ACT Form 72E Math Answer Explanation


Choice D

$$\frac{x}{72}=\frac{1\frac{3}{4}}{24} \\$$ $$ \Rightarrow x=5\frac{1}{4}$$

2015 December(72E) Math Question 2

Statistics Practice

- Mean, Median

2. The age, in years, of each of the first 6 presidents of the United States at his first inauguration was 57, 61, 57, 57, 58, and 57, respectively. Which of the following values is closest to the mean age, in years, of the first 6 presidents at their respective first inaugurations?

Choice H

$$\frac{57+61+57+57+58+57}{6}=57.8$$

2015 December(72E) Math Question 3

Word Problems Practice

- Temperature

3. The temperature F in degrees Fahrenheit is related to the temperature K in kelvins by the equation F = 1.8K — 459.67. Which of the following temperatures, in kelvins, is closest to 120 degrees Fahrenheit?

Choice A

$$120=1.8K-459.67 \\$$ $$ \Rightarrow K\approx 322$$

Choice F

$$-\frac{18}{3}x^{3-1}y^{2-1}=-6x^2y$$

Choice C

$$\sqrt{a}=9 \Rightarrow a=81$$

2015 December(72E) Math Question 6

Word Problems Practice

- Cost

6. Damon and 4 of his coworkers are having lunch. Each of the 5 people will pay for his or her own lunch, but they agree to divide the tip equally among themselves. The total for the 5 lunches is $80.00, and the group will add a tip of 15% of the total. Each person's portion of the tip will be how much?

Choice G

$$\frac{80\times15\%}{5}=2.4$$

Choice B

$$0.000000000087=8.73\times 10^{-11}$$

2015 December(72E) Math Question 8

Angles Practice

- Circles

8. The circular spinner dial for a new board game is divided into 6 congruent sectors. What is the arc measure, in degrees, of each sector?

Choice J

$$\frac{360}{6}=60$$

Choice E

$$\frac{4+45}{9}=\frac{49}{9}$$

2015 December(72E) Math Question 10

Matrices Practice

- Addition & Subtraction

10. 2[3	24]	2 + 3[-1_2]	= ?

Choice G

$$\begin{bmatrix}2\times1+3\times2 & 2\times2+3\times1 \\2\times3-3\times1 & 2\times4-3\times2 \end{bmatrix}=\begin{bmatrix}8 & 7 \\3 & 2 \end{bmatrix}$$

2015 December(72E) Math Question 11

Word Problems Practice

- Distance, Speed, Time

11. The speed of one motorcycle exceeds 4 times the speed of another motorcycle by 24 mph. The speed of the slower motorcycle is g mph. Which of the following expressions represents the speed of the faster motorcycle, in miles per hour?

Choice D

The speed of one motorcycle exceeds 4 times the speed of another motor cycle by 24 mph: 4g+24

Choice J

$$Area=\frac{12\times8}{2}=48$$

2015 December(72E) Math Question 13

Word Problems Practice

- Distance, Speed, Time

13. 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 data sets in the table.

2015 December(72E) Math Question 14

Word Problems Practice

- Distance, Speed, Time

14. Rajeev went on a delivery trip that began and ended at his truck terminal. He used all of the time during the trip driving, unloading, or resting. Rajeev began his trip on Tuesday at 7:00 a.m. when he left the terminal. During his driving time, he drove 1,200 miles at an average speed of 50 miles per hour. His driving time was twice as long as his unloading time, and his resting time was 30 hours. When did Rajeev end his delivery trip?

Choice H

$$Driving \ time = \frac{1200}{50}=24 \\$$ $$ Unloading \ time = 12 \\$$ $$ \Rightarrow Total \ time=24+12+30=66=24+24+18$$

Choice C

$$x+5y=3 \Rightarrow x=-5y+3$$

2015 December(72E) Math Question 16

Perimeter, Area, Volume Practice

- Other Shapes

16. The perimeter of a parallelogram is 80 inches, and the length of 1 side is 16 inches. If it can be determined, what are the lengths, in inches, of the other 3 sides?

Choice H

Thus the other side is also 16; the remaining 2 sides are: $$\frac{80-16-16}{2}=24$$

Choice D

$$6l+4s=26 \\$$ $$ 2l+4x=14 \\$$ $$ \Rightarrow l=3$$

2015 December(72E) Math Question 18

Probability Practice

- Combinations

18. How many different possible orders of pasta can a person get?

Choice K

$$2\times6\times5=60$$

2015 December(72E) Math Question 19

Word Problems Practice

- 1 Variable Equations

19. The Tully family also bought 5 salads priced at $2.00 per salad and 12 breadsticks priced at $1.50 for an order of 3 breadsticks. What was the total price of the pasta, salads, and breadsticks the Tully family bought, without tax and tip?

Choice C

$$26+2\times5+1.5\times \frac{12}{3}=42$$

2015 December(72E) Math Question 20

Angles Practice

- Parallel Lines

20. In the figure below, parallel lines AB and CD are cut by transversals AC and BD that intersect at E. Two angle measures are given. What is the measure of ZABD ?

Choice K

180-34=146

2015 December(72E) Math Question 21

Word Problems Practice

- 1 Variable Equations

21. The total cost, c dollars, for Main Street Orchestra to perform a concert at Milly's Auditorium is determined by c = r + 20m, where r is the rental fee, in dollars, of the auditorium and m is the number of orchestra members playing. The Friday night rental fee for Milly's Auditorium is $500. There will be 30 orchestra members playing in Friday night's concert. For the total price of exactly 200 tickets to equal the total cost of performing the concert, what should be the price of each ticket?

Choice E

$$200x=500+20\times30 \\$$ $$ \Rightarrow x=5.5$$

Choice K

$$\frac{9}{10+5+10+9+9+7}=18\%$$

Choice B

The length of the rectangle is w + 12; the area is $$w\cdot l=140 \\$$ $$ \Rightarrow w\cdot (w+12)=140$$

2015 December(72E) Math Question 24

Equation of a Line Practice

- Distance & Midpoint

24. What is the length, in coordinate units, of the line segment with endpoints (-8,4) and (4,9) in the standard (x,y) coordinate plane?

Choice H

$$\sqrt{(-8-4)^2+(4-9)^2}=\sqrt{169}$$

Choice E

(6,3+(7-3)+(7-3))=(6,11)

Choice H

$$Area = \pi r^2=\pi (\frac{6}{2})^2=9 …

2015 December(72E) Math Question 27

Statistics Practice

- Mean, Median

27. In a chemistry course, a student scored 99 on one test, 98 on another test, and 88 on each of the other tests. The student's test average for the course, where each test is weighted equally, is exactly 91. What is the total number of tests that the student has taken in the course?

Choice D

$$\frac{99+98+88\times(n-2)}{n}=91 \\$$ $$ \Rightarrow n=7$$

Choice J

Among the choices, only 60 …

2015 December(72E) Math Question 29

Quadratic Equations Practice

- Factoring

29. Which of the following expressions is a factor of the polynomial x^2 — x — 72 ?

Choice A

$$x^2-x-72=(x-9)(x+8)$$

2015 December(72E) Math Question 30

Perimeter, Area, Volume Practice

- Mixed Shapes

30. Javier will have a pool installed in his backyard. The interior of the pool is a right circular cylinder with a uniform depth of 5 feet and a radius of 8 feet. The maximum volume of water that can be in the pool is 75% of the volume of the pool. Which of the following values is closest to the maximum number of cubic feet of water that can be in the pool?

Choice F

$$Volume=\pi r^2 h=\pi (8^2)(5) \\$$ …

Choice E

$$\sin C= \frac{AB}{AC} \\ $$ …

Choice H

$$\angle R= \angle P=54° \\ …

Choice B

g(1) > 0

g(2) > 0

g(3) …

Choice J

$$b=3n \\$$ $$\Rightarrow \frac{1}{b}+\frac{1}{n}=\frac{1}{3n}+\frac{1}{n}=\frac{...}{3n}$$

Choice E

$$g=b+15 \\$$ $$ r=g+24=b+39 \\$$ …

2015 December(72E) Math Question 36

Angles Practice

- Other Shapes

36. The measures of 4 interior angles of a pentagon are 70°, 100°, 110°, and 135°, respectively. What is the measure of the 5th interior angle?

Choice K

$$(5-2)\times 180°-70°-100°-110°-135°=125°$$

2015 December(72E) Math Question 37

Perimeter, Area, Volume Practice

- Mixed Shapes

37. The midpoints of the sides of rectangle WXYZ are the vertices of rhombus ABCD. The dimensions of rectangle WXYZ are 6 cm by 8 cm. What is the perimeter, in centimeters, of rhombus ABCD ?

Choice A

$$WD=\frac{6}{2}=3 \\$$ $$ WA=\frac{8}{2}=4 \\$$ …

Choice G

OS=2r

Thus S(0, -2r)

Choice B

‘x varies directly with y’ …

2015 December(72E) Math Question 40

Quadratic Equations Practice

- Factoring

40. What is the set of real solutions for |x|^2— |x| — 2 = 0 ?

Choice G

$$(|x|-2)(|x|+1)=0 \\$$ $$ \Rightarrow |x|=2 …

Choice B

The line ST is going …

2015 December(72E) Math Question 42

Equation of a Line Practice

- Distance & Midpoint

42. What is the midpoint of ST ?

Choice H

$$(\frac{1+4}{2},\frac{2+0}{2})=(\frac{5}{2},1)$$

Choice D

$$\tan \angle STR = \frac{SR}{RT}=\frac{2}{3}$$

Choice F

SR is the radius which …

Choice D

A cube has 6 sides. …

2015 December(72E) Math Question 46

Graphing Properties Practice

- Intersection of Lines

46. When graphed in the standard (x,y) coordinate plane, the lines x = —3 and y = x — 5 intersect at what point?

Choice K

y=-3-5=-8

Choice B

$$y=\frac{4}{3}x+\frac{2}{3} \\$$ $$ \Rightarrow slope=\frac{4}{3} …

Choice K

K: |x|=x

When x is …

2015 December(72E) Math Question 49

Perimeter, Area, Volume Practice

- Other Shapes

49. In trapezoid ABCD illustrated below, AB is 8 units long, CD is 12 units long, and EF is 6 units long. Also, ZAEF and LDFE are right angles, What is the area of ABCD, in square units?

Choice A

$$Area = \frac{AB+CD}{2}\cdot EF=\frac{8+12}{2}\cdot 6=60$$

Choice J

$$x^2={\sqrt{a}}^2=a$$

Choice A

$$\tan\alpha=\frac{50-5}{80}=\frac{45}{80}$$

2015 December(72E) Math Question 52

Inequalities Practice

- Graphing Inequalities

52. One of the following inequalities is graphed below in the standard (x,y) coordinate plane. Which one?

Choice G

Firstly, one need to identify …

2015 December(72E) Math Question 53

Probability Practice

- More than One Event

53. A box contains 6 identically sized, solid-colored balls. One ball is green, 2 are yellow, and 3 are red. A ball is drawn at random and returned to the box, then a second ball is drawn at random. What is the probability that the first ball is red and the second ball is green?

Choice A

$$\frac{3}{6}\cdot\frac{1}{6}=\frac{1}{12}$$

Choice H

$$Area = (18-2x)(12-x)=2x^2-42x_216$$

2015 December(72E) Math Question 55

Trigonometry Practice

- Graphing Trig Functions

55. The graph of y = sin x in the standard (x,y) coordinate plane is reflected over the x-axis, shifted up a units, and then shifted left 0.5pi units. Which of the following equations represents the graph after the 3 transformations?

Choice B

The graph is reflected over …

Choice K

$$\frac{x^{24}}{x^6}\cdot x^2=x^{24+2-6}=x^{20}$$

Choice A

$$a\cdot b=-48 \\$$ $$ a+b=0 …

2015 December(72E) Math Question 58

Perimeter, Area, Volume Practice

- Other Shapes

58. As shown below, BE divides rectangle ACDF into 2 congruent trapezoids. The measure of BED is 45°. The lengths of BC, CD, and EF are given in feet. What is the area, in square feet, of rectangle ACDF ?

Choice K

72e

$$Area = 6\times(4+6+4)=84$$

Choice A

$$\frac{6}{7}\times \frac{1}{3}\times \frac{1}{100}=\frac{1}{350}$$ Choice A.

2015 December(72E) Math Question 60

Logarithms Practice

- Logarithm Expressions

60. For what real value of x, if any, is log(x+3)(x^2+ 3) = 2

Choice G

$$(x+3)^2=x^2+3 \\ $$ $$\Rightarrow x^2+6x+9=x^2+3 …