ACT Form 71A Math Answer Explanation



Choice C

$$0.75(120)+20(2)=90+40=130$$


Choice G

$$15=3\times5$$ $$6=2\times3$$ $$4=2\times2$$ $$2\times2\times3\times5=60$$


Choice D

$$(36+24)2=120$$


Choice J

$$180°-64°=116°$$


Choice B

$$\frac{1}{4}\times\frac{250}{10}=\frac{1}{4}\times25=6\frac{1}{4}$$


2012 December(71A) Math Question 6

Trigonometry

- Pythagorean Theorem & sin cos tan

6. The 2 cubes shown below have diagonals AB and CD.
respectively. The side lengths given are in feet. What is the ratio of the length of AB to the length of CD?

Choice F

$$3:12=1:4$$


Choice C

$$-(-1)+(-3)+2$$ $$=1-3+2$$ $$=0$$


Choice G

$$9+3x=27$$ $$3x=27-9=18$$ $$x=6$$ $$2x=12$$


2012 December(71A) Math Question 9

Word Problems

- 1 Variable Equations

9. A woman purchased 100 shares k at $5.00 per share. If each share rose first month decreased $0.08 the second month, and gained $0.03 the third month. what is the value of the woman's investment?

Choice A

$$100(5+0.1-0.08+0.03)$$ $$=100(5+0.05)$$ $$=505$$


2012 December(71A) Math Question 10

Word Problems

- Cost

10. At Acme Manufacturing Company, each employee's annual salary for next year will be 3.5% more than this years annual salary. An employee whose annual salary this year is $32, 000.00 will have what annual salary next year?

Choice J

$$$32000(1+0.035)$$ $$=$32000+$1120$$ $$=$33120$$


Choice E

$$\frac{1}{3}(\frac{5}{6}+\frac{1}{12}+\frac{1}{3})$$ $$=\frac{1}{3}(\frac{10}{12}+\frac{1}{12}+\frac{4}{12})$$ $$=\frac{1}{3}(\frac{15}{12})$$ $$=\frac{5}{12}$$


2012 December(71A) Math Question 12

Inequalities

- Number Line

12. Which of the following graphs shows the solution set for the inequality 4x-2≥6?

Choice H

$$4x-2\geq6$$ $$4x\geq8$$ $$x\geq2$$ $$\Rightarrow H.$$


2012 December(71A) Math Question 13

Word Problems

- 1 Variable Equations

13. The Hope-A- Lot Foundation is mailing brochures to
4, 000 prospective donors. The foundation's goal is to have proceeds of $1, 500 after paying $900 for the mailing. According to past mailings, the average donation was $20 per donor. Assuming this average.
how many of the prospective donors need to donate to reach the goal?

Choice C

$$1500+900=2400$$ $$2400\div20=120$$


2012 December(71A) Math Question 14

Angles

- Collinear Points & Triangles

14. Lines BC and DE are parallel, and transversals BD and intersect at A, as shown in the figure below.
Given that △ABC is an equilateral triangle, x=?

Choice H

$$\angle A=\angle B=\angle C=60°$$ $$x°=180°-60°=120°$$


2012 December(71A) Math Question 15

Quadratic Equations

- Factoring

15. What is the positive solution to the equation
6x2=30?

Choice E

$$16x^{2}=30$$ $$x^{2}=\frac{30}{16}$$ $$x=\sqrt{\frac{30}{16}}$$


Choice F

$$100(20)+30(40-20)$$ $$=2000+600$$ $$=2600$$ $$2600\div800=3\frac{1}{4}$$


2012 December(71A) Math Question 17

Statistics

- Mean, Median

Use the following information to answer questions 17-20.
An organization promoting good nutritional habits collected data on fat calories in foods from 9 fast-food restaurants in Mesa City. The values in the list below represent the number of fat calories in a small order of french fries at each of these fast-food restaurants
160,106,104,113,160,103,161,89,96
17. Based on the data listed, what is the median number of fat calories in a small order of french fries at these 9 restaurants?


2012 December(71A) Math Question 18

Graphing Properties

- Create Graphs or Figures

18. Which of the following bar graphs most accurately represents the data on the number of fat calories in a small order of french fries at the 9 fast-food restaurants?

Choice G

$$50~100\Rightarrow 89,96$$ $$101~150\Rightarrow 103,104,106,113$$ $$151~200\Rightarrow 160,160,161$$


Choice D

$$\frac{89}{x}=\frac{43}{100}$$ $$43x=8900$$ $$x=206.97$$


2012 December(71A) Math Question 20

Statistics

- Mean, Median

20. The organization collects data from 2 additional restaurants and includes the new data in the list. The each of the 2 additional restaurants is designated.es number of fat calories in a small order of french fries at by x and y, respectively. Which of the following expressions gives the average of this larger list of values?

Choice K

$$160+1006+104+113+160+103+161+89+96=1092$$ $$\Rightarrow\frac{1092+x+y}{9+2}=\frac{1092+x+y}{11}$$


2012 December(71A) Math Question 21

Probability

- One Event

21. A bag contains 8 red marbles, 5 yellow marbles, and
11 green marbles. How many additional red marbles must be added to the 24 marbles already in the bag so that the probability of randomly drawing a red marble is 3/5?

Choice B

$$\frac{3}{5}=\frac{8+r}{8+5+11+r}=\frac{8+r}{24+r}$$ $$3(24+r)=5(8+r)$$ $$72+3r=40+5r$$ $$32=2r$$ $$r=16$$


Choice G

$$(5x^{3}-4x^{2}+7x-3)-(3x^{3}+4x^{2}-3x+1)$$ $$=2x^{3}-8x^{2}+10x-4$$


2012 December(71A) Math Question 23

Graphing Properties

- Create Graphs or Figures

23. The table below gives the price to mail a single package through the United States Postal Service on August 30, 2007. The price depended on the weight of th e package.
Which of the following graphs best represents this information?

Choice E

$$\Rightarrow (E)$$


Choice J

$$f(-4)$$ $$=5(-4)^{2}-2(-4)+10$$ $$=5(16)+8+10$$ $$=80+8+10$$ $$=98$$


Choice D

$$\frac{6}{12}=0.5$$ $$\Rightarrow (D)$$


2012 December(71A) Math Question 26

Trigonometry

- Pythagorean Theorem & sin cos tan

26. Josh is standing in a pool and looking up at his friend Olivia. Olivia is lying on her stomach on the diving board looking at Josh. The horizontal and vertical distances, in meters, between Josh and olivia are given in the diagram below. What is the measure of the angle of elevation, 0, of Josh's line of sight?
(Note:Drawing is NOT to scale.)

Choice H

71a

$$\tan\theta=\frac{3}{8}$$ $$\theta=\arctan\frac{3}{8}$$


2012 December(71A) Math Question 27

Word Problems

- Distance, Speed, Time

27. Avari traveled the 2-mile trail from her house to Big Lake on her bicycle. She then traveled 3 times around the Big Lake Loop and returned home by the 2-mile trail. At the end of her bicycle ride, the trip odometer showed that she had traveled 22 miles. Which of the following equations, when solved, gives the distance Avari traveled once around Big Lake Loop, d miles?

Choice E

$$2+3d+2=22$$ $$\Rightarrow 4+3d=22$$


2012 December(71A) Math Question 28

Equation of a Line

- Equations that Describe a Line

28. In the standard(x,y) coordinate plane, what is the slope of the line with equation 3x+ 5y=67

Choice G

$$3x+5y=6$$ $$5y=-3x+6$$ $$y=-\frac{3}{5}+6$$ $$m=-\frac{3}{5}$$


Choice C

$$3x-12-2x+6=-5x-15+6$$ $$x-6=-5x-9$$ $$6x=-3$$ $$x=-\frac{1}{2}$$


Choice H

$$If\ x=\frac{1}{2}$$ $$(\frac{1}{2})^{3}=\frac{1}{8}$$ $$0\lt\frac{1}{8}\lt1$$ $$\Rightarrow …


Choice E

$$\frac{(2a)^{6}+3a^{4}-5a^{4}}{2a}$$ $$=\frac{8a^{6}-2a^{4}}{2a}$$ $$=4a^{5}-a^{3}$$


Choice J

$$1.28\times10^{4}-3.5\times10^{3}$$ $$=12.8\times10^{3}-3.5\times10^{3}$$ $$=9.3\times10^{3}$$


2012 December(71A) Math Question 33

Trigonometry

- Pythagorean Theorem & sin cos tan

33. In the right triangle shown below, the length of AC is
4 mm and the length of BC is 3 mm. For ∠A, the value of which of the following trigonometric expressions is 3/5?

Choice A

$$\overline{AB}^{2}=3^{2}+4^{2}=5^{2}$$ $$\overline{AB}=5$$ $$\sin A=\frac{3}{5}$$


Choice H

$$\frac{1}{12^{2}}:\frac{1}{6^{2}}$$ $$=\frac{1}{144}:\frac{1}{36}=1:4$$ $$l=10\times4=40$$


Choice C

$$3-(-2)=5$$ $$16-4=12$$ $$\sqrt{5^{2}+12^{2}}=\sqrt{25+144}=\sqrt{169}=13$$ $$\pi r^{2}=\pi …


2012 December(71A) Math Question 36

Trigonometry

- Pythagorean Theorem & sin cos tan

36. Royce plans to construct a triangular flower bed on the corner of his property where a sidewalk forms a right angle. The flower bed and the lengths, in feet, of 2 of its sides are shown in the figure below. The flower bed will be enclosed by a garden fence that is set up along a entire perimeter. To the nearest foot, how many feet garden fence will enclose the flower bed?


2012 December(71A) Math Question 37

Absolute Values

- Absolute Value Expressions

37. If  2w + 7 = | -2 |  how many different values are possible for w?

Choice B

$$2w+7=\mid-2\mid$$ $$\mid-2\mid=2$$ $$2w+7=2\Rightarrow w=-2.5$$


Choice K

71a

$$\Rightarrow (K)$$


2012 December(71A) Math Question 39

Equation of a Line

- Distance & Midpoint

39, The points A(12, 18) and B(-4, 2)lie in the standard(x,y)coordinate plane. What are the coordinates of the midpoint of AB?

Choice B

$$\frac{12+(-4)}{2}=4$$ $$\frac{18+2}{2}=10$$ $$\Rightarrow (4,10)$$


2012 December(71A) Math Question 40

Trigonometry

- Pythagorean Theorem & sin cos tan

40. For △ABC shown below, the length of BC is 50 mm Which of the following equations, when solved, will give the length, in millimeters, of AB
(Note:The law of sines states that given △XYZ,     )

Choice G

$$\frac{\sin 58°}{50}=\frac{\sin 68°}{\overline{AB}}$$


2012 December(71A) Math Question 41

Conics

- Circles

41. The vertices of △AOB are A(0, 6), O(0, 0), and B(8, 0)
as shown in the standard (x, y) coordinate plane below What are the coordinates of the center of the circle that circumscribes △AOB?

Choice D

71a

$$\frac{0+8}{2}=4$$ $$\frac{6+0}{2}=3$$


2012 December(71A) Math Question 42

Angles

- Circles

42. Circle P has a radius of 4 units and is in the standard
(x,y) coordinate plane. The set of all points in the coordinate plane that are 3 units from the center of Circle P is a circle that:

Choice J

71a


$$\frac{4x}{6x^{2}}=\frac{2}{3x}$$

A rational number is …


2012 December(71A) Math Question 44

Algebraic & Function Expressions

- Nested Functions

44. The functions f and g are defined as f(x)=3x+2 and g(x)=2x-1. What is the value of f(g(-2))?

Choice K

$$g(-2)=2(-2)-1=-5$$ $$f(-5)=3(-5)+2=-13$$


Choice B

$$36=6\times6$$ $$6\times4=24$$


2012 December(71A) Math Question 46

Word Problems

- Distance, Speed, Time

46. Carmen drove from blairtown to Ore City, a distance of 80 miles From Ore City she drove on to Janesville, and then drove back to Blairtown. The ratio of Carmens driving times on the first, second, and third segments of the trip, respectively, was 5:2:4, and she drove at the same average speed on each segment.
What was Carmen's total driving distance, in miles, for the 3 segments of the trip?

Choice F

$$Distance=V \times Time$$ $$1 segment:2 …


Choice C

$$180°-43°-32°=105°=\angle B$$ $$\overline{AC}\gt\overline{BC}\gt\overline{AB}$$


2012 December(71A) Math Question 48

Absolute Values

- Graphing Absolute Values

48. One of the following functions aphed in the standard (x,y) coordinate plane below. Which function is it?

Choice F

$$y=\mid x-1\mid-2$$ $$(1,-2)\Rightarrow -2=0-2$$


2012 December(71A) Math Question 49

Conics

- Ellipse

49. One of the following is an equation of the ellipse shown in the standard (x,y) coordinate plane below.
Which one?
(Note:The coordinate unit on the x axis is the same length as the coordinate unit on the y-axis.)

Choice E

71a

$$h\gt 0,k\gt 0$$ $$16=4^{2}$$ $$4=2^{2}$$ …


2012 December(71A) Math Question 50

Perimeter, Area, Volume

- Other Shapes

50. A regular pyramid with a square base is shown in the figure below. The slant height is v3 units and the length of the base edge is 2 units. What is the total length, in units, of all 8 edges of the pyramid?

Choice K

71a

$$2\times4+2\times4$$ $$=8+8$$ $$=16$$


Choice A

$$2x-3y=6---(1)$$ $$(1)\times3$$ $$\Rightarrow 6x-9y=18$$ $$-9=k$$ …


Choice F

$$2^{0}=1$$ $$x^{2}+1=0$$ $$x^{2}=-1$$ $$x=\pm I$$ …


2012 December(71A) Math Question 53

Angles

- Circles

Use the following information to answer questions 53-55.
In the figure below, a large circle with center o has a diameter AB that is 40 mm long. Point C lies on the large circle such that the measure of ∠ABC is 60° . A diameter of the small circle is AO.
53. What is the area, in square millimeters, of the small circle?

Choice D

71a

$$\pi r^{2}$$ $$=\pi 10^{2}$$ $$=100\pi$$


2012 December(71A) Math Question 54

Angles

- Circles

54. What is the length, in millimeters, of arc AB?

Choice G

$$\widehat{AB}$$ $$=2\pi r\times\frac{1}{2}$$ $$=2\pi(20)\times\frac{1}{2}$$ $$=20\pi$$


2012 December(71A) Math Question 55

Trigonometry

- Pythagorean Theorem & sin cos tan

55. The figure is placed in the standard (x, y) coordinate plane so that A has coordinates(-20, 0)and B coordinates(200).  What is the x-coordinate of C?

Choice D

71a

$$c\Rightarrow(10,\sqrt{3})$$


2012 December(71A) Math Question 56

Angles

- Circles

56. In the figure shown below, C, M, and N lie on the circle whose center is O, and ∠MON is a right angle, What is the sum of the measures of∠ CMO and ∠CNO?

Choice J

71a

$$90°\div2=45°$$


Choice E

71a

$$\frac{10L}{L}=10$$ $$(L-4)10=10L-40$$


2012 December(71A) Math Question 58

Sequences and Patterns

- General Sequences

58. For any integer n>0, the triangular number T, is the number of dots in a triangular array with n points on each side. The figure below shows the first 4 triangular numbers. What is the value of T64?

Choice K

$$T_{64}=1+2+3+...+64$$ $$=\frac{(1+64)64}{2}$$ $$=65\times32$$ $$=2080$$


2012 December(71A) Math Question 59

Quadratic Equations

- Factoring

59. For what integer k are both solutions of the equation x2+k+17=0 positive integers?

Choice A

$$x^{2}+kx+17=0$$ $$\Rightarrow (x-17)(x-1)=0$$ $$\Rightarrow x^{2}-x-17x+17=0$$ …


2012 December(71A) Math Question 60

Trigonometry

- Pythagorean Theorem & sin cos tan

60.In △ XYZ the measure of ∠x is 90°, the measure of ∠z is B, XY 12 units, and tan thete =4/9. What is the area of △XYZ , in square units?

Choice F

$$\overline{XY}=12$$ $$\tan\theta=\frac{4}{9}=\frac{\overline{XY}}{\overline{XZ}}=\frac{12}{\overline{XZ}}$$ $$4\times\overline{XZ}=9\times12$$ $$\overline{XZ}=27$$ $$Area=\overline{XZ}\times\overline{XY}\times\frac{1}{2}$$ …