ACT Form 73G Math Answer Explanation

2015 April(73G) Math Question 2

- Percents & Fractions

2. The table below shows the income earned by 5 students from selling tickets for a school fund-raiser. Each student earned the same amount for each ticket sold. Another student, Francis, also earned that same amount for each ticket sold, for an income earned of $123.25 from selling tickets. How many tickets did Francis sell?

Choice G

$$\frac{123.25}{\frac{55.25}{13}}=29$$

2015 April(73G) Math Question 3

- Radical Expressions

3. What is the value of the expression when x = —1 and m = —16 ?

Choice B

$$\sqrt{\frac{m}{x-3}}=\sqrt{\frac{-16}{-1-3}}=\sqrt{4}=2$$

2015 April(73G) Math Question 4

- Simplify Algebraic Expressions

4. For all real numbers a, b, and c, the expression ax — bx + cx can be written as the product of x and which of the following?

Choice H

$$ax-bx+cx=x(a-b+c)$$

2015 April(73G) Math Question 5

Word Problems

- Cost

5. Hai has $100 available to buy USB drives to back up data for his business computers. Each USB drive has a price of $8, and Hai will pay a sales tax of 7% of the total price of the USB drives. What is the maximum number of USB drives Hai can buy?

Choice A

$$\frac{$100}{$8\times 107\%}=11.68\approx 11$$

2015 April(73G) Math Question 6

- Pythagorean Theorem & sin cos tan

6. The lengths of the legs of a right triangle are 4 miles and 5 miles, respectively. Which of the following lengths, in miles, is closest to that of the hypotenuse of the right triangle?

Choice H

$$\sqrt{4^2+5^2}\approx6.4$$

2015 April(73G) Math Question 7

- LCM, GCF, Prime Numbers

7. What is the least common multiple of 50, 70, and 90 ?

Choice D

$$\frac{3150}{50}=63 \\$$ $$ \frac{3150}{70}=45 \\$$ $$ \frac{3150}{90}=35$$ Choice D.

2015 April(73G) Math Question 8

8. Susan makes holiday wreaths for 4 hours every Saturday. It takes her 20 minutes to make a small wreath and 30 minutes to make a large wreath. This Saturday, Susan will make twice as many large wreaths as small wreaths. How many of the large wreaths will she make this Saturday?

Choice J

Let the number of small wreaths be x, thus the number of large wreath 2x: $$30\times 2x+20\times x=4\times 60 \\$$ $$ \Rightarrow x=3 \\$$ $$ \Rightarrow 2x=6$$

Similar Triangles & Types of Triangles

2015 April(73G) Math Question 9

- Similar Triangles

9. In ABC below, D is on AB, E is on AC, and ED || CB. If it can be determined, what is the ratio of the area of ABC to the area of ADE ?

Choice E

The ratio ranges from 0 to 1.

2015 April(73G) Math Question 10

Equation of a Line

- Distance & Midpoint

10. On the real number line, point J is at —7 and point K is at —14. What is the distance between J and K ?

Choice H

Distance = |(-7)-(-14)|=7

2015 April(73G) Math Question 11

11. A system of equations is given below. What is the value of b in the (a,b) solution to the system?

Choice E

$$3b-7=b+1 \\ $$ $$\Rightarrow2b=8 \\$$ $$ \Rightarrow b=4$$

2015 April(73G) Math Question 12

12. The table below gives the total charge to rent a moving truck from each of 2 movers for various numbers of miles. For what number of miles would the total charge for renting a moving truck from Ben's be the same as the total charge for renting a moving truck from Ronnie's? (Note: There is a linear relationship between the number of miles and the total charge for both Ben's and Ronnie's.)

Choice J

For Ben's:

Total charge = Number of miles

For Ronnie's:

Total charge = 0.5 Number of miles +40

Number of miles = 0.5 Number of miles +40

Number of miles = 80

2015 April(73G) Math Question 13

- Circles

13. In the figure below, G is the center of the circle, LK is a diameter, H lies on the circle, J lies outside the circle on LK, and JM is tangent to the circle at M. Which of the following angles or minor arcs has the greatest degree measure?

Choice A

Apparently LM is greater than 90°, while the rest are either equal to or smaller than 90°.

2015 April(73G) Math Question 14

Probability

- One Event

Choice H

$$\frac{8}{6+8+g}=\frac{1}{3} \\$$ $$\Rightarrow g =10$$

2015 April(73G) Math Question 15

- Triangles, Squares, Rectangles, Circles

15. In the figure below, the border of a pool and the border of a patio surrounding that pool are similar rectangles. The given dimensions are in feet. What is the length of the pool, in feet?

Choice C

$$\frac{x}{60}=\frac{28}{40} \\$$ $$\Rightarrow x=42$$

2015 April(73G) Math Question 16

16. Marcie will paint the shaded region (1 pentagonal wall) of the outside of the playhouse shown below. All given dimensions are in feet. Each 1-quart container of paint will cover 50 square feet, and only 1 coat of paint will be applied. How many 1-quart containers of paint will Marcie need in order to cover the 1 pentagonal wall of the playhouse?

Choice G

$$Area \ of \ shaded \ region=12\times 6+ \frac{12\times3}{2}=80 \ square \ feet$$ Since one 1-quart container of paint will cover 50 square feet, 2 will be needed.

2015 April(73G) Math Question 17

- Scientific Notation

17. A certain computer performs 1.5 x 10^8 calculations per second. How many seconds would it take this computer to perform 6.0 x 10^16 calculations?

Choice D

$$\frac{6.0\times10^{16}}{1.5\times 10^{8}}=4.0\times10^8 \ seconds$$

2015 April(73G) Math Question 18

- Parallel Lines

Choice K

Since AB and DE are parallel, $$\angle A=\angle E$$ Also, since: $$\angle BCA=\angle DCE$$ It follows that $$\angle BAC=\angle CED$$

2015 April(73G) Math Question 19

Equation of a Line

- Distance & Midpoint

19. In the standard (x,y) coordinate plane, the point (2,-4) is the midpoint of the line segment with endpoints (8,-8) and:

Choice A

$$\frac{8+x}{2}=2 \\$$ $$ \Rightarrow x=-4\\ $$ $$\frac{-8+y}{2}=-4 \\ $$ $$\Rightarrow y=0$$

2015 April(73G) Math Question 20

20. The product of 2 positive integers is 78. The greater integer is 1 more than twice the lesser integer. What is the greater integer?

Choice J

Let the lesser integer be n: $$(2n+1)n=78 \\$$ $$ \Rightarrow n=6\\ $$ $$\Rightarrow 2n+1=13$$

2015 April(73G) Math Question 21

Probability

- One Event

Choice A

The set of numbers are: …

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2015 April(73G) Math Question 22

22. Noah bowled 3 games for his bowling team. In the second game, he scored 10 points more than in the first game; in the third game, he scored 10 points more than in the second game. His average score was 230 points for the 3 games. How many points did Noah score in the third game?

Choice J

Let the number of points …

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2015 April(73G) Math Question 23

Quadratic Equations

- Factoring

23. For what 2 values of x is the equation x2 — 5x + 6 = 0 true?

Choice E

$$x^2-5x+6=0 \\$$ $$ \Rightarrow (x-2)(x-3)=0 …

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2015 April(73G) Math Question 24

24. Observation of a certain bacteria colony has shown that its population of cells doubles every 3 hours. Given that the initial population of cells in this colony is about 8 million, which of the following values, in millions, would be closest to the number of cells in the bacteria colony after 15 hours?

Choice K

The colony is doubled 15/3=5 …

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2015 April(73G) Math Question 25

Probability

- Combinations

25. Audrey will take biology, algebra, and Spanish next year. Audrey will have 1 of the 3 teachers who teach biology, 1 of the 4 teachers who teach algebra, and I of the 2 teachers who teach Spanish. From among these 9 teachers, how many possibilities are there for Audrey's 3 teachers for the 3 classes?

Choice C

$$3\times4\times2=24$$

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2015 April(73G) Math Question 26

Matrices

- Addition & Subtraction

26. What value of x satisfies the matrix equation below?

Choice G

$$2x+1=9 \\$$ $$ \Rightarrow x=4$$

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2015 April(73G) Math Question 27

Statistics

- Mean, Median

27. Sophia's goal this past summer was to save an average of $5.00 per week for 10 weeks. She saved an average of $4.00 per week for the first 9 weeks and saved $12.00 for the 10th week. On average, how much more should Sophia have saved each week to reach her goal?

Choice B

$$5- \frac{4\times9 + 12}{10}=0.2$$

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2015 April(73G) Math Question 28

28. The rectangular deck on Sachi's house has a width of 4 yards and a length of 6 yards. Sachi remodels the deck by increasing both the length and width by the same amount. The area of her new deck is twice the area of her original deck. What is the length, in yards, of Sachi's new deck?

Choice H

$$(4+d)(6+d)=2\times4\times6 \\ $$ $$\Rightarrow d=2\\$$ …

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2015 April(73G) Math Question 29

- Pythagorean Theorem & sin cos tan

29. Nestor will bury one end of a cable 3 feet from the base of an antenna and attach the other end of the cable at a point on the antenna 8 feet above the ground, as shown below. When taut, the length of the exposed cable will be Ni. feet. Which of the following expressions represents the measure of the angle the taut cable will make with the level ground?

Choice C

$$\tan ?=\frac{8}{3}\\$$ $$\Rightarrow?=\tan ^{-1}(\frac{8}{3})$$

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2015 April(73G) Math Question 30

Conics

- Circles

30. The circle in the standard (x,y) coordinate plane below has center (-8.5,7.5) and has radius 5 coordinate units. Which of the following is an equation of this circle?

Choice J

The standard equation of a …

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2015 April(73G) Math Question 31

- Triangles, Squares, Rectangles, Circles

31. What is the area, in square coordinate units, of this circle?

Choice D

$$Area=\pi r^2=\pi 5^2=25 \pi$$

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2015 April(73G) Math Question 32

- Reflection, Rotation, Symmetry, Translation

32. The circle will be reflected across the y-axis. What will be the coordinates of the image of the center of the circle?

Choice K

When a point is reflected …

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2015 April(73G) Math Question 33

- Collinear Points & Triangles

33. In a plane, the distinct lines AB and CD intersect at A, where A is between C and D. The measure of BAC is 54°. What is the measure of BAD ?

Choice E

$$\angle BAD=180°-54°$$

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2015 April(73G) Math Question 34

- Graphing Trig Functions

34. For an angle with measure a in a right triangle, sin a = 180/181 and tan a = 180/19. What is the value of cos a?

Choice F

$$\tan \alpha =\frac{\sin\alpha}{\cos\alpha} \\$$ $$ …

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Number Types & Properties

2015 April(73G) Math Question 35

- Rational & Irrational Numbers and Integers

35. For which of the equations below is its solution an integer?

Choice E

I. n=19/3

II. n=4

III. …

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2015 April(73G) Math Question 36

- Exponent Expressions

Choice J

$$\frac{6x^{13}y^3}{2xy^{10}}=\frac{3x^{12}}{y^7}$$

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2015 April(73G) Math Question 37

- LCM, GCF, Prime Numbers

37. Consider all products xy such that x is divisible by 8 and y is divisible by 14. Which of the following whole numbers is NOT a factor of each product xy ?

Choice C

xy must be divisible by …

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Similar Triangles & Types of Triangles

2015 April(73G) Math Question 38

- Triangle Types

38. Shown in the standard (x,y) coordinate plane below is equilateral triangle A0C with coordinates A(a,b), C(4c,0), and O(0,0). In terms of c, what is a ?

Choice G

$$2a=4c \\$$ $$ \Rightarrow a=2c$$

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- Fixed Cost + Variable Cost

2015 April(73G) Math Question 39

Word Problems

39. On Monday, Jan and Diego opened separate bank accounts with initial deposits of $28.00 and $161.00, respectively. Every Monday after opening the accounts, Jan will add $18.25 to her account and Diego will withdraw $15.00 from his account. Which of the following equations, when solved, gives the number of weeks (w) after opening the accounts that Jan and Diego will have the same amount of money in their respective accounts? (Note: They make no other deposits or withdrawals, and no interest is applied to the money in the accounts.)

Choice D

By week w, Jan will …

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2015 April(73G) Math Question 40

40. Let p and q be real numbers such that p + q = 4, —pq = 12, and p > q. What is p?

Choice J

$$\Rightarrow q=4-p \\$$ $$ \Rightarrow …

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2015 April(73G) Math Question 41

- Write Function Expressions

41. The table below gives the weights, rounded to the nearest pound, at birth and at 1 year for 5 boys. A researcher models these weights as a linear function where the weight at 1 year is dependent on the weight at birth. Among the following models, which is best?

Choice A

As can be seen from …

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- Equations that Describe a Line

2015 April(73G) Math Question 42

Equation of a Line

42. In the standard (x,y) coordinate plane, what is the slope of the line that is perpendicular to the line 8x + 7y = 112?

Choice J

$$\Rightarrow 7y=-8x+112 \\$$ $$ \Rightarrow …

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2015 April(73G) Math Question 43

43. The circle below has a diameter of 8 centimeters. Which of the following is closest to the area, in square centimeters, of the square inscribed in the circle?

Choice B

$$d=\frac{8}{\sqrt{2}} \\$$ $$ \Rightarrow Area=d^2=(\frac{8}{\sqrt{2}})^2=32$$

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2015 April(73G) Math Question 44

- Circles

44. Points O(0,0) and B(0,3) below lie in the standard (x,y) coordinate plane. The collection of all points such that each is twice as far from B as from 0 forms a circle. The point (V3,0) is 1 point on the circle. What are the coordinates of the center of that circle?

Choice J

Since each point on the …

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2015 April(73G) Math Question 45

- Solve Algebraic Expressions

45. Given 3/x = 12 and x/y = 2, what is the value of y?

Choice A

$$\frac{3}{x}=12$$ $$ \Rightarrow x=\frac{1}{4} $$ …

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2015 April(73G) Math Question 46

Word Problems

- Temperature

Choice G

$$C=\frac{9}{5}C+32 $$ $$\Rightarrow C=-40$$

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2015 April(73G) Math Question 47

- Domain, Range, Asymptote, Undefined

47. The domain of f(x) = 2/(x^3-9x) is the set of all real numbers EXCEPT:

Choice E

The equation is not defined …

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2015 April(73G) Math Question 48

48. The figure shown below is composed of a rectangle and a semicircle. Points A and B are endpoints of both a side of the rectangle and a diameter of the semicircle. What is the perimeter, in feet, of the figure?

Choice G

$$\widehat{AB}=\frac{2\pi r}{2}=\pi r=3\pi \\$$ $$ …

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2015 April(73G) Math Question 49

- Other Shapes

49. What is the area, in square inches, of the parallelogram shown below?

Choice B

Area = 10 x 5 …

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2015 April(73G) Math Question 50

- Geometric Sequences

50. What is the sixth term of the geometric sequence whose second term is —4 and whose fifth term is 32 ?

Choice G

$$a_{5}=a_{2}\cdot k^3 \\$$ $$\Rightarrow 32=-4k^3 …

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2015 April(73G) Math Question 51

- Graphing Trig Functions

51. What are the values of 0, between 0 and 2pi, when tan 0 = —1 ?

Choice C

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2015 April(73G) Math Question 52

- Other Shapes

52. A can of soda pop has the shape of a right circular cylinder with an inside height of 6 inches and an inside diameter of 2 inches. When you pour the soda pop from the full can into a cylindrical glass with an inside diameter of 3 inches, about how many inches high is the soda pop in the glass? (Note: The volume of a right circular cylinder is itr2h.)

Choice F

$$\pi (\frac{2}{2})^2\cdot 6=\pi (\frac{3}{2})^2\cdot h …

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2015 April(73G) Math Question 53

53. Suppose that equally spaced dots are marked on each side of a regular polygon, with a dot at each vertex, and that the distance between consecutive dots is the same for all sides. The figure below shows 4 equally spaced dots per side, including a dot at each vertex, for an equilateral triangle. Which of the following expressions represents the number of dots for a regular polygon with n equally spaced dots, including one at each vertex, marked on each of its s sides?

Choice C

$$(n - 2)\cdot s +s=ns-s$$

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2015 April(73G) Math Question 54

54. A storage facility is currently offering a special rate to customers who sign contracts for 6 months or more. According to this special rate, the first month's rent is $1, and for each month after the first month, customers pay the regular monthly rental rate. The table below shows the storage unit sizes available, the floor dimensions, and the regular monthly rental rate. All the units have the same height. Dania will sign a contract to rent a Size 3 unit for 12 months at the current special rate. The amount Dania will pay for 12 months at the current special rate represents what percent decrease from the regular rental rate for 12 months?

Choice F

$$\frac{1+100\times11}{100\times12}-1=-8.25\%$$

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2015 April(73G) Math Question 55

- Percents & Fractions

55. Size 5 units can be subdivided to form other sizes of units. What is the greatest number of Size 1 units that can be formed from a single Size 5 unit?

Choice E

$$\frac{8\times16}{2\times4}=16$$

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2015 April(73G) Math Question 56

- Triangles, Squares, Rectangles, Circles

56. Janelle, the owner of the storage facility, is considering building new units that have floor dimensions larger than Size 5 units. She will use the floor area to determine the heating requirements of these larger units. For this calculation, Janelle will use the same relationship between the unit size number and the respective floor area for Sizes 1 through 5. Which of the following expressions gives the floor area, in square meters, of a Size x storage unit?

Choice H

Apparently, the relationship between size …

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2015 April(73G) Math Question 57

- Exponent Expressions

57. For how many integers x is the equation 3^(x + 1) = 9^(x-2) true?

Choice B

$$3^{x+1}=3^{3x-6} \\ $$ $$\Rightarrow x+1=3x-6 …

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2015 April(73G) Math Question 58

- Pythagorean Theorem & sin cos tan

58. In the standard (x,y) coordinate plane below, B is on the positive x-axis, the measure of AOB is 150°, and the length of AO is 1 coordinate unit. What are the coordinates of A ?

Choice F

$$\Rightarrow AC=\frac{1}{2}, \ CO=\frac{\sqrt{3}}{2} \\ …

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2015 April(73G) Math Question 59

- Polar Coordinates

Choice A

(3, 45°+360°n) represent the same …

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2015 April(73G) Math Question 60

- Domain, Range, Asymptote, Undefined

60. The equation y = (2x^2-18)/(X^2— 5x) has 2 vertical asymptotes and 1 horizontal asymptote. What is the horizontal asymptote?

Choice K

$$\Rightarrow y=\frac{2-\frac{18}{x^2}}{1-\frac{5}{x}}$$ When x approaches …

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