ACT Form 72F Math Answer Explanation



Choice E

$$\frac{4}{y}=0.4$$ $$\Rightarrow \frac{4}{y}\times y=0.4\times y$$ $$\Rightarrow 4=0.4y$$ $$\Rightarrow y=10$$


2019 June(72F) Math Question 2

Word Problems

- Cost

2. A motel manager's costs are 24% higher this year than they had been when the room rate was s60.00. If the room rate had increased by the same percent as the manager's costs, what would the room rate be this year?

Choice J

$$60\times1.24=74.4$$


2019 June(72F) Math Question 3

Word Problems

- 1 Variable Equations

3. Louis earns his regular pay of $10.00 per hour for up to 40 hours of work in a week. For each hour over
40 hours of work in a week, Louis is paid 1, times his regular pay. How much does Louis earn for a week in which he works 47 hours?

Choice C

$$$10\times40+$10\times(1\frac{1}{2})\times7$$ $$=$400+$105$$ $$=$505$$


Choice J

$$3x^{9}\cdot 5x^{9}$$ $$=3\times5\times x^{9+9}$$ $$=15x^{18}$$


2019 June(72F) Math Question 5

Perimeter, Area, Volume

- Other Shapes

5. The total surface area. T, of any right circular cone with a radius r and a slant heighi s, such as the cone hown below, can be determined by using the formula T=pi r 2+pi rs. if a cone has a 3-inch radius and a
5-inch slant height, what is its total surface area, in square inches?

Choice B

$$T=\pi r^{2}+\pi rs$$ $$T=\pi 3^{2}+\pi \times3\times5$$ $$=9\pi+15\pi$$ $$=24\pi$$


2019 June(72F) Math Question 6

Angles

- Parallel Lines

6. In the figure below, C lies on both AE and BD, AB and DE are parallel and congruent, and 2 angle measures are given. What is the measure of ∠ACB

Choice K

$$180°-45°-65°=70°$$


Choice B

$$35=5\cdot7$$ $$28=2\cdot2\cdot7$$ $$8=2\cdot2\cdot2$$ $$2\cdot2\cdot2\cdot5\cdot7=280$$


2019 June(72F) Math Question 8

Quadratic Equations

- Factoring

8. Which of the following polynomial equations has solutions -2 and 5

$$x=-2,5$$ $$f(x)=(x-5)(x+2)^{2}$$ $$f(-2)=0$$ $$f(5)=0$$


Choice F

$$60\div(3+5+7)=60\div15=4$$ $$4\times7=28$$


2019 June(72F) Math Question 10

Sequences and Patterns

- General Sequences

10. Zoe programs her calculator to evaluate a linear function, but she doesn't say what the function is When 9 is entered, the calculator displays the value 6.
When 12 is entered, the calculator displays the value 8. Which of the following expressions represents what the calculator will display when any number, n, is entered?

Choice F

$$f(9)=6$$ $$f(12)=8$$ $$F.\ f(x)=\frac{2}{3}n$$ $$f(9)=\frac{2}{3}\times9=6$$ $$f(12)=\frac{2}{3}\times12=8$$


2019 June(72F) Math Question 11

Equation of a Line

- Distance & Midpoint

11. Points C(2, 5)and D(8, I 1)lie in the standard(x,y) coordinate plane. What is the midpoint of CD?

Choice C

$$(\frac{2+8}{2},\frac{5+11}{2})=(5,8)$$


Choice F

$$1\times1\times\frac{1}{9}=\frac{1}{9}$$


Choice C

$$=3x^{2}+x^{2}-4x-6x-5-7$$ $$=4x^{2}-10x-12$$


2019 June(72F) Math Question 14

Logarithms

- Logarithm Expressions

14. For all positive integers a, let E represent the sentence a is even" and let P represent the sentence"a is prime. When a= 22,  which of the following statements is true?

Choice H

22 is even.

22 is not prime.


2019 June(72F) Math Question 15

Absolute Values

- Absolute Value Expressions

15.|6(-7)+4(8)|=?

Choice C

$$=\mid-42+32\mid=\mid-10\mid=10$$


Choice G

$$11x+6y=3$$ $$y=\frac{-11}{6}x+3$$


Choice A

$$g(2,4,3)$$ $$=4\times2\times4-3\times2\times (-3)^{2}$$ $$=32-54=-22$$


2019 June(72F) Math Question 18

Angles

- Circles

Use the following information to answer questions 18-21
A family plans to remodel their kitchen. They have a total budget of $45,000 to cover expenses in 6 categories. Not all the budget has been assigned. the budget amounts that have been assigned are shown in the table below.
Expense category    Budget amount 
Appliances                 $4,000
Cabinets                     $9,000
Flooring                      $5,000
Lighting                      $3,000
Labor                               ?
Other                               ?
Total budget             $45,000
18. In a circle graph illustrating the 6 budget amounts in the table, what should be the measure of the central angle of the Flooring sector?

Choice H

$$x°:360°=5000:45000$$ $$\Rightarrow x=40°$$


2019 June(72F) Math Question 19

Statistics

- Mean, Median

19. The amount budgeted for Appliances is the sum of the prices of 5 appliances -- 1 refrigerator, 1 dishwasher, 1 built-in cooktop, and 2 ovens(1 conventional and 1 microwave). What is the average price per appliance?

Choice A

$$4000\div5=800$$


$$3000:9000=1:3=\frac{1}{3}$$


Choice E

$$4000+9000+5000+3000=21000$$ $$45000-21000=24000$$ $$\frac{24000}{45000}=\frac{8}{15}=0.53333...$$


2019 June(72F) Math Question 22

Angles

- Other Shapes

22. In the figure below. 5 angle measures are given. The angle marked with a measure of y° is an exterior angle What is the value of y?

$$(2x-10)+y=180 \Rightarrow 2x+y=190$$ $$(x+25)+(2x+10)+(3x-25)+(2x-10)=360$$ $$\Rightarrow …


2019 June(72F) Math Question 23

Perimeter, Area, Volume

- Other Shapes

23. What is the area, in square coordinate units, of parallelogram ABCD shown in the standard (x,y)
coordinate plane below?

Choice D

$$7-2=5$$ $$9-4=5$$ $$8-0=8$$ $$5\times8=40$$


2019 June(72F) Math Question 24

Trigonometry

- Pythagorean Theorem & sin cos tan

24. Right triangle △ABC is shown below. The side lengths are given in centimeters. What is tan C?

Choice G

$$\tan C=\frac{8}{15}$$


Choice B

$$3a-5b=22---(1)$$ $$a+3b=-2---(2)$$ $$(2)\times3-(1)$$ $$\Rightarrow (3a+9b)-(3a-5b)=(-2)\times3-22$$ …


Choice F

$$3r+4p+50=545$$ $$\Rightarrow 3r+4p=495$$


2019 June(72F) Math Question 27

Sequences and Patterns

- General Sequences

27. The first 2 rectangles of a sequence of rectangles are shown below. The 1st rectangle is 4 inches long and I inch wide. The dimensions of the 2nd rectangle, and of each successive rectangle after the 2nd, are determined by continuing the following pattern: the length is 3 inches longer than the length of the previous rectangle, and the width is I inch longer than the width of the previous rectangle, What is the perimeter, in inches, of the 6th rectangle in the sequence?

Choice C

$$1st\ rectangle:\ 1\times2+4\times2=10$$ $$2nd\ rectangle:\ …


2019 June(72F) Math Question 28

Quadratic Equations

- Factoring

28. Which of the expressions below is a factor of the polynomial 2x+x-6x?
I. x
II. 2x+3
III. x-2

Choice F

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


2019 June(72F) Math Question 29

Probability

- One Event

29. Shown below. a circular dartboard has 4 sectors(red, white, blue, and green) whose areas are in the ratio of 1:2:3:4. respectively. Brad will throw 1 dart at the dartboard. and it will hit the dartboard at a random point contained in I of the sectors. What is the probability that the sector the dart hits is Not the blue sector?

Choice E

$$P(Not\ blue)$$ $$=1-P(Blue)$$ $$=1-\frac{3}{1+2+3+4}$$ $$=1-\frac{3}{10}$$ …


Choice J

$$3x+1=5x-9$$ $$\Rightarrow 2x=10$$ $$\Rightarrow x=5$$ …


2019 June(72F) Math Question 31

Trigonometry

- Pythagorean Theorem & sin cos tan

31. For right triangle △LMN below, cos L=7/6.   What sin N?

Choice C

72f

$$\cos L=\frac{a}{c}=\frac{7}{16}$$ $$\sin N=\frac{a}{c}=\frac{7}{16}$$


Choice K

$$(3.5+0.25n)(2500-100n)$$


Choice B

$$2\pi r=20$$ $$r=\frac{20}{2\pi}=\frac{10}{\pi}$$


2019 June(72F) Math Question 34

Trigonometry

- Pythagorean Theorem & sin cos tan

34. You're on a salvage ship in the Pacific Ocean when your ship,'s sonar locates a shipwreck at an angle of depression of 60, as shown in the figure below. After your ship travels 500 meters on the surface of the water to be directly over the wreck, how many meters down would you have to dive to reach the wreck?

Choice K

$$30°:60°:90°$$ $$\Rightarrow 1:\sqrt{3}:2$$ $$?=500\times\sqrt{3}=500\sqrt{3}$$


2019 June(72F) Math Question 35

Perimeter, Area, Volume

- Mixed Shapes

Use the following information to answer questions 35-38
Square ABCD, shown below, has side length 5 meters. The square is divided into 25 nonoverlapping congruent squares. Point P is the center of ABCD.
35. What is the length, in meters, of AC?

Choice C

$$45°:45°:90°$$ $$\Rightarrow 1:1:\sqrt{2}$$ $$\overline{AC}=5\times\sqrt{2}=5\sqrt{2}$$


2019 June(72F) Math Question 36

Perimeter, Area, Volume

- Mixed Shapes

36. The perimeter of 1 of the 25 congruent squares is how many meters?

Choice G

$$1\times4=4$$


Choice D

72f

$$450°=90°\times5$$


Choice J

72f


Choice C

$$\sqrt[3]{3x-9}+10=13$$ $$\Rightarrow \sqrt[3]{3x-9}=3$$ $$\Rightarrow (\sqrt[3]{3x-9})^{3}=3^{3}$$ …


2019 June(72F) Math Question 40

Word Problems

- % Interest

40. Ten years ago, Tara invested $2, 000 at 6% interest compounded monthly. Which of the following expressions represents todays value of the investment?

Choice J

$$=$2000(1+\frac{0.06}{12})^{120}$$


Choice A

$$2\div7$$ $$=0.285714285714...$$ $$=0.\overline{285714}$$


Choice K

$$\frac{60+65+70+75+80\times2+85\times3+90\times2+95\times4+100\times5}{20}$$ $$=\frac{60+65+70+75+160+255+180+380+500}{20}$$ $$=\frac{1475}{20}$$ $$=87.25$$


$$P=\frac{64G}{S}$$ $$\Rightarrow G=\frac{P\times S}{64}$$ $$\Rightarrow …


Choice K

x quarters,

x+5 dimes,

x+1 …


$$(3,9)\rightarrow(1,3)$$ $$(3k,9k)=(1,3)$$ $$k=\frac{1}{3}$$ $$(9,24)\Rightarrow(9\times\frac{1}{3},24\frac{1}{3})=(3,8)$$


2019 June(72F) Math Question 46

Equation of a Line

- Distance & Midpoint

46. Lindsay is designing a 5-foot-by-8-foot rectangular poster for her art class. She will cover both diagonals of the poster with straight iengths of yellow rope Which of the following values is closest to the total length, in feet, of the 2 yellow ropes Lindsay will need for the poster?

Choice F

72f

$$5^{2}+8^{2}=x^{2}=89$$ $$\Rightarrow x=\sqrt{89}\approx 9.43$$ $$9.43\times2=18.86$$


2019 June(72F) Math Question 47

Word Problems

- 1 Variable Equations

47. Robert writes a check for $20. When he records the check in his check register, he accidentally adds $20 to his balance instead of subtracting $20. which causes a discrepancy between what Roberts check register shows and what it should show. Because of his mistake. Roberts check register shows.

Choice E

$$x-20 \Rightarrow x+20$$ $$(x+20)-(x-20)=+40$$ $$$40\ …


2019 June(72F) Math Question 48

Word Problems

- Temperature

48. For a certain location in Grand Canyon National Park the highest temperature on record is 120° F and the lowest temperature on record is -18°F. Which of the statements below is necessarily true about the temperatures on record for this location?
I. The range of temperatures is 138.
II. The range of temperatures is 102F.
III. The median temperature is 69F

Choice F

The range of temperatures is …


Choice A

$$x+3=0$$ $$\Rightarrow x=-3$$


Choice G

$$Circle\ area=\pi r^{2}$$ $$\pi \times …


Choice D

$$\frac{3^{x}3^{4}}{(3^{4})^{5}}=\frac{1}{9}$$ $$\Rightarrow \frac{3^{x+4}}{3^{4\times5}}=\frac{1}{9}$$ $$\Rightarrow \frac{3^{x+4}}{3^{20}}=\frac{1}{9}=3^{-2}$$ …


Choice J

All real numbers except 7.


2019 June(72F) Math Question 53

Trigonometry

- Advanced sin, cos, tan functions

53. Green Camp and Blue Camp lie on opposite sides of Red Lake. A boat dock is located 1, 000 yards from Green Camp. The campers estimated the angles between these 3 locations to be as shown on the map below. Using these estimates, which of th e followin expressions gives the distance, in yards, between Green Camp and Blue Camp?

Choice D

72f

By the Law of Sines, …


2019 June(72F) Math Question 54

Graphing Properties

- Intersection of Lines

54. The graphs of the functions y=f(x)=(x-3)2+2 and y=g(x)=1/2x+1/2 , are shown in the standard(x,y) coordinate plane below. Which of the following is NOT true?

Choice G

$$F.\ f(x) > 0 \Rightarrow |f(x)| …


Choice D

To minimize 2/(y-x),

we simply …


Choice F

72f

$$\Rightarrow (-r,s)$$


2019 June(72F) Math Question 57

Matrices

- Multiplication

57. Which of the following matrices is equal to the matrix product

Choice E

$$=\begin{bmatrix}2 & -5 \\-3 & …


Choice G

72f

$$\overline{AE}=\overline{BE}$$ $$\overline{AG}=\overline{GE}$$ $$\Rightarrow \overline{EF'}:\overline{GH'}$$ $$=\overline{BE}:\overline{BG}$$ …


2019 June(72F) Math Question 59

Probability

- More than One Event

59. Suppose that a will be randomly selected from the set
{-3,-2,-1, 0, 1} and that b will be randomly selected from the set {-2,-1, 0, 1} . What is the probability that ab>0?

Choice D

$$ab\gt 0,$$ $$(-)(-)\gt 0,$$ $$(+)(+)\gt …


2019 June(72F) Math Question 60

Sequences and Patterns

- Logical Expressions

60. Distinct points A. B, C, and D are collinear, and B is between A and C. For D to be between A and C such that AD+ DB+ BC= AC, which of statements I-IV below must be true?
(Note: If B is between A and C, then AB +BC= AC.)
I.  AD= DB 
II. AD= BC 
III. B is between D and C 
IV. D is between B and C

Choice G

72f

B is between D and …