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

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

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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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?

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 Practice

- 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 Practice

- 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

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

Choice E

$$\Rightarrow (E)$$

Choice J

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

Choice D

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

Choice H

71a

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

2012 December(71A) Math Question 27

Word Problems Practice

- 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

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

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

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

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 …

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 Practice

- 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

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

Choice G

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

2012 December(71A) Math Question 41

Conics Practice

- 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 Practice

- 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 …

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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- Circles

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

Choice G

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

Choice D

71a

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

2012 December(71A) Math Question 56

Angles Practice

- 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 Practice

- 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 Practice

- Factoring

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

Choice A

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

Choice F

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