day one of the qualifying rounds, Rusty and Dale start from the same point, but their cars are reversed and each drives opposite ways. Rusty drives at a constant speed that is 8 feet per second faster than Dale’s constant speed. Rusty passes Dale for the first time in 150 seconds. Rusty drives at a constant rate of how many feet per second?

A. 16

B. 20

C. 24

D. 32

E. 40

50. In the qualifying rounds, Rusty averages 180 seconds per lap until he begins the last lap. He then goes into a lower gear. He averages 190 seconds per lap for this qualifying round. How many seconds does Rusty take to drive the final lap?

F. 155

G. 160

H. 185

J. 200

K. 260

51. Dale drives 6 laps in 90 minutes. At what average rate, in feet per hour, does Dale drive these 6 laps?

A. 400

B. 5,400

C. 10,000

D. 24,000

E. 48,000

52. Circle A has its center at point (−5,2) with a radius of 2, and circle B is represented by the equation (x + 4)² + (y−− 2)² = 9. Where is point (−2,2) located?

F. Inside circle A only

G. Inside circle B only

H. Inside both circle A and circle B

J. Outside both circle A and circle B

K. Cannot be determined from given information

53. A heart-shaped ornament is made from a square and two semicircles, each of whose diameter is a side of the square. The ornament is shown in the standard (x,y) coordinate plane below, where 1 coordinate unit represents 1 inch. The coordinates of six points on the border of the ornament are given. What is the perimeter, in inches, of the ornament?

A. 4 + 2π

B. 8 + 4π

C. 8 + 8π

D. 16 + 4π

E. 16 + 8π

54. A function f(x) is defined as even if and only if f(x) = f(−x) for all real values of x. Which one of the following graphs represents an even function f(x)?

55. In the standard (x,y) coordinate plane, point A is located at (w,w + 5) and point B is located at (4w,w−− 5). In coordinate units, what is the distance between A and B?

C. 9w² + 100

E. |w|

56. RST is a right triangle with side lengths of r, s, and t, as shown below. What is the value of cos² S + cos² R?

F. 1

57. In isosceles triangle ABC below, the measures of BAC and BCA are equal and . The diagonals of trapezoid DECA intersect at F. The lengths of and are 6 centimeters, the length of is 9 centimeters, and the length of is 27 centimeters. What is the length, in centimeters, of ?