ACT Form 73C Math Answer Explanation


2015 June(73C) Math Question 1

1. Which of the following expressions is equivalent to a(4 — a) — 5(a + 7) ?

Choice C

$$a(4-a)-5(a+7)=4a-a^2-5a-35=-a^2-a-35$$


2015 June(73C) Math Question 2

2. Which of the following inequalities orders the numbers 0.2, 0.03, and from least to greatest?

Choice G

$$\frac{1}{4}=0.25 \\$$ $$ \Rightarrow 0.03 \leq 0.2 \leq 0.25 \\$$ $$ \Rightarrow 0.03 \leq 0.2 \leq \frac{1}{4}$$


2015 June(73C) Math Question 3

3. If x^2 + 4 =29,then x^2— 4 =?

Choice C

$$x^2=29-4=25 \\$$ $$ \Rightarrow x^2-4=25-4=21$$


2015 June(73C) Math Question 4

4. In the figure below, QRS is equilateral, side QR is bisected by T, and side QS is bisected by U. What is the degree measure of TUS ?

Choice H

Since T and U bisect QR and QS, respectively, TU//RS. Thus, $$\angle TUQ=\angle S=60° \\$$ $$ \Rightarrow \angle TUS=180°-\angle TUQ=180°-60°=120°$$


2015 June(73C) Math Question 5

5. A line in the standard (x,y) coordinate plane passes through the points (-3,4) and (2,-5). The slope of the line:

Choice C

5. A line in the standard (x,y) coordinate plane passes through the points (-3,4) and (2,-5). The slope of the line:

The line is going downwards. Therefore the slope is negative.


2015 June(73C) Math Question 6

6. The square root of a certain number is approximately 9.2371. The certain number is between what 2 integers?

Choice K

$$\sqrt{81}=9 \\ $$ $$\sqrt{99}\approx\sqrt{100}=10$$ $$ \\ 9 \leq 9.2371 \leq 10 \\$$ $$ \Rightarrow 81\leq the \ number \leq 99$$


2015 June(73C) Math Question 7

7. A bag contains 10 pieces of flavored candy: 4 lemon, 3 strawberry, 2 grape, and 1 cherry. One piece of candy will be randomly picked from the bag. What is the probability the candy picked is NOT grape flavored?

Choice E

$$p=\frac{10-2}{10}=\frac{4}{5}$$


2015 June(73C) Math Question 8

8. When points A and B(-3,4) are graphed in the standard (x,y) coordinate plane below, the midpoint of AB will be (1,2). What will be the coordinates of point A ?

Choice K

$$\frac{x_{A}+(-3)}{2}=1 \\$$ $$ \Rightarrow x_{A}=5 \\$$ $$ \frac{y_{A}+4}{2}=2 \\$$ $$ \Rightarrow y_{A}=0 $$


2015 June(73C) Math Question 9

9. Andrea manages a company that currently has 116 customers, which is 8 more than twice the number of customers the company had 1 year ago. How many customers did the company have 1 year ago?

Choice B

$$2x+8=116 \\ $$ $$\Rightarrow x=54$$


2015 June(73C) Math Question 10

10. Joseph will have a 200-foot-long fence installed around his yard. The A+ Fence Company charges a $500.00 fee, plus a set amount per foot of fence. The A+ Fence Company has given Joseph an estimate of $2,200.00 to install the fence around his yard. What is the set amount per foot of fence?

Choice H

$$500+200x=$2,200 \\$$ $$ \Rightarrow x=$8.50$$


2015 June(73C) Math Question 11

11. Which of the following expressions is equivalent to (3x + 6)(2x — 1) ?

Choice D

$$(3x+6)(2x-1)=6x^2-3x+12x-6=6x^2+9x-6$$


2015 June(73C) Math Question 12

12. Carrie's Chocolate Shop and Tamika's Treat Shop both sell candy in boxes. The table below lists the price (the total amount the customer pays) of each box of candy sold at the shops. For each shop, there is a linear relationship between the price of a box of candies and the number of candies in that box. These are the only numbers of candies that can be purchased at the shops. Jeremy has $10.00 in quarters to spend on candy. What is the maximum number of quarters he would have left after paying for a box of 25 candies at Tamika's Treat Shop? (Note: Each quarter is worth $0.25.)

Choice J

A box of 25 candies at Tamika's Treat costs $4.25. Thus the money left is $10.00 - $4.25 = $5.75, which is 23 quarters: $$\frac{$5.75}{$0.25}=23$$


2015 June(73C) Math Question 13

13. At Tamika's Treat Shop, what is the average price per candy in a box of 20, to the nearest $0.01 ?

Choice B

$$\frac{$3.75}{20}=$0.1875\approx $0.19$$


2015 June(73C) Math Question 14

14. Which of the following equations gives the relationship between the price in dollars, c, and the number of candies, n, in a box of candies at Carrie's Chocolate Shop?

Choice F

Only the equation in F fulfils all the given prices and the number of candies given in the table.


2015 June(73C) Math Question 15

15. Mr. Chang made the true statement "At least 1 student in this class is a section leader in the band." Which of the following statements must be FALSE?

Choice E

The opposite of 'at least 1' is 'no one'.


2015 June(73C) Math Question 16

16. In the figure below, vertices D and F of ADEF lie on CG, the measure of LCDE is 148°, and the measure of EFG is 140°. What is the measure of DEF ?

2015 June(73C) Math Question 17

17. A company ships notepads in rectangular boxes that each have inside dimensions measuring 9 inches long, 9 inches wide, and 12 inches tall. Each notepad is in the shape of a cube with an edge length of 3 inches. What is the maximum number of notepads that will fit in 1 closed box?

Choice E

$$\frac{9\times9\times12}{3\times3\times3}=36$$


2015 June(73C) Math Question 18

18. The function f is defined as f(x) = —4x3 4x2. What is f(-4) ?

Choice J

$$f(-4)=-4(-4)^{3}-4(-4)^{2} =-4(-64)-4(16) =256-64 =192$$


2015 June(73C) Math Question 19

19. Which of the following (x,y) pairs is the solution for the system of equations x + 2y = 4 and —2x + y = 7 ?

Choice A

$$x+2y=4 \\ $$ $$-2x+y=7 \\$$ $$\Rightarrow x=-2, \ y=3$$


2015 June(73C) Math Question 20

20. The floor plan of Helena's kitchen is shown in the figure below; the given dimensions are in feet. Helena will install baseboard along the bottom of each wall (shown by solid lines in the floor plan). According to the floor plan, which of the following distances is closest to the perimeter, in feet, of Helena's kitchen?

Choice G

20. The floor plan of Helena's kitchen is shown in the figure below; the given dimensions are in feet. Helena will install baseboard along the bottom of each wall (shown by solid lines in the floor plan). According to the floor plan, which of the following distances is closest to the perimeter, in feet, of Helena's kitchen?

$$x=\sqrt{8^2+9^2}\approx12 \\$$ $$Perimeter = 8+10+12+19=49\approx50$$


2015 June(73C) Math Question 21

21. A 5-inch-by-7-inch photograph was cut to fit exactly into a 4-inch-by-6-inch frame. What is the area, in square inches, of the part of the photograph that was cut off?

Choice C

$$5\times7-4\times6=11$$


2015 June(73C) Math Question 22

22. In the figure shown below, AE || DC, AC and DE intersect at B, and the given lengths are in feet. What is the length, in feet, of BC ?

Choice H

$$\frac{BC}{AB}=\frac{BD …


2015 June(73C) Math Question 23

23. Given that x <= 4 and x + y >= 5, what is the LEAST value that y can have?

Choice D

When x=4, y can …


2015 June(73C) Math Question 24

24. The circle shown below has diameter AD, and points B and C lie on the circle. The measure of ZCAD is 300, and the measure of minor arc CD is 60°. What is the measure of minor arc AC ?

Choice J

24. The circle shown below has diameter AD, and points B and C lie on the circle. The measure of ZCAD is 300, and the measure of minor arc CD is 60°. What is the measure of minor arc AC ?


2015 June(73C) Math Question 25

25. One caution sign flashes every 4 seconds, and another caution sign flashes every 10 seconds. At a certain instant, the 2 signs flash at the same time. How many seconds elapse until the 2 signs next flash at the same time?

Choice D

Let the number of times …


2015 June(73C) Math Question 26

26. For all nonzero values of a and b, the value of which of the following expressions is always negative?

Choice K

Since a and b are …


2015 June(73C) Math Question 27

27. Graphed in the same standard (x,y) coordinate plane are a circle and a parabola. The circle has radius 3 and center (0,0). The parabola has vertex (-3,-2), has a vertical axis of symmetry, and passes through (-2,-1). The circle and the parabola intersect at how many

Choice C

27. Graphed in the same standard (x,y) coordinate plane are a circle and a parabola. The circle has radius 3 and center (0,0). The parabola has vertex (-3,-2), has a vertical axis of symmetry, and passes through (-2,-1). The circle and the parabola intersect at how many


2015 June(73C) Math Question 28

28. 40% of 250 is equal to 60% of what number?

Choice H

$$40\%\times250=60\%x \\$$ $$ \Rightarrow …


2015 June(73C) Math Question 29

29. Which of the following inequalities is equivalent to —2x— 6y> 2y— 4 ?

Choice A

$$-2x-6y>2y-4 \\$$ $$ \Rightarrow -2x>2y …


2015 June(73C) Math Question 30

30. For an angle with measure a in a right triangle, sin a = 40/41 and tan a = 40/9. What is the value of cos a ?

Choice F

$$ \tan\alpha=\frac{\sin\alpha …


2015 June(73C) Math Question 31

31. The perimeter of rectangle ABCD is 96 cm. The ratio of the side lengths AB:BC is 3:5. What is the length, in centimeters, of AB ?

Choice B

$$3k+5k+3k+5k=96 …


2015 June(73C) Math Question 32

32. For ABC shown below, base AC has a length of 16 inches and altitude BD has a length of 8 inches. The area of a certain square is equal to the area of ABC. What is the length, in inches, of a side of the square?

Choice G

$$d^2=\frac{16\times8 …


2015 June(73C) Math Question 33

33. In the figure shown below, ABCD is a rectangle, EFGH is a square, and CD is the diameter of a semicircle. Point K is the midpoint of CD. Point J is the midpoint of both AB and EF. Points E and F lie on AB. The 3 given lengths are in meters. The length of EH is what percent of the length of AD?

Choice B

$$\frac{3.6}{12}=30 \%$$


2015 June(73C) Math Question 34

34. What is the length, in meters, of JD?

Choice F

34. What is the length, in meters, of JD?

$$JD=\sqrt{5^2+12 …


2015 June(73C) Math Question 35

35. What is the length, in meters, of arc CD?

Choice B

$$length \ of arc\ \widehat{CD …


2015 June(73C) Math Question 36

36. The figure will be placed in the standard (x,y) coordinate plane so that K is at the origin, AB is parallel to the x-axis, and 1 meter equals 1 coordinate unit. Which of the following values could be the y-coordinate of H?

Choice H

12 - 3.6 = 8.4


2015 June(73C) Math Question 37

37. What is the length, in coordinate units, of the altitude from C to AB in ABC shown in the standard (x, y) coordinate plane below?

Choice A

4 - 1 = 3


2015 June(73C) Math Question 38

38. At a local post office, on average, 3 customers are in line when the post office closes each day. The probability, P, that exactly n customers are in line when the post office closes can be modeled by the equation P=. Given that e-3 0.05, which of the following values is closest to the probability that exactly 2 customers are in line when the post office

Choice J

$$P=\frac{3^ne^{-3 …


2015 June(73C) Math Question 39

39. What is the amplitude of the function f(x) = 1/2 cos(3x + pi)?

Choice B

The amplitude of f(x …


2015 June(73C) Math Question 40

40. A sphere with radius x inches has a volume of 12 cubic inches. What is the value of x ? (Note: The volume of a sphere with radius r is 4/3 Pi r^3.)

Choice H

$$\frac{4}{3}\pi x …


2015 June(73C) Math Question 41

41. For 20 quiz scores in a typing class, the table below gives the frequency of the scores in each score interval. Which score interval contains the median of the scores?

Choice D

The median will be the …


2015 June(73C) Math Question 42

42. The angle of elevation to the top of a tree from a spot on level ground 50 feet from the base of the tree is 40°. Which of the following is closest to the height of the tree, in feet? (Note: sin 40° = 0.64, cos 40° = 0.77, tan 40° = 0.84)

Choice H

$$\tan40°=\frac{height}{50}\\$$ $$ \Rightarrow …


2015 June(73C) Math Question 43

43. After visiting Florida State University during spring break, Francisco rents a car for 2 days to travel around Florida. He has $255 to spend on car rental for the 2 days. Sea Horse Car Rental charges $50 per day and $0.25 per mile. Ocean Blue Car Rental charges $60 per day and $0.20 per mile. Which company, if either, allows him to travel more miles for the 2 days, and how many miles more? (Note: Taxes are already included in the rental charges.)

Choice B

Sea Horse: $$\frac{255-50-50}{0 …


2015 June(73C) Math Question 44

44. The table below gives experimental data values for variables x and y. Theory predicts that y varies directly with x. Based on the experimental data, which of the following values is closest to the constant of variation? (Note: The variable y varies directly with the variable x provided that y = kx for some nonzero constant k, called the constant of variation.)

Choice G

0.05 fits the data …


2015 June(73C) Math Question 45

45. During a snowstorm, the relationship between the depth of accumulated snow, y inches, and the elapsed time, x hours, was modeled by the equation 2x — 5y = —5. One of the following graphs in the standard (x,y) coordinate plane models the equation for positive values of x and y. Which one?

Choice A

$$2x-5y=-5 \\ $$ $$\Rightarrow 2x+5 …


2015 June(73C) Math Question 46

46. Diana is baking bread, and the original recipe calls for 1 1/2 teaspoons of yeast and 2 1/2 cups of flour. Diana will use the entire contents of a packet that contains 2 1/4 teaspoons of yeast and will use the same ratio of ingredients called for in the original recipe. How many cups of flour will Diana use?

Choice J

$$\frac{x}{2\frac{1 …


2015 June(73C) Math Question 47

47. For all nonzero values of x,

Choice E

$$\frac{12x^6}{3x^2 …


2015 June(73C) Math Question 48

48. Four matrices are given bwlow. Which of the following matrix products is undefined?

Choice K

XZ is undefined.

https://www …


2015 June(73C) Math Question 49

49. A circle, 2 chords. and some lengths, in centimeters, are shown in the figure below, which is not drawn to scale. What is the value of x ? (Note: When two chords intersect, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.)

Choice D

$$2x=5\times7 \\$$ $$ \Rightarrow x …


2015 June(73C) Math Question 50

50. If the area of KLMN in the figure below is 12 square centimeters, what is the length, in centimeters, of LM ?

Choice F

50. If the area of KLMN in the figure below is 12 square centimeters, what is the length, in centimeters, of LM ?

$$\frac{[5+(5+x)]\cdot …


2015 June(73C) Math Question 51

51. A teacher assigns each of her 18 students a different integer from 1 through 18. The teacher forms pairs of study partners by using the rule that the sum of the pair of numbers is a perfect square. Assuming the 9 pairs of students follow this rule, the student assigned which number must be paired with the student assigned the number 1?

Choice B

The square of a number …


2015 June(73C) Math Question 52

52. Given that (x + 2) and (x — 1) are factors of the quadratic expression below, what are the values of a and b ?

Choice K

$$(x-1)(x+2)=x^{2 …


2015 June(73C) Math Question 53

53. The height above the ground, h units, of an object seconds after being thrown from the top of a building is given by the equation h = —2t2 + 10: + 48. An equivalent factored form of this equation shows that the object:

Choice E

53. The height above the ground, h units, of an object seconds after being thrown from the top of a building is given by the equation h = —2t2 + 10: + 48. An equivalent factored form of this equation shows that the object:

Since the given equation is …


2015 June(73C) Math Question 54

54. In the standard (x,y) coordinate plane below, the line through the points (-2,0) and (0,-4) is graphed. Which of the following values is the slope of any line that is in this plane and is perpendicular to the graphed line?

Choice J

The slopes of two lines …


2015 June(73C) Math Question 55

55. The shaded portion of the graph in the standard(x,y) coordinate plane below represents the solution set of one of the following systems of linear inequalities. Which one?

Choice A

The shaded area lies below …


2015 June(73C) Math Question 56

56. Whenever j and k are positive integers such that (square foot of 3)^j = 27^k, what is the value of j/k ?

Choice K

$$(\sqrt{3})^{j}=27^{k …


2015 June(73C) Math Question 57

57. A finite arithmetic sequence has 7 terms, and the first term is 3/4. What is the difference between the mean and the median of the 7 terms?

Choice A

For an arithmetic sequence, the …


2015 June(73C) Math Question 58

58. In the circle with center D shown below, the length of radius CD is 4 cm, the length of BC is 1 cm, and BC is perpendicular to radius AD at B. When ADC is measured in degrees, which of the following expressions represents the length, in centimeters, of AC ?

Choice F

$$sin\theta=\frac{1}{4 …


2015 June(73C) Math Question 59

59. Which of the following linear equations gives the vertical asymptote for the graph of y = 201x+ 202 / 203x + 204 in the standard (x,y) coordinate plane?

Choice D

The asymptote of a function …


2015 June(73C) Math Question 60

60. A right circular cone is intersected by a plane that is not parallel to the base and does not intersect the base or vertex, ps shown in the figure below. One of the following figures shows the shape of the intersection. Which figure is it?

Choice F

60. A right circular cone is intersected by a plane that is not parallel to the base and does not intersect the base or vertex, ps shown in the figure below. One of the following figures shows the shape of the intersection. Which figure is it?

Conic sections are the result …