# ACT Form 71C Math Answer Explanations

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*Explain to me:*

$$\frac{26200-17500}{4}=2175$$

Choice A.

1. A car was purchased 4 years ago for $26,200. The current value of the car is $17,500. What was the car’s average decrease in value per year?

$$6x-4x=7+3 \\ \Rightarrow 2x=10 \\ \Rightarrow x=5$$

Choice K.

2. For what value of x is 6x — 3 = 4x + 7 true?

11 is not a factor of 132 because 132 can not be divided by 11.

Choice B.

3. Which of the following integers is NOT a factor of 132?

Since the number of coins is 325: q+n=325

Since the total value of the coins is $56.25: 0.25q+0.05n=56.25

Choice F.

4. A vending machine only accepts quarters ($0.25) and nickels ($0.05). When the machine was emptied Friday afternoon, 325 coins were counted and had a value of $56.25. Which of the following systems of equations, when solved, gives the number of quarters, q, and the number of nickels, n ?

$$x+7.5+x=24 \\ \Rightarrow x=8.25$$

Choice A.

5. Carlotta is helping her grandfather center a large framed picture on his living room wall. As shown in the figure below, the rectangular wall is 24 feet long, and the rectangular framed picture is 4.5 feet high and 7.5 feet long. The left edge of the frame will be x feet from the left edge of the wall, and the right edge of the frame will be x feet from the right edge of the wall. What is the value of x? DO YOUR FIGURING HERE.

$$\angle C=180°-57°-57°=66°$$

Choice K.

6. In AABC, the sum of the measures of LA and LB is 57°. What is the measure of LC ?

$$a^2=32\times2 \\ \Rightarrow a=8$$

Choice B.

7. A square and a rectangle have the same area. The length of the rectangle is 32 centimeters, and the width of the rectangle is 2 centimeters. What is the length, in centimeters, of a side of the square?

$$4250=50a+1500 \\ \Rightarrow a=55$$

Choice H.

8. As a motivational speaker, Bree speaks at school assemblies, charging a school district for her travel costs and a fixed amount per assembly. Bree used the equation C = 50a + 1,500 to determine the charge of C dollars to speak at a assemblies in the Escambia City School District. Bree charged the district $4,250 to speak at school assemblies, How many assemblies did Bree Speak at in this district?

0 cannot be a factor of any number; 1 can be a factor of any number; a may not be necessarily be a factor of b, and vice versa; ab cannot be a factor be a or b.

Choice B.

9. Let a and 17 be positive prime numbers. Which of the following numbers must be a factor of both a and b ?

$$80-5(4^2-6)+6=36$$

Choice H.

10. What is the value of 80 – 5(x2 y) + y when x = 4 and y = 6 ?

$$h+s=10 \\ 12h+14s=128 \\ \Rightarrow h=6 , \ s= 4$$

Choice D.

11. At the Winter Weather Store, the price of 1 hat is $12 and the price of 1 scarf is $14. Jovon spent $128 to buy 10 items—a combination of hats and scarves.. How many hats did he buy?

$$\frac{5\times5}{5\times(5+7)}=\frac{5}{12}$$

Choice H.

12. Square ABEF and parallelogram ACDG are shown in the figure below. Points •E and F are on DG, B is on AC, and the lengths given are in inches. What is the ratio of the area of ABEF to the area of ACDG ?

$$6x-5\geq 7 \\ \Rightarrow 6x \geq 12 \\ \Rightarrow x \geq2$$

Choice B.

13. Which of the following graphs shows the solution set for the inequality 6x – 5 ?

$$w+w+5+w+w+5=60 \\ \Rightarrow w=12.5$$

Choice G.

14. The length of a rectangle is 5 inches longer than the width. The perimeter of the rectangle is 60 inches. What is the width of the rectangle, in inches?

$$\frac{10}{40}=25\%$$

Choice D.

15. Miriam conducted a survey of the students in her 8th-grade class to determine which of 7 Olympic sports were the most popular. Each student who responded to the survey selected 1 Olympic sport as his or her favorite. The circle graph below shows the number of students who selected each of the 7 Olympic sports. A total of 40 students responded to the survey. To the nearest 0.1%, what percent of the students who responded to the survey selected Basketball?

$$\frac{9}{5}=9:5$$

Choice H.

16. In this survey, what is the ratio of the number of students who selected Athletics to the number of students who selected Soccer?

$$\frac{1}{40}\times360°=9°$$

Choice D.

17. In the circle graph, what is the angle measure of the sector that represents the number of students who responded to the survey who selected Tennis?

24, 36, and 12 are multiples of both 3 and 4:

$$\frac{1+1+1}{8}=\frac{3}{8}$$

Choice F.

18. A spinner dial from a game is shown in the figure below. Each numbered sector of the circle has the same central angle measure. If the arrow on the spinner dial is spun randomly, what is the probability the arrow will point to a sector whose number is both a multiple of 3 and a multiple of 4 ?

$$\frac{15+20+65}{200}=50\%$$

Choice E.

19. The counselors at Lakewood High School interviewed 200 students to determine placement in 3 different honors courses: Math, Chemistry, and Literature. The numbers of these students placed in these honors courses are shown in the Venn diagram below. Of these 200 students, 10% were placed into all 3 honors courses, and 35% were placed into exactly 2 honors courses. What percent of the students were placed into exactly J. honors course?

$$4n=14m \\ \Rightarrow n=7,\ m=2 \\ \Rightarrow t=4\times7=28 \ seconds$$

Choice J.

20. One sign flashes every 4 seconds, and another sign flashes every 14 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?

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21. Which of the following transformations shifts all points graphed in the standard (x,y) coordinate. plane down 5 coordinate units?

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22. For AC shown below, B is on AC, the length of AB is 12 cm, and the length of BC is 20 cm. What is the distance, in centimeters, between C and the midpoint

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23. ‘Which of the following expressions is a factored form ofx2-5x+6?

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24. The sum of (-22 + 2x + 8) and which of the following polynomials is (5×2+ 3) ?

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25. The

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26. Ava starts with a long list of numbers, each of which she must multiply by 3 and then divide by She could get the same results by multiplying each number on her list by which of the following numbers?

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27. When 6x = 2y – 18 is graphed in the standard (x,y) coordinate plane, what is the y-intercept?

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28. Given functions f(x)= x — 5 and g(x)= 5 — x2, what is f(g(-4))?

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29. For right triangle AABC shown below, which of the following expressions has a value that is equal to cos A ?.

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30. The equation y = :2+-34 is graphed in the standard (x,y) coordinate plane below. No point on the graph has which of the following x-coordinates?

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31. The Wilson family sold a shipment of walnuts to a farming cooperative for $210 at the rate of $0.03 per pound. If their shipment averaged 12 walnuts per pound, which of the following is closeAt to the total number of walnuts in the shipment? •

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32. Over a 7-day period at Phil’s Fitness, the average attendance of the slowest day and the busiest day was 247 members. The average of the other 5 days was also 247 members. The total attendance for the 7-day period at Phil’s Fitness was how many members?

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33. Which of the following values is a zero of f(x) = 2x 5×2 — 12x ?

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34. The sum of a sequence of consecutive odd numbers, where the smallest term is 1, is always a perfect square. For example, 1 + 3 = 22 and 1 + 3 + 5 + 7 = 42- One of the sequences described above has a- sum of 144. What is the largest odd number in the sequence?

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35. What is the slope of the line that passes through the points (-2,5) and (3,-4) in the standard (x,y) coordinate plane?

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36. Among the following rational numbers, which has the greatest value?

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37. A parabola with equation y = ax2 is graphed in the standard (x,y) coordinate plane below. Point Z(p,q) lies on the parabola. For any point Z(p,q) on the parabola, which of the following points must also be on the parabola?

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38. What is the value of a if the coordinates of Z are (5, 100) ?

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39. Suppose that the parabola is rotated clockwise (c’) by 90° about the origin. Which of the following :`,is an equation for the parabola that results. froni this rotation?

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40. For all k > 0, I xl = I y = —k has how many (x,y) solutions?

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41. A Ferris wheel is turning at a constant speed during 1 of its rotations. Let t represent the time that has elapsed since the wheel started turning and let h represent the height above ground level of a certain seat on the wheel. The seat is at its minimum height at t = a seconds and is at its maximum height at t = b seconds. One of the following graphs represents the relationship between t and h during this rotation. Which one?

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42. In the figure below, X and Z are on line m, and Y is on line 1. The exterior angle to AXYZ at Z measures 160°. One of the following statements gives sufficient . additional information to find the measure of Lb. Which one?

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43. In the figure below, C lies in the interior of the acute angle LABD, and D lies in the interior of the acute angle LCBE. The measure of LABE is 75°, the measure of LABC is 25°, and the measure of LDBE is 35°. What is the measure of LABD ?

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44. Li is standing at point L on the north side of the small canyon shown in the figure below. As measured by line of sight, Li is 2 miles from an observation tower at 7′, and she is 3 miles from a scenic overlook at S. Li, the observation tower, and the scenic overlook are all at the same elevation. The measure of LS is 28°. Which of the following equations, when solved, gives the measure of LT ?

B.

This is what a bag of mulch looks like:

“When the mulch is uniformly spread 2 inches deep, each bag of mulch will cover 12 square feet.”Therefore, the volume per bag of mulch is（2 inches × 12 square feet ）。

The total volume of mulch needed by Mr Green is（3 inches × 10 feet × 26 feet）÷ 2= （3 inches × 130 square feet）。

The number of mulch needed is n =（3 inches × 130 square feet）÷ （2 inches × 12 square feet ）= 32.5 = 16.25 = 17

45. Mr. Green is buying mulch for his triangular-shaped garden shown below. When the mulch is uniformly spread 2 inches deep, each bag of mulch will cover 12 square feet. Mr. Green plans to uniformly spread his mulch 3 inches deep. Which of the following is closest to the number of bags of mulch Mr. Green will need for his garden?

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46. A child’s dose of medicine is often based on an adult’s dose of medicine. For some medicines, Cowling’s rule, d = D(a + 1) , relates the dose of d milligrams for a 24child of age a years to the adult dose of D milligrams. Which of the following expressions gives a in terms of

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47. For all x > 0, which of the following expressions is equivalent to log ((2x)-) ?

H.

It can be proved that ΔAED is similar to ΔADC. It follows that CD/AC=BE/AB.

48. In AACD shown below, BE is parallel to CD; and the given lengths are in feet. Given that CD denotes the length, in feet, of CD, which of the following proportions involving CD must be true?

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49. The graph of y = a sin bx is shown below for certain positive values of a and b. One of the following values is equal to a. Which one?

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50. The graph below illustrates the normal distribution curve. The percent of the data that falls within each standard deviation from the mean is given to the nearest 0.1%.

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51. Circles with centers A, B, and C, respectively, are mutually tangent, as shown below, and have radii of lengths a, b, and c, respectively. The lengths of AB, BC, and AC are 13 inches, 15 inches, and 26 inches, respectively. What is the value of a + b + c ?

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52. Which of the following sets of 3 lengths, in decimeters, are the side lengths of an obtuse triangle? (Note: An obtuse triangle has 1 angle whose measure is greater than 90° and less than 180°.)

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53. Angle A measures radians from its initial side to its terminal side. Angle B has the same initial side and terminal side as Angle A. Which of the following measures could be that of Angle B ?

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54. A polynomial in x has m nonzero terms. Another polynomial in x has n nonzero terms, where m < n. These polynomials are multiplied and all like terms are combined. The resulting polynomial in x has a maximum of how many nonzero terms?

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55. Hannah is building a fence along 3 sides of a rectangle, leaving 1 of the short sides unfenced. The fence has a post at each corner. The rectangle is 30 feet by 48 feet, measured from the centers of the corner posts. As shown in the figure below, adjacent posts of the fence are set every 6 feet, and there are 12 wooden slats between each pair of posts. How many wooden slats will Hannah use for the fence?

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56. Which of the following graphs best represents the system of inequalities below?

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57. In the diagram below, chord EF is parallel to diameter AB. The length of EF is 10 inches and the length of AB is 14 inches. What is the distance, in inches, from C, the center of the circle, to D, the midpoint of EF ?

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58. The entire graph of y =f(x) is shown in the standard (x,y) coordinate plane below. One of the following sets is the domain off. Which set?

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59. Let the values of real numbers a, b, and c be restricted by the 2 conditions given below.

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60. Let m and n be nonzero real numbers such that 2n+ 1= 2m. Which of the following is an expression for 2 3 in terms of m ?

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