ACT Form 71C Math Answer Explanation

2013 June(71C) Math Question 2

- Solve Algebraic Expressions

2. For what value of x is 6x — 3 = 4x + 7 true?

Choice K

$$6x-4x=7+3 \\ $$ $$\Rightarrow 2x=10 \\ $$ $$\Rightarrow x=5$$

Number Types & Properties

2013 June(71C) Math Question 3

- Rational & Irrational Numbers and Integers

3. Which of the following integers is NOT a factor of 132?

Choice B

11 is not a factor of 132 because 132 can not be divided by 11.

2013 June(71C) Math Question 4

- 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

2013 June(71C) Math Question 5

5. Carlotta is helping her grandfather center a large framed picture on his living room wall. As shown in the figure below, the rectangular wall is 24 feet long, and the rectangular framed picture is 4.5 feet high and 7.5 feet long. The left edge of the frame will be x feet from the left edge of the wall, and the right edge of the frame will be x feet from the right edge of the wall. What is the value of x? DO YOUR FIGURING HERE.

Choice A

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

2013 June(71C) Math Question 6

- Collinear Points & Triangles

6. In AABC, the sum of the measures of LA and LB is 57°. What is the measure of LC ?

Choice K

$$\angle C=180°-57°-57°=66°$$

2013 June(71C) Math Question 7

- Mixed Shapes

7. A square and a rectangle have the same area. The length of the rectangle is 32 centimeters, and the width of the rectangle is 2 centimeters. What is the length, in centimeters, of a side of the square?

Choice B

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

2013 June(71C) Math Question 8

- Fixed Cost + Variable Cost

8. As a motivational speaker, Bree speaks at school assemblies, charging a school district for her travel costs and a fixed amount per assembly. Bree used the equation C = 50a + 1,500 to determine the charge of C dollars to speak at a assemblies in the Escambia City School District. Bree charged the district $4,250 to speak at school assemblies, How many assemblies did Bree Speak at in this district?

Choice H

$$4250=50a+1500 \\$$ $$ \Rightarrow a=55$$

2013 June(71C) Math Question 9

- LCM, GCF, Prime Numbers

9. Let a and 17 be positive prime numbers. Which of the following numbers must be a factor of both a and b ?

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.

2013 June(71C) Math Question 10

- Solve Algebraic Expressions

10. What is the value of 80 – 5(x2 y) + y when x = 4 and y = 6 ?

Choice H

$$80-5(4^2-6)+6=36$$

2013 June(71C) Math Question 11

- 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

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

2013 June(71C) Math Question 12

- Mixed Shapes

12. Square ABEF and parallelogram ACDG are shown in the figure below. Points •E and F are on DG, B is on AC, and the lengths given are in inches. What is the ratio of the area of ABEF to the area of ACDG ?

Choice H

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

2013 June(71C) Math Question 13

Inequalities

- Number Line

13. Which of the following graphs shows the solution set for the inequality 6x – 5 ?

Choice B

$$6x-5\geq 7 \\ $$ $$\Rightarrow 6x \geq 12 \\$$ $$ \Rightarrow x \geq2$$

2013 June(71C) Math Question 14

14. The length of a rectangle is 5 inches longer than the width. The perimeter of the rectangle is 60 inches. What is the width of the rectangle, in inches?

Choice G

$$w+w+5+w+w+5=60 \\$$ $$ \Rightarrow w=12.5$$

2013 June(71C) Math Question 15

Probability

- 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

$$\frac{10}{40}=25\%$$

2013 June(71C) Math Question 16

- Proportions & Ratios

16. In this survey, what is the ratio of the number of students who selected Athletics to the number of students who selected Soccer?

Choice H

$$\frac{9}{5}=9:5$$

2013 June(71C) Math Question 17

- 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

- 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 ?
Back to top↑

Choice F

24, 36, and 12 are multiples of both 3 and 4: $$\frac{1+1+1}{8}=\frac{3}{8}$$

2013 June(71C) Math Question 19

- Percents & Fractions

19. The counselors at Lakewood High School interviewed 200 students to determine placement in 3 different honors courses: Math, Chemistry, and Literature. The numbers of these students placed in these honors courses are shown in the Venn diagram below. Of these 200 students, 10% were placed into all 3 honors courses, and 35% were placed into exactly 2 honors courses. What percent of the students were placed into exactly J. honors course?
Back to top↑

Choice E

$$\frac{15+20+65}{200}=50\%$$

2013 June(71C) Math Question 20

- 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?
Back to top↑

Choice J

$$4n=14m \\$$ $$ \Rightarrow n=7,\ m=2 \\$$ $$ \Rightarrow t=4\times7=28 \ seconds$$

2013 June(71C) Math Question 21

- Equations that Describe a Line

21. Which of the following transformations shifts all points graphed in the standard (x,y) coordinate. plane down 5 coordinate units?

Choice A

When the graph is shifted …

You are not logged in.

2013 June(71C) Math Question 22

- Distance & Midpoint

22. For AC shown below, B is on AC, the length of AB is 12 cm, and the length of BC is 20 cm. What is the distance, in centimeters, between C and the midpoint

Choice J

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

You are not logged in.

2013 June(71C) Math Question 23

Quadratic Equations

- Factoring

23. 'Which of the following expressions is a factored form ofx2-5x+6?

Choice A

$$x^2-5x+6=(x-3)(x-2)$$

You are not logged in.

2013 June(71C) Math Question 24

- Simplify Algebraic Expressions

24. The sum of (-22 + 2x + 8) and which of the following polynomials is (5x2+ 3) ?

Choice H

$$(-x^2+2x+8)+d=(5x^2+3) \\ $$ $$\Rightarrow d=7x^2-2x-5$$

You are not logged in.

2013 June(71C) Math Question 25

- Percents & Fractions

Choice D

$$\frac{3+\frac{1}{9}}{1+\frac{1}{18}}=\frac{\frac{3\times9+1}{9}}{\frac{1\times18+1}{18}}=2\frac{18}{19}$$

You are not logged in.

2013 June(71C) Math Question 26

- Percents & Fractions

26. Ava starts with a long list of numbers, each of which she must multiply by 3 and then divide by She could get the same results by multiplying each number on her list by which of the following numbers?

Choice K

$$\frac{\frac{4}{3}}{\frac{1}{6}}=\frac{4}{3}\times6=8$$

You are not logged in.

2013 June(71C) Math Question 27

- Equations that Describe a Line

27. When 6x = 2y - 18 is graphed in the standard (x,y) coordinate plane, what is the y-intercept?

Choice E

$$2y-18= 0\\ $$ $$\Rightarrow y …

You are not logged in.

2013 June(71C) Math Question 28

- Nested Functions

28. Given functions f(x)= x — 5 and g(x)= 5 — x2, what is f(g(-4))?

Choice H

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

You are not logged in.

- Pythagorean Theorem & sin cos tan

2013 June(71C) Math Question 29

Trigonometry

29. For right triangle AABC shown below, which of the following expressions has a value that is equal to cos A ?.

Choice B

$$\cos A = \frac{AC}{AB}=\sin B$$

You are not logged in.

- Domain, Range, Asymptote, Undefined

2013 June(71C) Math Question 30

Graphing Properties

30. The equation y = :2+-34 is graphed in the standard (x,y) coordinate plane below. No point on the graph has which of the following x-coordinates?

Choice G

$$x^2-4\neq0\\$$ $$ \Rightarrow x\neq2 \ …

You are not logged in.

2013 June(71C) Math Question 31

- Proportions & Ratios

31. The Wilson family sold a shipment of walnuts to a farming cooperative for $210 at the rate of $0.03 per pound. If their shipment averaged 12 walnuts per pound, which of the following is closeAt to the total number of walnuts in the shipment? •

Choice E

$$12\times \frac{210}{0.03}=84000$$

You are not logged in.

2013 June(71C) Math Question 32

Statistics

- 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

$$247\times 7=1729$$

You are not logged in.

2013 June(71C) Math Question 33

Quadratic Equations

- Factoring

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

Choice C

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

You are not logged in.

2013 June(71C) Math Question 34

Sequences and Patterns

- General Sequences

34. The sum of a sequence of consecutive odd numbers, where the smallest term is 1, is always a perfect square. For example, 1 + 3 = 22 and 1 + 3 + 5 + 7 = 42- One of the sequences described above has a- sum of 144. What is the largest odd number in the sequence?

Choice J

$$1+3+5+...+(2n-1)=n^2 \\$$ $$ \Rightarrow 1+3+5+...+23=12^2$$

You are not logged in.

2013 June(71C) Math Question 35

- Equations that Describe a Line

35. What is the slope of the line that passes through the points (-2,5) and (3,-4) in the standard (x,y) coordinate plane?

Choice E

$$\frac{-4-5}{3-(-2)}=-\frac{9}{5}$$

You are not logged in.

Number Types & Properties

2013 June(71C) Math Question 36

- Largest or Smallest Value

36. Among the following rational numbers, which has the greatest value?

Choice G

$$0.3\overline{4}=0.3444444...$$

You are not logged in.

2013 June(71C) Math Question 37

Conics

- Parabola

Choice B

The other point must be …

You are not logged in.

2013 June(71C) Math Question 38

- Solve Algebraic Expressions

38. What is the value of a if the coordinates of Z are (5, 100) ?

Choice H

$$100=a5^2 \\$$ $$ \Rightarrow a=4$$

You are not logged in.

2013 June(71C) Math Question 39

Conics

- Parabola

Choice B

$$x=ay^2$$ is open to the …

You are not logged in.

- Absolute Value Expressions

2013 June(71C) Math Question 40

Absolute Values

40. For all k > 0, I xl = I y = —k has how many (x,y) solutions?

Choice F

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

You are not logged in.

- Create Graphs or Figures

2013 June(71C) Math Question 41

Graphing Properties

41. A Ferris wheel is turning at a constant speed during 1 of its rotations. Let t represent the time that has elapsed since the wheel started turning and let h represent the height above ground level of a certain seat on the wheel. The seat is at its minimum height at t = a seconds and is at its maximum height at t = b seconds. One of the following graphs represents the relationship between t and h during this rotation. Which one?

Choice A

Since when t is a, …

You are not logged in.

2013 June(71C) Math Question 41

Choice A

Since when t is a, …

You are not logged in.

2013 June(71C) Math Question 42

- 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°; …

You are not logged in.

2013 June(71C) Math Question 43

- 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 …

You are not logged in.

- Advanced sin, cos, tan functions

2013 June(71C) Math Question 44

Trigonometry

44. Li is standing at point L on the north side of the small canyon shown in the figure below. As measured by line of sight, Li is 2 miles from an observation tower at 7', and she is 3 miles from a scenic overlook at S. Li, the observation tower, and the scenic overlook are all at the same elevation. The measure of LS is 28°. Which of the following equations, when solved, gives the measure of LT ?

Choice F

Generally:

$$\frac{\sin T}{LS}=\frac{\sin S}{LT}$$

You are not logged in.

2013 June(71C) Math Question 45

45. Mr. Green is buying mulch for his triangular-shaped garden shown below. When the mulch is uniformly spread 2 inches deep, each bag of mulch will cover 12 square feet. Mr. Green plans to uniformly spread his mulch 3 inches deep. Which of the following is closest to the number of bags of mulch Mr. Green will need for his garden?

Choice B

This is what a bag …

You are not logged in.

2013 June(71C) Math Question 46

- Write Algebraic Expressions

46. A child's dose of medicine is often based on an adult's dose of medicine. For some medicines, Cowling's rule, d = D(a + 1) , relates the dose of d milligrams for a 24child of age a years to the adult dose of D milligrams. Which of the following expressions gives a in terms of

Choice H

$$H=\frac{24d}{D}-1$$

You are not logged in.

2013 June(71C) Math Question 47

Logarithms

- Logarithm Expressions

47. For all x > 0, which of the following expressions is equivalent to log ((2x)-) ?

Choice D

$$\log_{}{((2x)^\frac{1}{2})}=\frac{1}{2}\log_{}{(2x)}=\frac{1}{2}\log_{}{2}+\frac{1}{2}\log_{}{x}$$

You are not logged in.

Similar Triangles & Types of Triangles

2013 June(71C) Math Question 48

- Similar Triangles

48. In AACD shown below, BE is parallel to CD; and the given lengths are in feet. Given that CD denotes the length, in feet, of CD, which of the following proportions involving CD must be true?

Choice H

It can be proved that …

You are not logged in.

- Graphing Trig Functions

2013 June(71C) Math Question 49

Trigonometry

49. The graph of y = a sin bx is shown below for certain positive values of a and b. One of the following values is equal to a. Which one?

Choice A

The range of the graph …

You are not logged in.

2013 June(71C) Math Question 50

Statistics

- 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

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

You are not logged in.

2013 June(71C) Math Question 51

- Mixed Shapes

51. Circles with centers A, B, and C, respectively, are mutually tangent, as shown below, and have radii of lengths a, b, and c, respectively. The lengths of AB, BC, and AC are 13 inches, 15 inches, and 26 inches, respectively. What is the value of a + b + c ?
Back to top↑

Choice C

$$a+b+c=\frac{AB+BC+AC}{2}=\frac{13+15+26}{2}=27$$

You are not logged in.

Similar Triangles & Types of Triangles

2013 June(71C) Math Question 52

- Triangle Types

52. Which of the following sets of 3 lengths, in decimeters, are the side lengths of an obtuse triangle? (Note: An obtuse triangle has 1 angle whose measure is greater than 90° and less than 180°.)

Choice K

$$8^2+11^2<16^2 $$ $$\Rightarrow obtuse …

You are not logged in.

2013 June(71C) Math Question 53

- 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 \\ …

You are not logged in.

2013 June(71C) Math Question 54

Quadratic Equations

- 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, …

You are not logged in.

2013 June(71C) Math Question 55

55. Hannah is building a fence along 3 sides of a rectangle, leaving 1 of the short sides unfenced. The fence has a post at each corner. The rectangle is 30 feet by 48 feet, measured from the centers of the corner posts. As shown in the figure below, adjacent posts of the fence are set every 6 feet, and there are 12 wooden slats between each pair of posts. How many wooden slats will Hannah use for the fence?

Choice D

$$\frac{48}{6}\times12+\frac{48}{6}\times12+\frac{36}{6}\times12-12=252$$

You are not logged in.

2013 June(71C) Math Question 56

Inequalities

- Intersection of Lines

Choice G

Only graph G fulfils the …

You are not logged in.

2013 June(71C) Math Question 57

- 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

$$CD=\sqrt{CF^2-DF^2}=\sqrt{7^2-5^2}=2\sqrt{6}$$

You are not logged in.

- Domain, Range, Asymptote, Undefined

2013 June(71C) Math Question 58

Graphing Properties

58. The entire graph of y =f(x) is shown in the standard (x,y) coordinate plane below. One of the following sets is the domain off. Which set?

Choice K

The domain of f is …

You are not logged in.

2013 June(71C) Math Question 59

Inequalities

- 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 …

You are not logged in.

Exponents, Radicals, & Scientific Notation

2013 June(71C) Math Question 60

- Exponent Expressions

60. Let m and n be nonzero real numbers such that 2n+ 1= 2m. Which of the following is an expression for 2 3 in terms of m ?

Choice K

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

You are not logged in.