ACT Form 71E Math Answer Explanation


Choice C

$$42:49=\frac{42}{49}$$ $$=\frac{7\times6}{7\times7}=\frac{6}{7}$$

2013 December(71E) Math Question 2

Quadratic Equations Practice

- Factoring

2. The only solution to the equation (x-2)(x-10)=c  is x=6. what is c?

Choice F

$$x=6,(6-2)(6-10)=4(-4)=-16=c$$

2013 December(71E) Math Question 3

Word Problems Practice

- 1 Variable Equations

3. Samantha, Larry, and Maria own shares of stock in the Plentiful Peanuts company. Samantha owns 50 shares, Larry owns 30 shares, and Maria owns 70 shares Today, the value of 1 share of Plentiful Peanuts stock is $4.05. What is the total value of Samantha's Larry's, and Maria's shares of Plentiful Peanuts stock?

Choice E

$$(50+30+70)4.05=150\times4.05=607.5$$

Choice J

$$=5\times3\times p^{6+2}=15p^{8}$$

2013 December(71E) Math Question 5

Word Problems Practice

- Cost

5. Simeon ordered graduation announcements from Nouncements Notes. He was charged $35.00 for his announcements and was charged 8% of that amount for shipping. How much was Simeon charged for his announcements and shipping?

Choice D

$$35(1+0.08)=35+2.8=37.8$$

2013 December(71E) Math Question 6

Probability Practice

- One Event

6. A jar contains only 11 red balls, 9 yellow balls, 5 green balls, and n white balls. Each ball is a solid color What is the probability that a ball randomly chosen from the jar is yellow?

Choice J

$$P(Yellow)$$ $$=\frac{9}{11+9+5+n}$$ $$=\frac{9}{25+n}$$

Choice H

$$303,488,509=3.03\times10^{8}$$

2013 December(71E) Math Question 9

Angles Practice

- Parallel Lines

9. In the figure below, AB is parallel to CD, and FG Intersects AB and CD at J and K, respectively. which of the following equations must be true?

Choice D

$$x°+y°=180°$$

2013 December(71E) Math Question 10

Angles Practice

- Vertex Angles & Bisectors

10. In the figure below, AB is congruent to BC, and AE intersects BF at C. What is the measure of ∠B?

Choice J

$$\angle BCA=\angle ECF=31°$$ $$\angle BAC=\angle BCA=31°$$ $$\angle ABC=180°-31°-31°=118°$$

Choice D

$$(7+\frac{1}{5})\div(1+\frac{1}{10})$$ $$=\frac{36}{5}\times\frac{10}{11}$$ $$=\frac{72}{11}$$ $$=6\frac{6}{11}$$

Choice H

$$6\times10=60$$ $$60=2\times2\times3\times5$$

2013 December(71E) Math Question 13

Word Problems Practice

- Fixed Cost + Variable Cost

13. Kaya rented a boat from a marina for a fee of S50, plus s3 for every gallon of gas she used. when she returned the boat, Kaya was given a bill of $98. She had kept track of her gas usage and thought that the bill was in error-specifically, that the bill was $21 more than it should have been. If Kaya was correct, how many gallons of gas did Kaya use during her rental?

Choice C

$$98-50=48$$ $$48\div3=16$$ $$21\div3=7$$ $$16-7=9$$

Choice H

$$7\times4+4\times10=28+40=68$$

Choice B

$$=\mid-2\mid-\mid-6\mid=2-6=-4$$

Choice F

$$96\times\frac{28}{100}=n$$ $$\Rightarrow \frac{28}{100}=\frac{n}{96}$$

2013 December(71E) Math Question 17

Equation of a Line Practice

- Distance & Midpoint

17. In the standard(x, y) coordinate plane, the point (2, -6) is the midpoint of the line segment with endpoints(8,-8)and:

Choice B

$$8-2=6$$ $$-8-(-6)=-2$$ $$\Rightarrow(2-6,-6-(-2))=(-4,-4)$$

Choice K

$$g(-3)=2\times(-3)^{2}-3(-3)$$ $$=2\times9+9$$ $$=27$$

2013 December(71E) Math Question 19

Inequalities Practice

- Number Line

19. The graph below is that of the solution set to one of the following statements. Which one?

Choice A

$$x\lt9,x\gt-1$$

2013 December(71E) Math Question 20

Inequalities Practice

- Inequality Expressions

20. The inequality 5(x+ 2)>6(x-5)is equivalent to which of the following inequalities?

Choice K

$$\Rightarrow 5x+10\gt6x-30$$ $$\Rightarrow 40\gt x$$

Choice C

$$3^{2}+x^{2}=5^{2}$$ $$\Rightarrow x=4$$ $$3^{2}+5^{2}=y^{2}$$ $$\Rightarrow …

Choice F

$$\sqrt{(7-2)^{2}+(3-3)^{2}}$$ $$=\sqrt{5^{2}+0^{2}}$$ $$=5$$

Choice E

71e

$$13+7+5+2=27$$

Choice G

$$Day\ 1=750-0=750$$ $$Day\ 2=875-750=125$$ $$Day\ …

Choice A

$$\overline{BC}=\sqrt{15^{2}-9^{2}}=12$$ $$\sin A=\frac{12}{15}=\frac{4}{5}$$

2013 December(71E) Math Question 26

Quadratic Equations Practice

- Factoring

26. The expression a2-2a+ 1 is equivalent to:

Choice F

$$(a-1)(a-1)$$ $$=a^{2}-a-a+1$$ $$=a^{2}-2a+1$$

Choice B

$$r=18\div2=9$$ $$2\pi\times9=18\pi$$

2013 December(71E) Math Question 28

Inequalities Practice

- Number Line

28. Which of the following number line graphs represents the solution set of the equation x+2=3?

Choice K

$$x^{2}+2=3$$ $$x^{2}=1$$ $$x=1,-1$$

Choice D

$$54\times24=1200$$ $$120\div225=5.3333...$$

Choice K

$$(60+34)2=194\times2=188$$

Choice B

$$1\ yard = 3\ feet$$ …

2013 December(71E) Math Question 32

Probability Practice

- More than One Event

32. A fair coin has 2 sides : 1 side is heads and 1 side is tails. A fair cube has 6 sides:2 sides are solid green, 2 sides are solid yellow, and 2 sides are solid red. The coin and the cube are each tossed once. what is the probability they will land so that the side facing up on  the coin is tails and the side facing up on the cube is yellow?

Choice H

$$\frac{1}{2}\times\frac{2}{6}=\frac{1}{6}$$

Choice C

$$r\times r\times r^{2}=128 $$ $$\Rightarrow …

Choice G

$$x^{2}-4\neq0 $$ $$\Rightarrow x^{2}\neq4 $$ …

Choice A

71e

2013 December(71E) Math Question 36

Equation of a Line Practice

- Distance & Midpoint

36. When a certain map is drawn in the standard(x,y)
coordinate plane, one city has coordinates(5, 6) and another city has coordinates(9, 12)。 If 1 coordinate unit corresponds to 25 miles, which of the following is closest to the straight-line distance, in miles, between these 2 cities?

Choice H

$$\sqrt{(9-5)^{2}+(12-6)^{2}}$$ $$=\sqrt{4^{2}+6^{2}}$$ $$=\sqrt{16+36}$$ $$=\sqrt{52}\approx 7.2$$ …

2013 December(71E) Math Question 37

Angles Practice

- Other Shapes

37. In the figure below, the measures of 4 angles of pentagon ABCDE are given. What is the measure of 
∠A

Choice D

$$n=5,180°(n-2)$$ $$\Rightarrow 180°(5-2)=540°$$ $$540°-120°-120°-80°-80°=140°$$

Choice K

$$\sin ?=\frac{3.7}{5.4}$$ $$?=\frac{3.7}{5.4}\sin^{-1}$$

Choice D

71e

$$30°:60°:90°$$ $$=1:\sqrt{3}:2$$ $$=20:20\sqrt{3}:40$$ $$h=20\sqrt{3}\approx34.64...$$

Choice H

$$\frac{35}{25}$$ $$=\frac{42}{30}$$ $$=\frac{7}{5}$$ $$=\frac{\overline{AB}}{40}$$ $$\Rightarrow …

Choice B

71e

$$4:\sqrt{52}=x:\sqrt{13}$$ $$\Rightarrow x=2$$ $$2^{2}+y^{2}=(\sqrt{13})^{2}$$ $$=4+y^{2}=13$$ …

Choice J

$$\frac{50\times3+40\times2}{5}$$ $$=\frac{230}{5}$$ $$=46$$

Choice B

71e

2013 December(71E) Math Question 44

Quadratic Equations Practice

- FOIL

44. Min and Blanca answered the same homework question with different, yet equivalent, expressions.
Min's expression was (x+3)-x. Blanca said her expression was a simplified form of Mins expression Which of the following expressions could be Blanca's?

Choice G

$$(x+3)^{2}-x^{2}$$ $$=x^{2}+3x+3x+9-x^{2}$$ $$=6x+9$$

2013 December(71E) Math Question 45

Angles Practice

- Circles

45 Points A through F lie on the circle shown below so that the distance between any 2 adjacent points is equal. The circle has diameter 2 inches. What is the length,in inches, of AB?

Choice B

71e

$$60°:60°:60°$$ $$=1:1:1$$ $$\overline{AB}=\frac{9}{2}\times\frac{1}{2}=\frac{9}{4}$$

Choice K

$$a=\frac{1}{\sqrt{b}}$$ $$(\frac{1}{\sqrt{b}})^{2}\times b=\frac{1}{b}\times b=1$$

Choice A

$$3x+2y=5$$ $$2y=-3x+5$$ $$y=\frac{-3}{2}x+\frac{5}{2}$$ $$m'=\frac{-3}{2}$$ $$m\times …

Choice F

$$\frac{7-x}{2}=n$$ $$7-x=2n$$ $$x=7-2n$$ $$n=1,x=5$$ $$n=2,x=3$$ …

2013 December(71E) Math Question 49

Quadratic Equations Practice

- Factoring

49. Miles sells Baby Laugh dolls. The number of dolls, x, that Miles sells in I year depends on the price he sells them for that year p dollars per doll. The equation x= 52-2p, where 0<p< 26, gives the relationship between x and p. Miles's revenue is the money collected from selling x dolls. The maximum revenue Miles can make from the sale of Baby Laugh dolls in I year occurs at what price per doll?

Choice A

71e

$$Revenue$$ $$=xp$$ $$=(52-p)p$$ $$=52p-p^{2}$$ $$=-1(p^{2}-52p)$$ …

Choice J

$$6\div4=\frac{3}{2}$$ $$9\div6=\frac{3}{2}$$ $$9\times\frac{3}{2}=\frac{27}{2}=13.5$$

Choice A

$$V=V_{0}(1-r)^{t}$$ $$\frac{V}{V_{0}}=(1-r)^{t}$$ $$\sqrt[t]{\frac{V}{V_{0}}}=1-r$$ $$r=1-\sqrt[t]{\frac{V}{V_{0}}}$$

2013 December(71E) Math Question 52

Word Problems Practice

- % Interest

52. A used car with a purchase price of $20,000 has constant annual rate of decrease in value of o According to the model, what is the value of the car, to the nearest dollar, at the end of 3 years?

Choice H

$$V=V_{0}(1-r)^{t}$$ $$V=20000(1-0.1)^{3}$$ $$=20000(0.9)^{3}$$ $$=20000(0.729)$$ $$=14580$$

2013 December(71E) Math Question 53

Word Problems Practice

- Cost

53. A used car has a constant annual rate of decrease in value of 0.075. According to the model, which of the following expressions gives the number of years after purchase for the car to reach a value that is 50% of its purchase price?

Choice E

$$V=V_{0}(1-r)^{t}$$ $$\frac{1}{2}=1(1-0.075)^{t}=(1-0.075)^{t}$$ $$\log\frac{1}{2}=\log(1-0.075)^{t}$$ $$t=\frac{\log\frac{1}{2}}{\log(1-0.075)}$$

Choice F

$$\overline{BC}=x$$ $$a^{2}=7^{2}+16^{2}-2\times7\times 16\times 0.81$$ $$a^{2}=49+256-181.44=123.56$$ …

Choice D

$$A.\ 1:3=40:120$$ $$B.\ 2:5=48:120$$ $$C.\ …

2013 December(71E) Math Question 56

Absolute Values Practice

- Graphing Absolute Values

56. The graph of y=f(x) is shown in the standard(x,y)
coordinate plane below. One of the following graphs is that of y = | f(x)|. Which graph is it?

Choice K

Absolute value describes the distance …

2013 December(71E) Math Question 57

Word Problems Practice

- 1 Variable Equations

57. Fifteen cars containing a total of 60 people crossed a toll bridge. Each of the 15 cars contained at least person but no more than 5 people. At most how many cars contained exactly 3 people?

Choice C

$$3\times15=45$$ $$60-45=15$$ $$5-3=2$$ $$15\div2=7...1$$ $$15-8=7$$

2013 December(71E) Math Question 58

Trigonometry Practice

- Graphing Trig Functions

58. Let thete be the radian angle measure that satisfies sin 2 thete - sin thete=- for 0<thete<pi/2  What is cos thete?

Choice J

$$\sin \theta=x$$ $$x^{2}-x=-\frac{1}{4}$$ $$x^{2}-x+\frac{1}{4}=0$$ $$4(x^{2}-x+\frac{1}{4})=0$$ …

Choice B

$$y=\frac{4}{3}x-2---(1)$$ $$(1)\times6\Rightarrow 6y=8x-12$$ $$8x-6y=12$$ $$\frac{12}{8}=\frac{3}{2}$$ …

2013 December(71E) Math Question 60

Perimeter, Area, Volume Practice

- Mixed Shapes

60. As shown in the figure below, D is the center of the circle, and right triangle AABD intersects the circle at C and F. Point C is the midpoint of BD, which is 12 cm long. The shaded region inside the circle and outside the triangle has an area of 32T square centimeters.
what is the measure of ∠A?

Choice H

71e

$$\pi r^{2}=\pi 6^{2}=36\pi$$ $$\frac{32\pi}{36\pi}=\frac{320}{360}$$ $$360-320=40$$ …