ACT Form 71C Math Answer Explanation


2013 June(71C) Math Question 1

Statistics Practice

- 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 Practice

- 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

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

Choice K

$$\angle C=180°-57°-57°=66°$$

Choice B

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

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 Practice

- 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

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

Choice H

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

2013 June(71C) Math Question 13

Inequalities Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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 …

Choice J

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

2013 June(71C) Math Question 23

Quadratic Equations Practice

- 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

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

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 Practice

- 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 Practice

- Factoring

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

Choice C

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

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 Practice

- 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 Practice

- 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 …

Choice F

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

Choice A

Since when t is a, …

2013 June(71C) Math Question 42

Angles Practice

- 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 Practice

- 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 …

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 Practice

- 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 …

Choice A

The range of the graph …

2013 June(71C) Math Question 50

Statistics Practice

- 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

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

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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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

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