ACT Form 72E Math Answer Explanation


2016 April(72E) Math Question 1

1. A cookie recipe that yields 24 cookies requires 1 3/4 cups of butter, When the ingredients in this recipe are increased proportionally, how many cups of butter are required for the recipe to yield 72 cookies?

Choice D

$$\frac{x}{72}=\frac{1\frac{3}{4}}{24} \\$$ $$ \Rightarrow x=5\frac{1}{4}$$


2015 December(72E) Math Question 2

2. The age, in years, of each of the first 6 presidents of the United States at his first inauguration was 57, 61, 57, 57, 58, and 57, respectively. Which of the following values is closest to the mean age, in years, of the first 6 presidents at their respective first inaugurations?

Choice H

$$\frac{57+61+57+57+58+57}{6}=57.8$$


2015 December(72E) Math Question 3

3. The temperature F in degrees Fahrenheit is related to the temperature K in kelvins by the equation F = 1.8K — 459.67. Which of the following temperatures, in kelvins, is closest to 120 degrees Fahrenheit?

Choice A

$$120=1.8K-459.67 \\$$ $$ \Rightarrow K\approx 322$$


2015 December(72E) Math Question 4

4. For nonzero values of x and y, which of the following expressions is equivalent to — 18x^3y^2 / 3xy?

Choice F

$$-\frac{18}{3}x^{3-1}y^{2-1}=-6x^2y$$


2015 December(72E) Math Question 5

5. If square root of a = b and b = 9, a=?

Choice C

$$\sqrt{a}=9 \Rightarrow a=81$$


2015 December(72E) Math Question 6

6. Damon and 4 of his coworkers are having lunch. Each of the 5 people will pay for his or her own lunch, but they agree to divide the tip equally among themselves. The total for the 5 lunches is $80.00, and the group will add a tip of 15% of the total. Each person's portion of the tip will be how much?

Choice G

$$\frac{80\times15\%}{5}=2.4$$


2015 December(72E) Math Question 7

7. The number 0.000 000 000 087 3 is equivalent to which of the following expressions?

Choice B

$$0.000000000087=8.73\times 10^{-11}$$


2015 December(72E) Math Question 8

8. The circular spinner dial for a new board game is divided into 6 congruent sectors. What is the arc measure, in degrees, of each sector?

Choice J

$$\frac{360}{6}=60$$


2015 December(72E) Math Question 9

9. When 5 4/9 is written as an improper fraction in lowest terms, the numerator of the fraction is:

Choice E

$$\frac{4+45}{9}=\frac{49}{9}$$


2015 December(72E) Math Question 10

10. 2[3	24]	2 + 3[-1_2]	= ?

Choice G

$$\begin{bmatrix}2\times1+3\times2 & 2\times2+3\times1 \\2\times3-3\times1 & 2\times4-3\times2 \end{bmatrix}=\begin{bmatrix}8 & 7 \\3 & 2 \end{bmatrix}$$


2015 December(72E) Math Question 11

11. The speed of one motorcycle exceeds 4 times the speed of another motorcycle by 24 mph. The speed of the slower motorcycle is g mph. Which of the following expressions represents the speed of the faster motorcycle, in miles per hour?

Choice D

The speed of one motorcycle exceeds 4 times the speed of another motor cycle by 24 mph: 4g+24


2015 December(72E) Math Question 12

12. In the Coaltown High School cafeteria, the student council is laying a triangular carpet in a corner that is designated to be a lounge. The carpet is a right triangle with the 2 shorter sides having lengths of 12 feet and 8 feet, as shown in the figure below. What is the area, in square feet, of the carpet?

Choice J

$$Area=\frac{12\times8}{2}=48$$


2015 December(72E) Math Question 13

13. Students studying motion observed a cart rolling at a constant rate along a straight line. The table below gives the distance, d feet, the cart was from a reference point at 1-second intervals from t = 0 seconds to t = 5 seconds.

Choice C

Only the equation in C fulfils all data sets in the table.


2015 December(72E) Math Question 14

14. Rajeev went on a delivery trip that began and ended at his truck terminal. He used all of the time during the trip driving, unloading, or resting. Rajeev began his trip on Tuesday at 7:00 a.m. when he left the terminal. During his driving time, he drove 1,200 miles at an average speed of 50 miles per hour. His driving time was twice as long as his unloading time, and his resting time was 30 hours. When did Rajeev end his delivery trip?

Choice H

$$Driving \ time = \frac{1200}{50}=24 \\$$ $$ Unloading \ time = 12 \\$$ $$ \Rightarrow Total \ time=24+12+30=66=24+24+18$$


2015 December(72E) Math Question 15

15. For the equation 3x + 15y = 9, which of the following expressions represents x in terms of y ?

Choice C

$$x+5y=3 \Rightarrow x=-5y+3$$


2015 December(72E) Math Question 16

16. The perimeter of a parallelogram is 80 inches, and the length of 1 side is 16 inches. If it can be determined, what are the lengths, in inches, of the other 3 sides?

Choice H

Thus the other side is also 16; the remaining 2 sides are: $$\frac{80-16-16}{2}=24$$


2015 December(72E) Math Question 17

17. The Tully family and the Quan family ate dinner together at Eugenio's Pasta Restaurant. An order of pasta comes in 1 of 2 sizes, large or small, and consists of 1 of 6 types of pasta and 1 of 5 types of sauce. The table below gives the number of large and small orders of pasta bought by each family, and the price each family paid for their orders of pasta (without tax and tip). The price is the same for each order of a given size. What is the price of a large order of pasta?

Choice D

$$6l+4s=26 \\$$ $$ 2l+4x=14 \\$$ $$ \Rightarrow l=3$$


2015 December(72E) Math Question 18

18. How many different possible orders of pasta can a person get?

Choice K

$$2\times6\times5=60$$


2015 December(72E) Math Question 19

19. The Tully family also bought 5 salads priced at $2.00 per salad and 12 breadsticks priced at $1.50 for an order of 3 breadsticks. What was the total price of the pasta, salads, and breadsticks the Tully family bought, without tax and tip?

Choice C

$$26+2\times5+1.5\times \frac{12}{3}=42$$


2015 December(72E) Math Question 20

20. In the figure below, parallel lines AB and CD are cut by transversals AC and BD that intersect at E. Two angle measures are given. What is the measure of ZABD ?

Choice K

180-34=146


2015 December(72E) Math Question 21

21. The total cost, c dollars, for Main Street Orchestra to perform a concert at Milly's Auditorium is determined by c = r + 20m, where r is the rental fee, in dollars, of the auditorium and m is the number of orchestra members playing. The Friday night rental fee for Milly's Auditorium is $500. There will be 30 orchestra members playing in Friday night's concert. For the total price of exactly 200 tickets to equal the total cost of performing the concert, what should be the price of each ticket?

Choice E

$$200x=500+20\times30 \\$$ $$ \Rightarrow …


2015 December(72E) Math Question 22

22. An experiment consisted of rolling a 6-sided cube with the digits 1 through 6 on its faces, 1 digit per face. The cube was rolled 50 times, and after each roll, the number appearing on the top face was recorded. The number of times each digit was recorded is represented in the bar graph shown below. In what percent of the total number of rolls did a 5 appear on the top face of the cube?

Choice K

$$\frac{9}{10+5+10+9+9+7}=18\%$$


2015 December(72E) Math Question 23

23. The length of a rectangle is 12 feet longer than the rectangle's width. The area of the rectangle is 140 square feet. The width of the rectangle, w feet, is the positive solution to which of the following equations?

Choice B

The length of the rectangle is w + 12; the area is $$w\cdot l=140 \\$$ $$ \Rightarrow w\cdot (w+12)=140$$


2015 December(72E) Math Question 24

24. What is the length, in coordinate units, of the line segment with endpoints (-8,4) and (4,9) in the standard (x,y) coordinate plane?

Choice H

$$\sqrt{(-8-4)^2+(4-9)^2 …


2015 December(72E) Math Question 25

25. The point (6,3) and the line y = 7 are graphed in the standard (x,y) coordinate plane below. After the point has been reflected across the line, what are the coordinates of the point's image?

Choice E

(6,3+(7-3)+(7-3))=(6 …


2015 December(72E) Math Question 26

26. The diameter of a circle is 6 feet. What is the area, in square feet, of the circle?

Choice H

$$Area = \pi r^2=\pi …


2015 December(72E) Math Question 27

27. In a chemistry course, a student scored 99 on one test, 98 on another test, and 88 on each of the other tests. The student's test average for the course, where each test is weighted equally, is exactly 91. What is the total number of tests that the student has taken in the course?

Choice D

$$\frac{99+98+88\times …


2015 December(72E) Math Question 28

28. A cash prize will be given to the winning group of the talent show at Aspen High School. The winning group will receive a cash prize that, when divided equally among the students in that group, gives each student in the group a whole-dollar share of the prize. Considering that a group of 2, 3, 4, or 5 students could win the show, what is the least possible amount that the cash prize can be?

Choice J

Among the choices, only 60 …


2015 December(72E) Math Question 29

29. Which of the following expressions is a factor of the polynomial x^2 — x — 72 ?

Choice A

$$x^2-x-72=(x-9)(x+8)$$


2015 December(72E) Math Question 30

30. Javier will have a pool installed in his backyard. The interior of the pool is a right circular cylinder with a uniform depth of 5 feet and a radius of 8 feet. The maximum volume of water that can be in the pool is 75% of the volume of the pool. Which of the following values is closest to the maximum number of cubic feet of water that can be in the pool?

Choice F

$$Volume=\pi r^2 h …


2015 December(72E) Math Question 31

31. In ABC shown below, sin C = 4/5 and the length of AB is 10 inches. What is the length, in inches, of AC ?

Choice E

$$\sin C= \frac{AB}{AC …


2015 December(72E) Math Question 32

32. In isosceles triangle PQR shown below, PQ = QR and the measure of P is 54°. If it can be determined, what is the measure of Q ?

Choice H

$$\angle R= \angle P=54 …


2015 December(72E) Math Question 33

33. A function is defined by g(a) = —2a +7, and its domain is the set of integers from 1 through 30. inclusive. For how many values of a is g(a) negative?

Choice B

g(1) > 0

g(2 …


2015 December(72E) Math Question 34

34. Given that n is a positive integer and b is 3 times n, what is the least common denominator, in terms of n. for the addition of 1/b and 1/n ?

Choice J

$$b=3n \\$$ $$\Rightarrow \frac{1 …


2015 December(72E) Math Question 35

35. Nichelle has 84 solid-colored disks that are either red, blue, or green. She lines them up on the floor and finds that there are 24 more red disks than green and 15 more green disks than blue. How many red disks does she have?

Choice E

$$g=b+15 \\$$ $$ r=g …


2015 December(72E) Math Question 36

36. The measures of 4 interior angles of a pentagon are 70°, 100°, 110°, and 135°, respectively. What is the measure of the 5th interior angle?

Choice K

$$(5-2)\times 180°-70°-100 …


2015 December(72E) Math Question 37

37. The midpoints of the sides of rectangle WXYZ are the vertices of rhombus ABCD. The dimensions of rectangle WXYZ are 6 cm by 8 cm. What is the perimeter, in centimeters, of rhombus ABCD ?

Choice A

$$WD=\frac{6}{2}=3 …


2015 December(72E) Math Question 38

38. In the standard (x,y) coordinate plane below, a circle has a radius of r coordinate units and passes through the origin, O. The circle has diameter OS, where S lies on the negative y-axis. In terms of r, what are the coordinates of S ?

Choice G

OS=2r

Thus S(0 …


2015 December(72E) Math Question 39

39. Let k be a constant of proportionality and let w, x, y, and z be positive real number variables. In which of the following equations does x vary directly with y, directly with the square of w, and inversely with z ?

Choice B

‘x varies directly with y …


2015 December(72E) Math Question 40

40. What is the set of real solutions for |x|^2— |x| — 2 = 0 ?

Choice G

$$(|x|-2)(|x|+1)=0 …


2015 December(72E) Math Question 41

41. In the standard (x,y) coordinate plane below, R is located at (1,0), S is located at (1,2), and T is located at (4,0) to form right triangle ❑RST The given lengths are in coordinate units. What is the slope of ST ?

Choice B

The line ST is going …


2015 December(72E) Math Question 42

42. What is the midpoint of ST ?

Choice H

$$(\frac{1+4}{2},\frac …


2015 December(72E) Math Question 43

43. Which of the following expressions gives the measure of STR ?

Choice D

$$\tan \angle STR = \frac{SR …


2015 December(72E) Math Question 44

44. Right triangle RST will be rotated about the x-axis to form a right circular cone. How long, in coordinate units, is the radius of the cone's base?

Choice F

SR is the radius which …


2015 December(72E) Math Question 45

45. A cube has a total surface area of 216 square centimeters. Which of the following expressions gives the area, in square centimeters, of a single face of the cube?

Choice D

A cube has 6 sides …


2015 December(72E) Math Question 46

46. When graphed in the standard (x,y) coordinate plane, the lines x = —3 and y = x — 5 intersect at what point?

Choice K

y=-3-5=-8


2015 December(72E) Math Question 47

47. In the standard (x,y) coordinate plane, which of the following lines is perpendicular to the line 3y = 4x + 2 ?

Choice B

$$y=\frac{4}{3}x …


2015 December(72E) Math Question 48

48. For every negative real value of x, all of the following statements are true EXCEPT:

Choice K

K: |x|=x

When x …


2015 December(72E) Math Question 49

49. In trapezoid ABCD illustrated below, AB is 8 units long, CD is 12 units long, and EF is 6 units long. Also, ZAEF and LDFE are right angles, What is the area of ABCD, in square units?

Choice A

$$Area = \frac{AB+CD}{2 …


2015 December(72E) Math Question 50

50. Let a be a prime number greater than 10,000 and let x = square root of a. Which of the following expressions represents a rational number?

Choice J

$$x^2={\sqrt{a}}^2 …


2015 December(72E) Math Question 51

51. Melanie is standing 80 feet from the launch site of a hot-air balloon when the balloon lifts off from the ground and rises vertically. Melanie's horizontal line of sight is 5 feet above the ground. When the bottom of the balloon is 50 feet above the ground, as shown below, which of the following expressions gives the angle that Melanie's horizontal line of sight makes with her line of sight to the bottom of the balloon?

Choice A

$$\tan\alpha=\frac{50-5}{80 …


2015 December(72E) Math Question 52

52. One of the following inequalities is graphed below in the standard (x,y) coordinate plane. Which one?

Choice G

Firstly, one need to identify …


2015 December(72E) Math Question 53

53. A box contains 6 identically sized, solid-colored balls. One ball is green, 2 are yellow, and 3 are red. A ball is drawn at random and returned to the box, then a second ball is drawn at random. What is the probability that the first ball is red and the second ball is green?

Choice A

$$\frac{3}{6}\cdot\frac …


2015 December(72E) Math Question 54

54. The dimensions shown below are in feet. What is the area, in square feet, of the shaded rectangle?

Choice H

$$Area = (18-2x)(12-x)=2x^2-42x_216$$


2015 December(72E) Math Question 55

55. The graph of y = sin x in the standard (x,y) coordinate plane is reflected over the x-axis, shifted up a units, and then shifted left 0.5pi units. Which of the following equations represents the graph after the 3 transformations?

Choice B

The graph is reflected over …


2015 December(72E) Math Question 56

56. For all positive real numbers x, which of the following expressions is equivalent to ?

Choice K

$$\frac{x^{24}}{x^6 …


2015 December(72E) Math Question 57

57. Two numbers have a product of —48 and a sum of 0. What is the lesser of the 2 numbers?

Choice A

$$a\cdot b=-48 \\$$ $$ a …


2015 December(72E) Math Question 58

58. As shown below, BE divides rectangle ACDF into 2 congruent trapezoids. The measure of BED is 45°. The lengths of BC, CD, and EF are given in feet. What is the area, in square feet, of rectangle ACDF ?

Choice K

58. As shown below, BE divides rectangle ACDF into 2 congruent trapezoids. The measure of BED is 45°. The lengths of BC, CD, and EF are given in feet. What is the area, in square feet, of rectangle ACDF ?

$$Area = 6\times(4+6 …


2015 December(72E) Math Question 59

59. What is 1/3% of 6/7 ?

Choice A

$$\frac{6}{7}\times \frac …


2015 December(72E) Math Question 60

60. For what real value of x, if any, is log(x+3)(x^2+ 3) = 2

Choice G

$$(x+3)^2=x^2 …