ACT Form 72G Math Answer Explanation


2014 December(72G) Math Question 1

Word Problems Practice

- Cost

1. Bella sells real estate. She is paid 3% of the sale price of every house she sells. What is Bella paid for selling a house at the sale price of $250,000 ?

Choice D

$$$250,000\times 3\%=$7,500$$

2014 December(72G) Math Question 2

Quadratic Equations Practice

- Factoring

2. For what value(s) of x is (x + 3)(x — 4) = 0 true?

Choice F

Solving for the equation, we have x = -3 and 4.

Choice B

$$\frac{1}{6}\approx0.167$$

Choice H

$$2x+1=5\Rightarrow x=2 \\ $$ $$\Rightarrow (x+3)^3=(2+3)^3=5^3=125$$

2014 December(72G) Math Question 5

Graphing Properties Practice

- Vectors

5. What vector is the result of adding the vectors (1,-2), (2,3), and (-4,2) ?

Choice B

<1+2-4,-2+3+2>=<-1,3>

Choice H

$$\frac{2}{3}x=6\\$$ $$ \Rightarrow 2x=18 \\ $$ $$\Rightarrow x=9$$

2014 December(72G) Math Question 7

Statistics Practice

- Mean, Median

7. In Summit City, the daily low temperatures, in degrees Fahrenheit (°F), during the first week in December were —7, —11, 15, 23, 45, 23, and 2. To the nearest 1°F, what was the mean daily low temperature for that week?

Choice B

$$\frac{-7-11+15+23+45+23+2}{7}=13°C$$

2014 December(72G) Math Question 8

Inequalities Practice

- Inequality Expressions

8. Assume that a, b, and c are real numbers. If a > b, and (a + c) > (b + c), then what is true of c ?

Choice K

$$(a+c)>(b+c) \\$$ $$\Rightarrow a>b$$

Choice C

$$7y=-2x-12 \\$$ $$ \Rightarrow y=-\frac{2}{7}x-\frac{12}{7}$$

Choice K

$$h=\sqrt{4^2+8^2}=\sqrt{80}$$

2014 December(72G) Math Question 11

Word Problems Practice

- Distance, Speed, Time

11. Students studying motion observed a cart rolling at a constant rate along a straight line. The table below gives the distance, d feet, the cart was from a reference point at 1-second intervals from t = 0 seconds to t = 5 seconds.

Choice C

Only the equation in C fulfils all sets of d and t in the table.

Choice G

$$\frac{30}{24}=\frac{40}{x} \\ $$ $$\Rightarrow x=32$$

2014 December(72G) Math Question 13

Probability Practice

- One Event

13. A box contains 3 red, 2 white, and 5 blue beads, each of which is a solid color. Miguel will randomly remove 1 bead from the box, record the color of the bead, and return the bead to the box. Miguel will complete this process 50 times. How many times should Miguel expect to record a bead color that is NOT white?

Choice D

$$50\times \frac{3+5}{3+2+5}=40$$

Choice F

$$3(a+6b)=c \\ $$ $$\Rightarrow a+6b=\frac{c}{3} \\$$ $$ \Rightarrow a=\frac{c}{3}-6b$$

Choice E

$$\sqrt{2x}=10 \\$$ $$ \Rightarrow 2x=100 \\ $$ $$\Rightarrow x=50$$

2014 December(72G) Math Question 16

Quadratic Equations Practice

- Factoring

16. A high school ice hockey conference that consists of 10 teams requires that every conference team must play each of the other conference teams at least once in a season. For n teams in the conference, the number of conference games played in a season must then be at least n 2— n . What is the minimum number of conference games that must be played in a season for this conference?

Choice H

$$\frac{10^2-10}{2}=45$$

Choice A

$$\tan\alpha=\frac{\sin\alpha}{\cos\alpha} \\ $$ $$\Rightarrow \cos\alpha=\frac{\sin\alpha}{\tan\alpha}=\frac{\frac{112}{113}}{\frac{112}{15}}=\frac{15}{113}$$

Choice H

$$\angle A = \frac{180°-75°}{2}=52.5°$$

Choice B

$$150=300(\frac{t+1}{24}) \\$$ $$ \Rightarrow t\approx 11$$

Choice F

72g

As can be seen from the graph, only (7,2) is 3 cm from (4,2).

2014 December(72G) Math Question 21

Angles Practice

- Circles

21. The Ferris wheel at a state fair has a radius of 60 feet, rotates at a constant speed, and completes 1 rotation in 4 minutes. How many degrees does the Ferris wheel rotate in 30 seconds?

Choice D

$$\frac{30}{60\times4}\times360°=45°$$

Choice F

"Odetta spends$8.75 on the purchase …

2014 December(72G) Math Question 23

Perimeter, Area, Volume Practice

- Other Shapes

23. The perimeter of a parallelogram is 76 inches, and the length of 1 side is 10 inches. If it can be determined, what are the lengths, in inches, of the other 3 sides?

Choice C

Since it is a parallelogram, the length of the other side is 10 inches, and that: $$\frac{76-10-10}{2}=28$$

2014 December(72G) Math Question 24

Word Problems Practice

- Cost

24. John told Raquel that if he spent up to $30 from his savings account, his savings account would have at least 3/4 as much in it as it has now. From John's statement, Raquel can deduce that the least amount of money that John could have in his savings account now is:

Choice K

$$x-30\geq \frac{3}{4}x \\ $$ $$\Rightarrow …

2014 December(72G) Math Question 25

Probability Practice

- One Event

25. At East High School, the 20 cast members of the spring musical sold 2 types of tickets—student and adult—for the Friday night and Saturday night performances. The price of each student ticket was $2 and the price of each adult ticket was $3. The table below gives the number of tickets sold, by type and by night. The stem-and-leaf plot below shows the number of tickets, regardless of type, sold by each of the 20 cast members. The auditorium where the spring musical will be performed has 10 seats in the 1st (front) row. Each row behind the 1st row has 4 more seats than does the row in front of it. Suppose 1 cast member will be picked at random from the 20 cast members who sold tickets to receive a prize. What is the probability of picking a cast member who sold more than 30 tickets?

Choice A

$$\frac{4}{20}=\frac{1}{5}$$

2014 December(72G) Math Question 26

Word Problems Practice

- 1 Variable Equations

26. For which night was the total amount collected for the tickets greater, and by how many dollars was it greater?

Choice J

$$100\times2+110\times3-122\times2-90\times3=16$$

Choice A

$$10+4\times7=38$$

Choice H

$$\frac{CD}{DE}=\frac{AB}{BC} \\$$ $$ \Rightarrow \frac{CD}{4}=\frac{4}{8} …

Choice A

$$AD=AB+BD=AC-BC+BD=32 \\$$ $$ \Rightarrow 15-BC+20=32 …

Choice H

$$\frac{1.8\times10^9}{3.6\times10^6}=500$$

Choice C

$$\frac{9\times12}{2}=54$$

2014 December(72G) Math Question 32

Word Problems Practice

- Cost

32. What was the total cost, including sales tax, of the fertilizer, plants, and mulch that the landscaper purchased?

Choice G

$$(15.5+0.7\times50+4\times3)\times1.08=67.5$$

Choice E

72g

Choice K

$$h=\sqrt{9^2+12^2}=15 \\ $$ $$\Rightarrow 9+12+15=36$$

Choice D

$$\frac{AB}{AC}=\frac{BE}{CD} \\ $$ $$\Rightarrow \frac{AB}{20}=\frac{5}{15} …

Choice G

$$2\pi r=2\pi \frac{d}{2}=50 \\$$ $$ …

2014 December(72G) Math Question 37

Equation of a Line Practice

- Distance & Midpoint

37. Which of the following values is closest to the distance, in inches, between point A and point B on the ruler shown below? (Note: The ruler shown is NOT actual size.)
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Choice C

$$3.75-0.25-\frac{0.25}{2}=3.375$$

Choice J

n-m=r+6-(r+5)=1

2014 December(72G) Math Question 39

Angles Practice

- Parallel Lines

39. In the figure below, lines l and m are parallel. Lines a and b intersect l at the same point. What is the value of x?

Choice C

$$(4x+12)+(3x+18)+(5x-18)=12x+12=180 \\$$ $$ x= 14$$

2014 December(72G) Math Question 40

Perimeter, Area, Volume Practice

- Mixed Shapes

40. The line y = 1/2x + 3 and the line y = 3x — 12 intersect at (6,6), as shown in the standard (x,y) coordinate plane below. The shaded region is bounded by these 2 lines, the x-axis, and the y-axis. What is the area of the shaded region, in square coordinate units?

Choice G

$$Area = 6\times6-\frac{6\times3}{2}-\frac{6\times2}{2}=21$$

2014 December(72G) Math Question 41

Angles Practice

- Vertex Angles & Bisectors

41. In the figure below, AD, BE, and CF all intersect at point Z. The measures of 2 angles are given. What is the measure of CZE ?

Choice E

$$\angle CZE=180-51=129$$

2014 December(72G) Math Question 42

Word Problems Practice

- Distance, Speed, Time

42. Emma and Kristin each ran 10 laps (4,000 meters) around a 400-meter track without stopping. Emma ran at a constant speed of 250 meters per minute. Kristin ran at a constant speed of 200 meters per minute. Both Emma and Kristin began running at the same instant. How many laps did Kristin have left to run when Emma had completed her run?

Choice J

$$\frac{4000-200\times\frac{4000}{250}}{400}=2$$

Choice D

Lines with the same slope …

Choice G

$$4a=8\times3 \\$$ $$ \Rightarrow a=6$$

Choice C

$$\frac{2}{6}=\frac{k}{12} \\$$ $$ \Rightarrow k=4 …

Choice K

The fact that 7 is …

2014 December(72G) Math Question 47

Quadratic Equations Practice

- Factoring

47. Which of the following binomials is a factor of 3x^2 + 11x — 4 ?

Choice E

$$3x^2+11x-4=(x+4)(3x-1)$$

Choice G

$$\tan53°=\frac{60}{x} \\$$ $$ \Rightarrow x\approx …

2014 December(72G) Math Question 49

Statistics Practice

- Mean, Median

49. Each of 20 students in a class took a test and received a whole number score. The median of the scores was 87. None of the students received a score of 87, and 30% of the students received scores of 90 or above. How many students received scores of 88 or 89 ?

Choice B

$$10-20\times30\%=4$$

2014 December(72G) Math Question 50

Perimeter, Area, Volume Practice

- Other Shapes

50. The radius and height of the right circular cylinder shown below are given in centimeters. Which of the following is closest to the volume, in cubic centimeters, of the cylinder?

Choice K

$$\pi r^2 h=\pi 10^2 \times7\approx …

2014 December(72G) Math Question 51

Logarithms Practice

- Logarithm Expressions

51. What value of x satisfies the equation below?

Choice D

$$\log_{2^4}{x}=-\frac{3}{4} \\$$ $$ \Rightarrow \frac{1}{4}\log_{2}{x}=-\frac{3}{4} …

Choice K

y must be negative even …

Choice E

72g

Choice G

$$\cos(\alpha+\beta)=\cos(60°)=\frac{1}{2}$$

2014 December(72G) Math Question 55

Matrices Practice

- Determinants

55. The determinant of any 2 x 2 matrix is ad - bc. The determinant of is equal to 0. What are all possible values of x?

Choice A

$$(x+3)(x-2)-7\times2=0 \\$$ $$ \Rightarrow (x+5)(x-4)=0 …

Choice K

$$|x|+1\leq 2 \\$$ $$ \Rightarrow …

2014 December(72G) Math Question 57

Trigonometry Practice

- Graphing Trig Functions

57. Given that 2 sin a = 2 and 2 cos(- b) = 2, which of the following could be a value, in radians, of a + b ?

Choice D

$$\sin \alpha =1 \Rightarrow \alpha …

2014 December(72G) Math Question 58

Conics Practice

- Ellipse

58. One of the following equations determines the graph in the standard (x,y) coordinate plane below. Which one?

Choice F

The center of the ellipse …

Choice A

$$0.5^3\times(6\times8)=6$$

Choice J

$$f(g(x))=g(f(x)) \\$$ $$ \Rightarrow 2(3x+b)+5=3(2x+5)+b …