# ACT Form 73G Math Answer Explanations

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*Explain to me:*

$$|3(-2)+4|=|-6+4|=|-2|=2$$

Choice B.

1. |3(-2) + 4| = ?

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

Choice G.

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?

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

Choice B.

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

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

Choice H.

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?

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

Choice A.

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?

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

Choice H.

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?

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

Choice D.

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

Let the number of small wreaths be x, thus the number of large wreath 2x:

$$30\times 2x+20\times x=4\times 60 \\ \Rightarrow x=3 \\ \Rightarrow 2x=6$$

Choice J.

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?

The ratio ranges from 0 to 1.

Choice E.

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 ?

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

Choice H.

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 ?

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

Choice H.

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

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

Choice J.

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.)

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

Choice A.

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?

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

Choice H.

14. A bowl contains 6 red beads, 8 black beads, and a number of green beads. There are no other beads in the bowl. The probability of randomly choosing a black bead from the bowl is 1/3. How many green beads are in the bowl?

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

Choice C.

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?

$$Area \ of \ shaded \ region=12\times 6+ \frac{12\times3}{2}=80 \ square \ feet$$

Since one 1-quart container of paint will cover 50 square feet, 2 will be needed.

Choice G.

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?

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

Choice D.

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?

Since AB and DE are parallel, $$\angle A=\angle E$$

Also, since: $$\angle BCA=\angle DCE$$

It follows that

$$\angle BAC=\angle CED$$

Choice K.

18. In the figure below, AE and BD intersect at C, and AB || DE. Which of the following angles must have the same measure as BAC?

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

Choice A.

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

Let the lesser integer be n:

$$(2n+1)n=78 \\ \Rightarrow n=6\\ \Rightarrow 2n+1=13$$

Choice J.

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?

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21. A set of numbers consists of all the odd integers that are greater than 1 and less than 21. What is the probability that a number picked at random from the set will be divisible by 3 ?

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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?

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

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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?

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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?

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

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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?

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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?

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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?

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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?

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

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

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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 ?

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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?

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

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36. Whenever x and y are nonzero,

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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 ?

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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 ?

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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.)

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

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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?

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

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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?

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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?

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

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46. Temperatures measured in degrees Fahrenheit (F) are related to temperatures measured in degrees Celsius (C) by the formula F =IC + 32. There is 1 value of x for which x degrees Fahrenheit equals x degrees Celsius. What is that value?

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

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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?

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

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

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

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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.)

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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?

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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?

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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?

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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?

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

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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 ?

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59. Which of the following polar coordinates represents the same location as (3, 45°) ?

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

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