ACT Form 71G Math Answer Explanation



2013 April(71G) Math Question 1

Word Problems

- 1 Variable Equations

1. Harry is paid a regular hourly wage of $12. 50 per hour for working up to and including 40 hours in I week.
For each additional hour he works in a week, harry is paid twice his regular hourly wage. Harry worked
46 hours this week. What is his pay for this week?
(Note:Amounts are before taxes and benefits are deducted.)

Choice C

$$40\times$12.5+6\times$12.5\times2$$ $$=500+150$$ $$=650$$


Choice K

$$\frac{4(x-6)}{3}=16$$ $$\Rightarrow 4(x-6)=16\times3=48$$ $$\Rightarrow 4x-24=48$$ $$\Rightarrow 4x=72$$ $$\Rightarrow x=18$$


Choice A

$$2x+8=30$$ $$x=11$$


Choice J

$$\Rightarrow 6x-3=3x+1$$ $$\Rightarrow 3x=4$$ $$\Rightarrow x=\frac{4}{3}$$


2013 April(71G) Math Question 6

Probability

- Combinations

6. Tristan has 5 pairs of shoes, 6 pairs of pants, and
5 shirts, which can be worn in any combination.He needs to choose a clothes combination to wear to the school dance. How many different combinations consisting of I of his 5 pairs of shoes, I of his 6 pairs of pants, and I of his 5 shirts are possible for Tristan to wear to the dance?

Choice K

$$5\times6\times5=150$$


2013 April(71G) Math Question 7

Word Problems

- Temperature

7. In Arkansas in the twentieth century, the highest recorded temperature was 120 F and the lowest recorded temperature was-29 F. This highest recorded temperature was how many degrees Fahrenheit greater than this lowest recorded temperature?

Choice E

$$120-(-29)=149$$


Choice G

$$=\frac{3\times4+1\times5}{3\times1-4\times5}$$ $$=\frac{12+5}{3-20}$$ $$=\frac{17}{-17}$$ $$=-1$$


2013 April(71G) Math Question 9

Absolute Values

- Graphing Absolute Values

9. The function y=2 | x-3 | is graphed in the standard
(x, y) coordinate plane below
One of the following graphs in the standard (r,y)
coordinate plane shows the result of shifting the function up 4 coordinate units. Which graph?

Choice E

$$x=3,y=0\Rightarrow (3,0)$$ $$x=0,y=6\Rightarrow (0,6)$$ $$(3,0+4)=(3,4)$$ $$(0,6+4)=(0,10)$$


Choice G

$$21=3\times7$$ $$6=2\times3$$ $$4=2\times2$$ $$\Rightarrow 2\times2\times3\times7=84$$


2013 April(71G) Math Question 11

Statistics

- Mean, Median

11. The average of 5 numbers is 89. What is the 5th number if the first 4 of the numbers are 78.92.96
and 94?

Choice A

$$89\times5=445$$ $$445-78-92-96-94=85$$


Choice H

$$\frac{2520}{12\times14}=15$$


2013 April(71G) Math Question 13

Word Problems

- 1 Variable Equations

13. At Hamburger Heaven, Corissa paid less than $15 for her order of x hamburgers and x bags of french fries Each hamburger cost h dollars, and each bag of french fries cost f dollars. Which of the following expressions represents the amount of money, in dollars, that Corissa should have received back after she paid for her order with $15?
(Note:There is no tax on food at Hamburger Heaven.)

Choice D

$$15-(x\times f+x\times h)$$ $$=15-x(f+h)$$


2013 April(71G) Math Question 14

Absolute Values

- Absolute Value Expressions

14.  |8(-6)+3(2)|=?

Choice H

$$=\mid-48+6\mid=\mid-42\mid=42$$


Choice E

$$The\ area\ of\ a\ circle=\pi r^{2}$$ $$\Rightarrow \pi R^{2}-3\pi r^{2}$$


2013 April(71G) Math Question 16

Angles

- Parallel Lines


Choice H

$$y=50$$ $$x+y=180$$ $$\Rightarrow x=130$$ $$x-y=130-50=80$$


2013 April(71G) Math Question 17

Inequalities

- Number Line

17. Which of the following graphs shows the solution set for the inequality 3x-5>=7?

Choice B

$$3x-5\geq 7$$ $$\Rightarrow 3x\geq7+5$$ $$\Rightarrow x\geq \frac{7+5}{3}$$ $$\Rightarrow x\geq \frac{12}{3}$$ $$\Rightarrow x\geq 4$$


Choice K

$$s(-3)=3(-3)^{2}-5$$ $$=3(9)-5$$ $$=27-5$$ $$=22$$


2013 April(71G) Math Question 19

Angles

- Circles

19. A circle with center C is shown below. Points W,X,Y, and Z lie on the circle. The measure of ∠wcy is 100° the measure of∠ XCZ is 80°, and the measure of
∠ WCZ is150°. What is the measure of ∠XCY?

Choice C

71g

$$\angle WCZ-\angle WCY$$ $$150°-100°=50°=\angle YCZ$$ $$\angle XCY=\angle XCZ-\angle YCZ$$ $$ =80°-50°=30°$$


Choice G

$$500\div200=2.5$$ $$6\div2.5=2.4$$


Choice A

$$\frac{27}{5}=5.4$$ $$\frac{24}{6}=4$$ $$\frac{20}{4}=5$$ $$4\times500=2000$$


Choice H

$$(7+6)s+(8+6.5)f+(5.5+5)b$$ $$=13s+14.5f+10.5b$$


Choice B

$$5a=3---(1)$$ $$2a+3b=5---(2)$$ $$(2)\times5-(1)\times2$$ $$\Rightarrow (10a+15b)-(10a)=5\times5-3\times2$$ …


Choice H

$$24\times150=3600$$ $$4\times60=240$$ $$3600\div240=15$$


Choice E

$$7-(-4)x^{2}-6x+(5-3)$$ $$=11x^{2}-6x+2$$


2013 April(71G) Math Question 26

Graphing Properties

- Interpret Graphs or Figures

26. For what real number value of a is the equation ... true?

Choice J

$$(x^{2})^{3}(x^{4})^{5}=x^{a}$$ $$=x^{2\times3+4\times5}=x^{a}$$ $$6+20=a$$ $$a=26$$


Choice D

$$\frac{0.003}{100}=0.0003$$


2013 April(71G) Math Question 28

Probability

- One Event

Use the following information to answer uestions 28-30
Each of the 200 people in a random sample of the
2,500 people at the mall today was asked which, if any, of the following types of pets he or she owns:bird, cat, dog or fish. All 200 people answered the question. The answers were tallied, and the exact percents of people who own the pets are shown in the diagram below.
28. Because this was a random sample, the percents in the sample are the most likely estimates for the corresponding percents among all the people at the mall today. What estimate does this give for the number of people at the mall today who own dogs but none of the other 3 types of pets?

Choice H

$$2500\times0.15=375$$


2013 April(71G) Math Question 29

Probability

- One Event

29. What percent of the people in the random sample own exactly 1 type of the 4 types of pets?

Choice D

$$45+5+15+5=70$$


2013 April(71G) Math Question 30

Probability

- One Event

30. Suppose 25 additional people at random were asked the question, with the following answers:15 own fish only, 5 own a cat and a dog only, and 5 own a cat,a dog, and a bird only. Among all 225 people asked what fraction own fish b ut none of the other 3 types of pets?

Choice F

$$200\times0.45=90$$ $$\Rightarrow \frac{90}{200}=0.45$$ $$\Rightarrow \frac{90+15}{200+25}=\frac{105}{225}$$


2013 April(71G) Math Question 31

Angles

- Collinear Points & Triangles

31. Triangle AABC and collinear points A, C, and D are shown in the figure below. The measure of ∠A is 43°
the measure of ∠BCA is(4y)°, and the measure of
∠ BCD is(11y)°. What is the measure of ∠B?

Choice C

$$4y+11y=15y$$ $$y=\frac{180}{15}=12$$ $$4y=48$$ $$\angle B=180°-43°-48°=89°$$


2013 April(71G) Math Question 32

Conics

- Circles

32. In the standard (x,y)coordinate plane, what are the coordinates of the center of the circle with equation
(x-v5)2+(y-1)2=1?

Choice F

$$x-\sqrt{5}=0$$ $$x=\sqrt{5}$$ $$y-1=0$$ $$y=1$$


2013 April(71G) Math Question 33

Trigonometry

- Pythagorean Theorem & sin cos tan

33. Anoki wants to determine the height of a vertical lighthouse, shown below. He measures the angle of elevation to the top of the lighthouse at a point 103 feet along level ground from the center of the base of the lighthouse. The angle of elevation is 31. Which of the following expressions gives the best approximation of the height of the lighthouse, in feet?

Choice E

$$\tan 31°=\frac{?}{103}$$ $$?=\tan 31°103$$


Choice J

y=4 is a line parallel …


Choice B

$$2x+3y=12---(line\ l)$$ $$\Rightarrow y=-\frac{2}{3}x+4$$ $$y=(-\frac{2}{3}\times2)x+(4-3)$$ …


2013 April(71G) Math Question 36

Trigonometry

- Graphing Trig Functions

36. The graph of y= tan x is shown in the standard (x,y coordinate plane below. What is the period of tan x?

Choice H

71g


Choice C

$$5r\times2r=90$$ $$10r^{2}=90$$ $$r^{2}=9$$ $$r=3$$ $$3\times5=15$$


2013 April(71G) Math Question 38

Graphing Properties

- Interpret Graphs or Figures

38. Josey rode her bicycle 4 km at a constant speed, beginning and ending at her home. A graph, with be distance traveled plotted along the y-axis and elapsed time during the ride plotted along the x-axis, was constructed for the values of y from 0 km through 4 km.
The shape of the graph can best be described as a:

Choice G

As time elapses, the distance …


Choice B

$$2\times2\times(1\frac{1}{4})+(1\frac{1}{4})(1\frac{1}{4})$$ $$5+\frac{25}{16}$$ $$=6+\frac{9}{16}$$


Choice K

71g


2013 April(71G) Math Question 41

Inequalities

- Graphing Inequalities

41. One of the following graphs in the standard (x,y) coordinate plane is the graph of y> ax+ b for some positive a and negative b. Which graph?

Choice D

71g


Choice K

$$\frac{2x-y}{x+y}=\frac{3}{4}$$ $$4(2x-y)=3(x+y)$$ $$8x-4y=3x+3y$$ $$5x=7y$$ $$\frac{x}{y}=\frac{7}{5}$$


Choice C

$$8:7=5:? $$ $$\frac{8}{7}=\frac{5}{?}$$ $$8\times?=5\times7$$ $$?=\frac{35}{8}=4\frac{3}{8}$$


Choice H

The formula is BMI = …


2013 April(71G) Math Question 45

Trigonometry

- Pythagorean Theorem & sin cos tan

45. In the circle shown below, radius AC is 15 inches long, chord BE is 10 inches long, and AC is perpendicular to BE at D. How many inches long is AD?

Choice E

71g

$$15^{2}-5^{2}=\overline{AD}^{2}$$ $$225-25=200=\overline{AD}^{2}$$ $$\overline{AD}=\sqrt{200}=10\sqrt{2}$$


2013 April(71G) Math Question 46

Perimeter, Area, Volume

- Other Shapes

46. A pyramid composed of 4 congruent triangular sides and a square base is shown"unfolded"in the standard
(r,y) coordinate plane below, Points A, B, and Care vertices of I of the triangular sides. What is the total surface area, in square coordinate units, of the pyramid?

Choice G

71g

$$\triangle\times4+\Box$$ $$\Rightarrow\triangle=10\times12\times\frac{1}{2}=60$$ $$\Rightarrow\Box =10\times10=100$$ $$\triangle\times4+\Box=60\times4+100=340$$


2013 April(71G) Math Question 47

Perimeter, Area, Volume

- Other Shapes

47. The side lengths of the flat, trapezoidal ceiling of a bedroom are given in the figure below. Marie will paint the entire ceiling with I coat of paint, using paint that has a price of $8 per quart and is sold only by the full quart. Each quart of paint covers an area of 90 square feet with 1 coat of paint. What is the total price of the paint that Marie needs to buy?

Choice D

$$(12+24)\times16\times\frac{1}{2}$$ $$=36\times16\times\frac{1}{2}$$ $$=288$$ $$\frac{288}{90}=3...18$$

Marie …


2013 April(71G) Math Question 48

Angles

- Other Shapes

48. In the figure below, F lies on EG, and the measure of ZE is (3a). Which of the following inequalities is true?

Choice F

$$3a°\lt90°$$ $$a\lt30$$


2013 April(71G) Math Question 49

Word Problems

- 1 Variable Equations

49. Each contestant at a math competition starts with
30 points. A contestant earns 10 points for each question answered correctly and loses 5 points for each question answered incorrectly. Sammi answered twice as many questions correctly as incorrectly, finishing with 150 points. How many questions did Sammi answer correctly

Choice C

$$Correct\rightarrow 2x$$ $$Incorrect\rightarrow x$$ $$30+10(2x)-5(x)=150$$ …


Choice F

$$f(x)=\frac{2x+b}{x+a}$$ $$=\frac{2x}{x+a}+\frac{b}{x+a}$$ $$=\frac{2}{1+\frac{a}{x}}+\frac{\frac{b}{x}}{1+\frac{a}{x}}$$ $$\lim_{\frac{a}{x}\rightarrow0}\frac{2}{1+\frac{a}{x}}=2$$ $$\lim_{\frac{b}{x}\rightarrow0}\frac{\frac{b}{x}}{1+\frac{a}{x}}=0$$


2013 April(71G) Math Question 51

Number Types & Properties

- Order of Values

51. On the real number line. -0.423 is between n/100 and
(n+1)/100  for some integer n. What is the value of n?

Choice B

$$-0.43\lt-0.423\lt-0.42$$ $$\Rightarrow -\frac{43}{100}\lt-0.423\lt-\frac{42}{100}$$ $$n=-43$$


2013 April(71G) Math Question 52

Statistics

- Mean, Median

52. The stem-and-leaf plot below shows the number of daily credit card sales at Fancy Fabrics during a 34-day period. What is the median number of daily credit card sales

Choice F

71g

$$\frac{51+53}{2}=52$$


2013 April(71G) Math Question 53

Angles

- Circles

53. Angle A has a measure of r radians. Angle A and Angle B are coterminal. Angle B could have which of the following measures?

Choice D

$$\frac{25}{3}\pi=\frac{25\times180°}{3}=1500°$$ $$1500°=360°\times4+60°$$


2013 April(71G) Math Question 54

Number Types & Properties

- Complex Number

54. Which of the following complex numbers equals
(6-7i)(x+6i)?

Choice J

$$(6-7i)(\pi+6i)$$ $$=6\pi+36i-7i\pi+42$$ $$=(J)$$


2013 April(71G) Math Question 55

Quadratic Equations

- Factoring

55. If r= 4 is one solution to the equation x'-ax-12=0
then the other solution is:

Choice A

$$x=4,$$ $$4^{2}-4a-12=0$$ $$\Rightarrow 16-4a-12=0$$ $$\Rightarrow …


2013 April(71G) Math Question 56

Trigonometry

- Pythagorean Theorem & sin cos tan

56. For all x such that tan x 0, the expression seex.sin x tan x is equivalent to which of the following?
(Note:sec x-
cos.r, tan x= sinx cos r

Choice J

$$=\frac{\frac{1}{(\cos x)^{2}}\cdot\sin x}{\frac{\sin x}{\cos x}}$$ …


2013 April(71G) Math Question 57

Perimeter, Area, Volume

- Other Shapes

57. All quadrilaterals in one of the following categoric have diagonals that are congruent. Which category?

Choice E

71g


2013 April(71G) Math Question 58

Trigonometry

- Pythagorean Theorem & sin cos tan

58. Three line segments are graphed in the standard (x,y)
coordinate plane below. Line segment AB has endpoints A(2, 0)and B(4, 0), A'B is the image of AB after a rotation counterclockwise ()  by 120 about the origin, and A"B is A'B projected onto the x-axis What is the length, in coordinate units, of A"B"?

Choice F

71g

$$30°:60°:90°$$ $$=1:\sqrt{3}:2$$


2013 April(71G) Math Question 59

Sequences and Patterns

- Arithmetic Sequences

59. Consecutive terms of a certain arithmetic sequence have a positive common difference. The sum of the first 3 terms of the sequence is 120. Which of the following values CANNOT be the first term of the arithmetic sequence?

Choice E

$$n\gt 0,$$ $$x,x+n,x+2n=120$$ $$\Rightarrow 3x+3n=120$$ …


Choice F

71g

$$f(x)=\sqrt[3]{x+2}$$ $$=(x+2)^{-3}$$ $$f^{-1}(x)=x^{3}-2$$