ACT Form 71C Math Answer Explanation



2013 June(71C) Math Question 1

Statistics

- Mean, Median

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?

Choice A

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


Choice K

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


Choice B

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


2013 June(71C) Math Question 4

Word Problems

- 2 Variable Addition

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 ?

Choice F

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 A

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


2013 June(71C) Math Question 6

Angles

- Collinear Points & Triangles

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

Choice K

$$\angle C=180°-57°-57°=66°$$


2013 June(71C) Math Question 7

Perimeter, Area, Volume

- Mixed Shapes

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?

Choice B

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


2013 June(71C) Math Question 8

Word Problems

- Fixed Cost + Variable Cost

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?

Choice H

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


Choice B

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 H

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


2013 June(71C) Math Question 11

Word Problems

- 2 Variable Addition

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?

Choice D

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


2013 June(71C) Math Question 12

Perimeter, Area, Volume

- Mixed Shapes

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 ?

Choice H

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


2013 June(71C) Math Question 13

Inequalities

- Number Line

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

Choice B

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


Choice G

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


2013 June(71C) Math Question 15

Probability

- One Event

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?

Choice D

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


Choice H

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


2013 June(71C) Math Question 17

Angles

- Circles

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?

Choice D

$$\frac{1}{40}\times360°=9°$$


2013 June(71C) Math Question 18

Probability

- More than One Event

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 ?
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Choice F

24, 36, and 12 are multiples of both 3 and 4: $$\frac{1+1+1}{8}=\frac{3}{8}$$


Choice E

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


2013 June(71C) Math Question 20

Word Problems

- Interesection

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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Choice J

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


Choice A

When the graph is shifted …


2013 June(71C) Math Question 22

Equation of a Line

- Distance & Midpoint

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

Choice J

$$\frac{AB}{2}+BC=\frac{12}{2}+20=26$$


2013 June(71C) Math Question 23

Quadratic Equations

- Factoring

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

Choice A

$$x^2-5x+6=(x-3)(x-2)$$


Choice H

$$(-x^2+2x+8)+d=(5x^2+3) \\ $$ $$\Rightarrow d=7x^2-2x-5$$


Choice D

$$\frac{3+\frac{1}{9}}{1+\frac{1}{18}}=\frac{\frac{3\times9+1}{9}}{\frac{1\times18+1}{18}}=2\frac{18}{19}$$


Choice K

$$\frac{\frac{4}{3}}{\frac{1}{6}}=\frac{4}{3}\times6=8$$


Choice E

$$2y-18= 0\\ $$ $$\Rightarrow y …


Choice H

$$g(-4)=5-(-4)^2=5-16=-11 \\$$ $$ f(g(-4))=f(-11)=-11-5=-16$$


2013 June(71C) Math Question 29

Trigonometry

- Pythagorean Theorem & sin cos tan

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

Choice B

$$\cos A = \frac{AC}{AB}=\sin B$$


Choice G

$$x^2-4\neq0\\$$ $$ \Rightarrow x\neq2 \ …


Choice E

$$12\times \frac{210}{0.03}=84000$$


2013 June(71C) Math Question 32

Statistics

- Mean, Median

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?

Choice J

$$247\times 7=1729$$


2013 June(71C) Math Question 33

Quadratic Equations

- Factoring

33. Which of the following values is a zero of f(x) = 2x 5x2 — 12x ?

Choice C

$$2x^3-5x^2-12x=0\\ $$ $$\Rightarrow x(x-4)(2x+3)=0\\$$ $$ …


2013 June(71C) Math Question 34

Sequences and Patterns

- General Sequences

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?

Choice J

$$1+3+5+...+(2n-1)=n^2 \\$$ $$ \Rightarrow 1+3+5+...+23=12^2$$


Choice E

$$\frac{-4-5}{3-(-2)}=-\frac{9}{5}$$


Choice G

$$0.3\overline{4}=0.3444444...$$


2013 June(71C) Math Question 37

Conics

- Parabola

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?

Choice B

The other point must be …


Choice H

$$100=a5^2 \\$$ $$ \Rightarrow a=4$$


2013 June(71C) Math Question 39

Conics

- Parabola

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?

Choice B

$$x=ay^2$$ is open to the …


2013 June(71C) Math Question 40

Absolute Values

- Absolute Value Expressions

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

Choice F

$$k>0\Rightarrow-k<0 \Rightarrow |x|=|y| …


2013 June(71C) Math Question 41

Graphing Properties

- Create Graphs or Figures

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?

Choice A

Since when t is a, …


2013 June(71C) Math Question 42

Angles

- Collinear Points & Triangles

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?

Choice G

b + e = 160°; …


2013 June(71C) Math Question 43

Angles

- Vertex Angles & Bisectors

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 ?

Choice C

$$\angle ABE=\angle ABC+\angle CBD+\angle DBE …


2013 June(71C) Math Question 44

Trigonometry

- Advanced sin, cos, tan functions

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 ?

Choice F

Generally:

$$\frac{\sin T}{LS}=\frac{\sin S}{LT}$$


Choice B

71c

This is what a bag …


Choice H

$$H=\frac{24d}{D}-1$$


2013 June(71C) Math Question 47

Logarithms

- Logarithm Expressions

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

Choice D

$$\log_{}{((2x)^\frac{1}{2})}=\frac{1}{2}\log_{}{(2x)}=\frac{1}{2}\log_{}{2}+\frac{1}{2}\log_{}{x}$$


Choice H

It can be proved that …


2013 June(71C) Math Question 49

Trigonometry

- Graphing Trig Functions

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?

Choice A

The range of the graph …


2013 June(71C) Math Question 50

Statistics

- General Statistics

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%.

Choice F

$$\frac{74.4-69}{2.7}=2$$


2013 June(71C) Math Question 51

Perimeter, Area, Volume

- Mixed Shapes

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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Choice C

$$a+b+c=\frac{AB+BC+AC}{2}=\frac{13+15+26}{2}=27$$


Choice K

$$8^2+11^2<16^2 $$ $$\Rightarrow obtuse …


2013 June(71C) Math Question 53

Angles

- Circles

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 ?

Choice D

$$\frac{9}{2} \pi=\frac{1}{2} \pi+4 \pi \\ …


2013 June(71C) Math Question 54

Quadratic Equations

- Factoring

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?

Choice K

When two polynominals are multiplied, …


Choice D

$$\frac{48}{6}\times12+\frac{48}{6}\times12+\frac{36}{6}\times12-12=252$$


2013 June(71C) Math Question 56

Inequalities

- Intersection of Lines

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

Choice G

Only graph G fulfils the …


2013 June(71C) Math Question 57

Angles

- Circles

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 ?

Choice C

71c

$$CD=\sqrt{CF^2-DF^2}=\sqrt{7^2-5^2}=2\sqrt{6}$$


Choice K

The domain of f is …


2013 June(71C) Math Question 59

Inequalities

- Inequality Expressions

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

Choice A

$$a=b-5 \\$$ $$ b\geq c+3 …


Choice K

$$2^{n+3}=2^{n+1}\cdot2^2=2m\cdot 2^2=8m$$