ACT Form 72C Math Answer Explanation


2014 June(72C) Math Question 1

1. The monthly fees for single rooms at 5 colleges are $370, $310, $380, $340, and $310, respectively. What is the mean of these monthly fees?

Choice C

To calculate the mean of n numbers, take the sum of the n numbers and divide it by n. $$\frac{$370+$310+$380+$340+$310}{5}=$342$$


2014 June(72C) Math Question 2

2. Disregarding sales tax, how much will you save when you buy a $12.00 compact disc that is on sale for 25% off?

Choice H

A discount is a percentage that is subtracted from a number.

For example, a 10% discount of $30 is $3 (10% converted to a decimal is .10 and .10 x 30 is 3).

$12 x 25% = $3


2014 June(72C) Math Question 3

3. Given f = cd3, f= 450, and d = 10, what is c

Choice A

$$450 = c\cdot 10^3 \\$$ $$ \Rightarrow x=0.45$$


2014 June(72C) Math Question 4

4. Jorge's current hourly wage for working at Denti Smiles is $12.00. Jorge was told that at the beginning of next, month, his new hourly wage will be an increase of 6% of his current hourly wage. What will be Jorge's new hourly wage?

Choice H

% increase = Increase ÷ Original Number × 100

$12 × 106% = $12.72


2014 June(72C) Math Question 5

5. The first term is 1 in the geometric. sequence 1, —3, 9, —27, .... What is the SEVENTH, term of the geometric sequence?

Choice E

In a Geometric Sequence each term is found by multiplying the previous term by a constant.

The constant for the sequence in question is -3. Therefore the sequence is:

1, -3, 9, -27, 81, -243, 729, ...


2014 June(72C) Math Question 6

6. If /a = b and b= 36, a = ?

Choice K

A square root of a number a is a number y such that y^2 = a; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is a. $$\sqrt{a} = 36 \\ ⇒a = 36^2 = 1296$$


2014 June(72C) Math Question 7

7. The shipping rate for customers of Ship Quick consists of a fee per box and a price per pound for each box. The table below gives the fee and the price per pound for customers shipping boxes of various weights.

Choice C

$10 + $0.65 × 15 = $19.75


2014 June(72C) Math Question 8

8. The table below shows the number of cars Jing sold each month last year. What is the median of the data in the table?

Choice K

The median of a set of numbers is the middle number in the set (after the numbers have been arranged from least to greatest) -- or, if there are an even number of data, the median is the average of the middle two numbers.

In descending order, the set of numbers is:

13, 15, 16, 19, 19, 22, 25, 25, 26, 27, 28, 29

Median = (22 + 25) / 2 = 23.5


2014 June(72C) Math Question 9

9. 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

A relationship of direct proportionality that, when plotted on a graph, traces a straight line. In linear relationships, any given change in an independent variable will always produce a corresponding change in the dependent variable.

Thus, d = at + k

14 = 0⋅a + k

20 = 1⋅a + k

k = 14

a = 6

d = 6t + 14


2014 June(72C) Math Question 10

10. If x + 5/8 = 5/24, then x = ?

Choice K

x = 5/24 - 5/8 = 5/24 - 15/24 = - 10/24 = - 5/12


2014 June(72C) Math Question 11

11. The absolute value of which of the following numbers is the greatest?

Choice A

The absolute value of a number is its distance from zero on a number line . For instance, and have the same absolute value (). So, the absolute value of a positive number is just the number itself, and the absolute value of a negative number is its opposite.

|-0.4|= 0.4

|-0.042| = 0.042

|-0.0048| = 0.0048

|0.04| = 0.04

|0.047| = 0.047


2014 June(72C) Math Question 12

12. In the figure below, C is the intersection of AD and BE. If it can be determined, what is the measure of LBAC ?

Choice G

12. In the figure below, C is the intersection of AD and BE. If it can be determined, what is the measure of LBAC ?

The sum of measures of three angles of any triangle is invariably equal to 180°.

Vertical Angles are angles opposite each other when two lines cross. They are always equal.

∠A = 180° - 35° - 45° = 100°


2014 June(72C) Math Question 13

13. This month, Kami sold 70 figurines in 2 sizes. The large figurines sold for $12 each, and the small figurines sold for $8 each. The amount of money he received from the sales of the large figurines was equal to the amount of money he received from the sales of the small figurines. How many large figurines did Kami sell this month?

Choice B

Let the number of large figurines be L;

Let the number of small figurines be S;

L + S = 70

12⋅L = 8⋅S

L = 28, S= 42


2014 June(72C) Math Question 14

14. Given that √2x — 11 = 1, x = ?

Choice K

√(2x) = 1+11 = 12

⇒ 2x = 144

⇒ x = 72


2014 June(72C) Math Question 15

15. Antwan drew the circle graph below describing his time spent at school in 1 day. His teacher said that the numbers of hours listed were correct, but that the central angle measures for the sectors were not correct. What should be the central angle measure for the Core subjects sector?

Choice C

A circle has a total of 360 degrees all the way around the center.

So if that central angle determining a sector has an angle measure of 60 degrees, then the sector takes up 60/360 or 1/6, of the degrees all the way around.

$$\frac{4}{4+1+1+3}\times 360°= 160°$$


2014 June(72C) Math Question 16

16. The area of a rectangular sheet of paper is 32 square inches. The length of the sheet of paper is twice its width. What is the perimeter, in inches, of the sheet of paper?

Choice K

To find the area of a rectangle, multiply the length by the width. The formula is: A = L * W where A is the area, L is the length, W is the width, and * means multiply.

The perimeter of a rectangle is equal to the sum of all the sides. However, since a rectangle's opposite sides are congruent, we only need to know the length and width.

L × W = 32

L = 2W

⇒ W = 4, L = 8

⇒ Perimeter = (4 + 8) × 2 = 24


2014 June(72C) Math Question 17

17. A car accelerated from 88 feet per second (fps) to 220 fps in exactly 3 seconds. Assuming the acceleration was constant, what was the car's acceleration, in feet per second per second, from 88 fps to 220 fps ?

Choice C

Given the final speed, vf (which is 220 fps), and the initial speed, vi (which is 88 fps), and you know the time needed (3 seconds), you can find the acceleration, a. Because vf – vi = a⋅t

vf – vi = a⋅t

⇒ 220 - 88 = 3a

⇒ a = 44


2014 June(72C) Math Question 18

18. In scientific notation, 670,000,000 + 700,000,000 = ?

Choice J

Scientific notation is a way of writing very large or very small numbers. A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10. For example, 650,000,000 can be written in scientific notation as 6.5 ✕ 108

670,000,000 + 700,000,000 = 1,370,000,000 = 1.37 ✕ 109


2014 June(72C) Math Question 19

19. In a plane, the distinct lines AB and CD intersect at A, where A is between C and D. The measure of ∠BAC is 47°. What is the measure of ∠BAD ?

Choice D

19. In a plane, the distinct lines AB and CD intersect at A, where A is between C and D. The measure of ∠BAC is 47°. What is the measure of ∠BAD ?

The angle measure of a straight line is 180 degrees.


2014 June(72C) Math Question 20

20. Which of the following expressions is equivalent to (3x)^2/x^5 ?

Choice H

Rule name Rule Example
Product rules a na m = a n+m 23 ⋅ 24 = 23+4 = 128
a nb n = (a b) n 32 ⋅ 42 = (3⋅4)2 = 144
Quotient rules a n / a m = a nm 25 / 23 = 25-3 = 4
a n / b n = (a / b) n 43 / 23 = (4/2)3 = 8
Power rules (bn)m = bn⋅m (23)2 = 23⋅2 = 64
bnm = b(nm) 232 = 2(32)= 512
m√(bn) = b n/m 2√(26) = 26/2 = 8
b1/n = nb 81/3 = 38 = 2
Negative exponents b-n = 1 / bn 2-3 = 1/23 = 0.125
Zero rules b0 = 1 50 = 1
0n = 0 , for n>0 05 = 0
One rules b1 = b 51 = 5
1n = 1 15 = 1
Minus one rule (-1)n = -1 , n odd v
(-1)n = 1 , n even
(-1)5 = -1
Derivative rule (xn) = nx n-1 (x3) = 3⋅x3-1
Integral rule xndx = xn+1/(n+1)+C x2dx = x2+1/(2+1)+C

$$\Rightarrow \frac{(3x)^2}{x^5}=\frac{9x^2}{x^5}=\frac{9}{x^{5-2}}=\frac{9}{x^{3}}$$


2014 June(72C) Math Question 21

21. Whenever 4x + 7 = 2x – g, which of the following expressions must be equal to x ?

Choice A

⇒ 4x - 2x = -g - 7

⇒ 2x …


2014 June(72C) Math Question 22

22. For trapezoid ABCD shown below, AB || DC, the measures of the interior angles are distinct, and the measure of ∠D is x°. What is the degree measure of ∠A in terms of x ?

Choice F

When parallel lines get crossed …


2014 June(72C) Math Question 23

23. Which of the following expressions is equivalent to 1/2 y^2(6x + 2y + 12x -2y) ?

Choice A

$$\require{cancel} \frac{1}{2 …


2014 June(72C) Math Question 24

24. Sara and Behzad are saving to make a down payment on a house. With an initial deposit of $8,000, they have opened an account that compounds interest at an annual rate of 2.1%. Assuming that Sara and Behzad make no additional deposits or withdrawals, which of the following expressions gives the dollar value of the account 4 years after the initial deposit? (Note: For an account with an initial deposit of P dollars that compounds interest at an annual rate of r%, the value of the account t years after the initial deposit is P(1 + 00) dollars.)

Choice F

Plugging in the numbers in …


2014 June(72C) Math Question 25

25. Right triangle △RST has its right angle at vertex S. The length of ST is 6.0 feet and the length of RS is 2.5 feet. Which of the following values is closest to the length, in feet, of RT ?

Choice D

25. Right triangle △RST has its right angle at vertex S. The length of ST is 6.0 feet and the length of RS is 2.5 feet. Which of the following values is closest to the length, in feet, of RT ?

The Pythagorean theorem states that …


2014 June(72C) Math Question 26

26. An artist makes a profit of (500p — p^2) dollars from selling p paintings. What is the fewest number of paintings the artist can sell to make a profit of at least $60,000 ?

Choice H

The phrase "At least" means …


2014 June(72C) Math Question 27

27. Last month, Lucie had total expenditures of $900. The pie chart below breaks down these expenditures by category. The category in which Lucie's expenditures were greatest is what percent of her total expenditures, to the nearest 1 % ?

Choice B

Among the categories of expenditures …


2014 June(72C) Math Question 28

28. When the system of inequalities below is graphed in the standard (x,y) coordinate plane, one of the following graphs is that of the solution set of the system. Which graph?

Choice G

To graph a linear inequality …


2014 June(72C) Math Question 29

29. Janelle is loading a truck by using a ramp, as shown below. The ramp is 8 feet long, and the end of the ramp that is resting on the truck is 2.5 feet above the level ground. Which of the following expressions gives the angle of inclination of the ramp?

Choice B

To obtain an angle from …


2014 June(72C) Math Question 30

30. What is the perimeter, in inches, of the isosceles right triangle shown below, whose hypotenuse is 8√2 inches

Choice K

An Isosceles Right Triangle is …


2014 June(72C) Math Question 31

31. The equation y = ax^2 + bx + c is graphed in the standard (x,y) coordinate plane below for real values of a, b, and c. When y = 0, which of the following best describes the solutions for x ?

Choice C

31. The equation y = ax^2 + bx + c is graphed in the standard (x,y) coordinate plane below for real values of a, b, and c. When y = 0, which of the following best describes the solutions for x ?

As we can see from …


2014 June(72C) Math Question 32

32. The radius of the base of the right circular cone shown below is 5 inches, and the height of the cone is 7 inches. Solving which of the following equations gives the measure, 0, of the angle formed by a slant height of the cone and a radius?

Choice G

To obtain an angle from …


2014 June(72C) Math Question 33

33. A formula to estimate the monthly payment, p dollars, on a short-term loan is ... where a dollars is the amount of the loan, r is the annual interest rate expressed as a decimal, and y years is the length of the loan. When a is multiplied by 2, what is the effect on p ?

Choice D

The formula can be rearranged …


2014 June(72C) Math Question 34

34. To make a 750-piece jigsaw puzzle more challenging, a puzzle company includes 5 extra pieces in the box along with the 750 pieces, and those 5 extra pieces do not fit anywhere in the puzzle. If you buy such a puzzle box, break the seal on the box, and immediately select 1 piece at random, what is the probability that it will be 1 of the extra pieces?

Choice J

The probability of selecting an …


2014 June(72C) Math Question 35

35. The length of a rectangle is 3 inches more than twice the width of the rectangle. The perimeter of the rectangle is 36 inches. What is the width of the rectangle, in inches?

Choice B

The perimeter of a rectangle …


2014 June(72C) Math Question 36

36. The solid shown below is composed of a right circular cylinder and a right circular cone with base diameters and heights given in centimeters. The cylinder and the cone have equal base diameters. What is the volume, in cubic centimeters, of the solid?

Choice H

Plugging the data in the …


2014 June(72C) Math Question 37

37. Gianna is converting a 12-foot-by-15-foot room in her house to a craft room. Gianna will install tile herself but will have CC Installations build and install the cabinets. The scale drawing shown below displays the location of the cabinets in the craft room (0.25 inch represents 2 feet). Cabinets will be installed along one of the 12-foot walls from floor to ceiling, and 4 cabinets that are each 3 feet tall will be installed in the middle of the room. These are the only cabinets that will be installed, and each of them will be 2 feet wide and 2 feet deep. CC Installations has given Gianna an estimate of $2,150.00 for building and installing the cabinets. A 15-foot wall is how many inches long in the scale drawing?

Choice B

It is given in the …


2014 June(72C) Math Question 38

38. Gianna will install tile on the portion of the floor that will NOT be covered by cabinets. What is the area, in square feet, of the portion of the floor that will NOT be covered by cabinets?

Choice H

38. Gianna will install tile on the portion of the floor that will NOT be covered by cabinets. What is the area, in square feet, of the portion of the floor that will NOT be covered by cabinets?

As we can see from …


2014 June(72C) Math Question 39

39. CC Installations' estimate consists of a $650.00 charge for labor, plus a fixed charge per cabinet. The labor charge and the charge per cabinet remain the same for any number of cabinets built and installed. CC Installations would give Gianna what estimate if the craft room were to have twice as many cabinets as Gianna is planning to have?

Choice D

The cabinet charge out of …


2014 June(72C) Math Question 40

40. Trapezoid ABCD is graphed in the standard (x,y) coordinate plane below. What is the slope of CD ?

Choice G

Given any two points on …


2014 June(72C) Math Question 41

41. When ABCD is reflected over the y-axis to A'B'C'D', what are the coordinates of D'?

Choice A

The rule for a reflection …


2014 June(72C) Math Question 42

42. Which of the following vertical lines cuts ABCD into 2 trapezoids with equal areas?

Choice K

42. Which of the following vertical lines cuts ABCD into 2 trapezoids with equal areas?

A line perpendicular to the …


2014 June(72C) Math Question 43

43. Given that for some real number a, what is x + z?

Choice D

43. Given that for some real number a, what is x + z?

To multiply a matrix by …


2014 June(72C) Math Question 44

44. The points E(6,4) and F(14,12) lie in the standard (x,y) coordinate plane shown below. Point D lies on EF between E and F such that the length of EF is 4 times the length of DE. What are the coordinates of D ?

Choice G

44. The points E(6,4) and F(14,12) lie in the standard (x,y) coordinate plane shown below. Point D lies on EF between E and F such that the length of EF is 4 times the length of DE. What are the coordinates of D ?

With the supplemented lines, we …


2014 June(72C) Math Question 45

45. A certain triangle has a perimeter of x meters. One side of the triangle is 60 meters long, another side is 1/3 the length of the perimeter, and the third side is 1/4 the length of the perimeter. What is the perimeter, in meters, of the triangle?

Choice E

The perimeter of a triangle …


2014 June(72C) Math Question 46

46. The difference (larger minus smaller) between 2 numbers is 15. If n represents the larger number, which expression below represents the average (arithmetic mean) of the 2 numbers?

Choice K

To calculate the mean of …


2014 June(72C) Math Question 47

47. 4/√2+2/√3=?

Choice B

47. 4/√2+2/√3=?

To add fractions there are …


2014 June(72C) Math Question 48

48. A square in the standard (x,y) coordinate plane has vertices at (1,3), (2,1), (4,2), and (3,4). Where do the diagonals of the square intersect?

Choice J

48. A square in the standard (x,y) coordinate plane has vertices at (1,3), (2,1), (4,2), and (3,4). Where do the diagonals of the square intersect?

The center of rectangle is …


2014 June(72C) Math Question 49

49. The shaded region in the graph below represents the solution set to which of the following systems of inequalities?

Choice A

To graph inequalities, first, graph …


2014 June(72C) Math Question 50

50. In the standard (x,y) coordinate plane, line a contains the points (-4,2) and (-1,-3), and line b contains the points (3,0) and (7,0). At what point does line a intersect line b ?

Choice F

50. In the standard (x,y) coordinate plane, line a contains the points (-4,2) and (-1,-3), and line b contains the points (3,0) and (7,0). At what point does line a intersect line b ?

It should be noted that …


2014 June(72C) Math Question 51

51. In the figure below, the side lengths and the length of an altitude of parallelogram BCDE are given in feet. What is the area, in square feet, of BCDE ?

Choice D

The area of a parallelogram …


2014 June(72C) Math Question 52

52. The square below is divided into 3 rows of equal area. In the top row, the region labeled A has the same area as the region labeled B. In the middle row, the 3 regions have equal areas. In the bottom row, the 4 regions have equal areas. What fraction of the square's area is in a region labeled A ?

Choice K

52. The square below is divided into 3 rows of equal area. In the top row, the region labeled A has the same area as the region labeled B. In the middle row, the 3 regions have equal areas. In the bottom row, the 4 regions have equal areas. What fraction of the square's area is in a region labeled A ?

$$A_{1}=\frac{1}{2 …


2014 June(72C) Math Question 53

53. Which of the following is a quadratic equation that has

Choice A

Factoring 9x2+12x+4 …


2014 June(72C) Math Question 54

54. Bonkosi mixes 60 milliliters of Solution A with 40 milliliters of Solution X. Solution A has a 40% hydrochloric acid concentration; Solution X has an unknown hydrochloric acid concentration. When Bonkosi tests the resulting 100-milliliter solution, she finds that it has a 36% hydrochloric acid concentration. What is the hydrochloric acid concentration of Solution X ?

Choice H

$$40\% \times 60 + a\times40 …


2014 June(72C) Math Question 55

55. What are the real number values of x that make the equation 4√x^12 = x^3 true?

Choice E

The square root must be …


2014 June(72C) Math Question 56

56. The functions y = sin x and y = sin(x + a) + b, for constants a and b, are graphed in the standard (x,y) coordinate plane below. The functions have the same maximum value. One of the following statements about the values of a and b is true. Which statement is it?

Choice F

It can be observed that …


2014 June(72C) Math Question 57

57. Which of the following number line graphs shows the solution set to the inequality |x — 5| < -1?

Choice E

The absolute value of a …


2014 June(72C) Math Question 58

58. The sides of an acute triangle measure 14 cm, 18 cm, and 20 cm, respectively. Which of the following equations, when solved for 0, gives the measure of the smallest angle of the triangle?

Choice J

The smallest angle must be …


2014 June(72C) Math Question 59

59. For all values of x where the expression is defined,

Choice D

To reduce a fraction to …


2014 June(72C) Math Question 60

60. Mr. Martin wants to plant 7 trees evenly spaced along a straight fence 300 feet long, with 1 of the trees at each end of the fence. About how many feet apart should he plant the trees?

Choice J

60. Mr. Martin wants to plant 7 trees evenly spaced along a straight fence 300 feet long, with 1 of the trees at each end of the fence. About how many feet apart should he plant the trees?

As we can see from …