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# ACT Form 74F Math Answer Explanations

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Explain to me:

$$\frac{x}{5}=\frac{6}{3} \\ \Rightarrow x= 10$$

Choice C.

1. Marcus’s favorite casserole recipe requires 3 eggs and makes 6 servings. Marcus will modify the recipe by using 5 eggs and increasing all other ingredients in the recipe proportionally. What is the total number of servings the modified recipe will make?

$$p=\frac{1}{35-3}=\frac{1}{32}$$

Choice K.

2. The 35-member History Club is meeting to choose a student government representative. The members decide that the representative, who will be chosen at random, CANNOT be any of the 3 officers of the club. What is the probability that Hiroko, who is a member of the club but NOT an officer, will be chosen?

$$2x+7=15 \\ \Rightarrow x= 4$$

Choice B.

3. For what value of x is the equation 22′ 7 = 215 true?

$$f(3)=5(3)^2-7(4\times 3+3)=-60$$

Choice J.

4. Let the function f be defined as f(x) = 5×2 — 7(4x + 3). What is the value of f(3) ?

$$p = \frac{8}{5+7+8}=\frac{8}{20}=\frac{2}{5}$$

Choice D.

5. A wallet containing 5 five-dollar bills, 7 ten-dollar bills, and 8 twenty-dollar bills is found and returned to its owner. The wallet’s owner will reward the finder with 1 bill drawn randomly from the wallet. What is the probability that the bill drawn will be a twenty-dollar bill? DO YOUR FIGURING HERE.

$$40+2n=35+3n \\ \Rightarrow n=5$$

Choice H.

6. The ABC Book Club charges a $40 monthly fee, plus$2 per book read in that month. The Easy Book Club charges a $35 monthly fee, plus$3 per book read in that month. For each club, how many books must be read in 1 month for the total charges from each club to be equal?

Choice D.

7. In parallelogram ABCD below, AC is a diagonal, the measure of %ABC is 40°, and the measure of LACD is 570. What is the measure of LCAD ?

$$\frac{8x-3}{x}=\frac{8\times \frac{1}{2}-3}{\frac{1}{2}}=2$$

Choice G.

8. When x = 2, what is the value of 8x-3

$$x=\frac{3+1}{2}=2 \\ y=\frac{8+(-4)}{2}=2$$

Choice D.

9. In the standard (x,y) coordinate plane, what is the midpoint of the line segment that has endpoints (3,8) and (1,-4) ?

Choice G.

10. The fluctuation of water depth at a pier is shown in the figure below. One of the following values gives the positive difference, in feet, between the greatest water depth and the least water depth shown in this graph. Which value is it?

$$slope=\frac{-5-1}{2-(-2)}=-\frac{3}{2}$$

Choice D.

11. What is the slope of the line through (-2,1) and (2,-5) in the standard (x,y) coordinate plane?

$$(x-30)\times 17 = 221 \\ \Rightarrow x = 43$$

Choice H.

12. In Cherokee County, the fine for speeding is $17 for each mile per hour the driver is traveling over the posted speed limit. In Cherokee County, Kirk was fined$221 for speeding on a road with.a posted speed limit of 30 mph. Kirk was fined for traveling at what speed, in miles per hour?

$$x=\frac{12}{8}=\frac{3}{2} \\ y=\frac{22-10}{2}=6 \\ \Rightarrow x +y=7\frac{1}{2}$$

Choice B.

13. What is the sum of the solutions of the 2 equations DO YOUR FIGURING HERE.

H.

Median = Average = 420 ÷ 5 = 84

→ sum of the 4 scores that are NOT the median = sum – median = 420 – 84 = 336

14. The average of 5 distinct scores has the same value as

$$||-8+4|-|3-9||=|4-6|=2$$

Choice D.

15. What is the value of the expression below?

$$x^{\frac{2}{3}}=\sqrt[3]{x^2}$$

Choice K.

16. Which of the following expressions is equivalent to x3 ?

B.

By rearranging the equation into slope intercept form, we have:

y =(4/7)x-(5/7)

Therefore the slope is 4/7.

17. In the standard (x,y) coordinate plane, what is the slope of the line given by the equation 4x = 7y + 5 ?

B.

By Pythagorean theorem, AB² = 32²+24² so AB = 40. Since the midpoint of a line segment divides it into two segments of equal length, the midpoint of AB is 20 inches from point A.

18. For which of the following conditions will the sum of integers in and n always be an odd integer?

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19. The lengths of the 2 legs of right triangle AABC shown below are given in inches. The midpoint of AB is how many inches from A ?

K.

It is not given that the triangle is a right triangle, so you can’t use the Pythagorean theorem. The length of DF ranges from, not inclusively, (√30-3) to (√30+3).

20. In DEF, the length of DE is VT:31 inches, and the

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21. Laura plans to paint the 8-foot-high rectangular walls of her room, and before she buys paint she needs to know the area of the wall surface to be painted. Two walls are 10 feet wide, and the other 2 walls are 15 feet wide. The combined area of the 1 window and the 1 door in her room is 60 square feet. What is the area, in square feet, of the wall surface Laura plans to paint?

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22. The length of a rectangle is 5 inches longer than the width. The perimeter of the rectangle is 40 inches. What is the width of the rectangle, in inches?

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23. 8% of 60 is + of what number?

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24. Armin is trying to decide whether to buy a season pass to his college basketball team’s 20 home games this season. The cost of an individual ticket is $14, and the cost of a season pass is 8175. The season pass will admit Armin to any home basketball game at no additional cost. What is the minimum number of home basketball games Armin must attend this season in order for the cost of a season pass to be less than the total cost of buying an individual ticket for each game he attends? Back to top↑ Want to see the full ACT 72E Math Answer Explanations? Be a “Sponsor Member” to see this and all other answer explanations we have completed thus far in their entirety. 25. 4.8x KC 1.6x 10-ll Back to top↑ H. Want to see the full ACT 72E Math Answer Explanations? Be a “Sponsor Member” to see this and all other answer explanations we have completed thus far in their entirety. 26. A circle in the standard (x,y) coordinate plane has center C(-1,2) and passes through A(2,6). Line segment AB is a diameter of this circle. What are the coordinates of point B ? Back to top↑ Want to see the full ACT 72E Math Answer Explanations? Be a “Sponsor Member” to see this and all other answer explanations we have completed thus far in their entirety. 27. Which of the following expressions is a factor of x3 – 64 ? Back to top↑ Want to see the full ACT 72E Math Answer Explanations? Be a “Sponsor Member” to see this and all other answer explanations we have completed thus far in their entirety. 28. The average of a list of 4 numbers is 90.0. A new list of 4 numbers has the same first 3 numbers as the original list. but the fourth number in the original list is 80, and the fourth number in the new list is 96. What is the average of this new list of numbers? Back to top↑ Want to see the full ACT 72E Math Answer Explanations? Be a “Sponsor Member” to see this and all other answer explanations we have completed thus far in their entirety. 29. The number a is located at —2.5 on the number line below. Back to top↑ J. There is 7/9 of the original pizza remaining. Thus, each brother will get (1/3)(7/9) = 7/27 of the original pizza. 30. Maria ordered a pizza. She ate only 2/9 of it and gave the remaining pizza to her 3 brothers. What fraction of the whole pizza will each of Maria’s brothers receive, if they share the remaining pizza equally? Back to top↑ Want to see the full ACT 72E Math Answer Explanations? Be a “Sponsor Member” to see this and all other answer explanations we have completed thus far in their entirety. 31. The number 1,001 is the product of the prime numbers 7, 11, and 13. Knowing this, what is the prime factorization of 30,030 ? Back to top↑ G. Area of the scale drawing = (28 + 40) · 16 / 2 = 544 square inches 32. What is the area, in square inches, of the scale drawing of the park? Back to top↑ $$Length \ of green \ side=\sqrt{16^2+12^2}=20 \\ \Rightarrow Perimeter = 20+28+16+40=104 \ inches = 104\times 1.5 \ feet = 156 \ feet$$ Choice E. 33. Mikea’s proposal includes installing a fence on the perimeter of the park. What is the perimeter, in feet, of the park? Back to top↑ Want to see the full ACT 72E Math Answer Explanations? Be a “Sponsor Member” to see this and all other answer explanations we have completed thus far in their entirety. 34. The length of the south side of the park is what percent of the length of the north side? Back to top↑ C. The area of the walkway can be arrived at by deducting the area of bigger rectangular (30’ x 36’) from that of the smaller rectangular (30’ x 24’): Area of walkway = 30’ x 36’ – 30’ x 24’ = 360 square feet 35. Mr. Smith plans to build a 3-foot-wide walkway around the outside of the cabin, as shown in the floor plan. What will be the area, in square feet, of the top surface of the walkway? Back to top↑ J. There are 3 rooms so 3 fans will cost 3($52) = $156. There are 3 small and 1 large windows so the total cost of the curtains will be 3($39.50)+1($79) =$197.50. Therefore, the total cost is $156+$197.50 = $353.50. 36. Mrs. Smith will install a ceiling fan in each room of the cabin and will place curtains over the 4 windows. Each of the ceiling fang has a price of$52.00. The price of curtains for each small window (S) is \$39.50, and the price of curtains for the large window (L) is twice that for the small window. Based on this information, which of the following values is closest to the total price Mrs. Smith will pay for curtains and ceiling fans?

A.

Because the chance of raining on a particular day is independent of the chance of rain on another day, the probability that it rains two consecutive days is (.2)2 = .04

37. Mr. and Mrs. Smith plan to roof the cabin on 2 consecutive days. Assuming that the chance of rain is independent of the <lay, what is the probability that it will rain both days ?

K.

An irrational number cannot be written as a ratio of two integers. (F) evaluates to √(1/4) = 1/2. (G) evaluates to √(4) = 2 = 2/1. (H) evaluates to 2 = 2/1. (J) evaluates to √(16) = 4 = 4/1. (K) cannot be simplified further and cannot be written as a ratio of two integers.

38. Which of the following expressions, when evaluated, equals an irrational number?

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39. A line through the origin and (10,4) is shown in the standard (x,y) coordinate plane below. The acute angle between the line and the positive x-axis has measure 0. What is the value of tan 0 ?

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40. The equation 12x — 81 + 3 = 5 has 2 solutions. Those solutions are equal to the solutions to which of the following pairs of equations?

A.

There are 12 students who have scores between 65-70 but 13 who have scores in that range plus the range of 71-80 (totalling a range of 65-80). Thus, there are 13-12 = 1 student with scores between 71-80.

41. The frequency chart below shows the cumulative number of Ms. Hernandez’s science students whose test scores fell within certain score ranges. All test scores are whole numbers.

G.
For this sound, I = 1000K, so d = 10log(1000K/K) = 10(3) = 30

42. The number of decibels, d, produced by an audio source can be modeled by the equation d = 10 log(7), where I is the sound intensity of the audio source and K is a constant. How many decibels are produced by an audio source whose sound intensity is 1,000 times the value of K ?

$$score = 1\times 80 \times 75 \% + 2 \times60 \times90 \%+3\times60 \times 25\%=213$$

Choice C.

43. Mario plays basketball on a town league team. The table below gives Nlario’s scoring statistics for last season. flow many points did Mario score playing basketball last season!

The graph is moved to the right for n units whenever x is substitute for x – n.

Choice F.

44. The graph of y Ix — 61 is in the standard (x,y)

A.

The volume of the toy soldier can be found by deducting the volume of the rectangular container after the submerging of the toy soldier from that before:

Volume of toy soldier = 8 x 6 x 6.6 – 8 x 6 x 4 = 125

45. Toby wants to find the volume of a solid toy soldier. He fills a rectangular container 8 cm long, 6 cm wide, and 10 cm high with water to a depth of 4 cm. Toby totally submerges the toy soldier in the water. The height of the water with the submerged toy soldier is 6.6 cm. Which of the following is closest to the volume, in cubic centimeters, of the toy soldier?

J.

Volume of the box = 18³
Volume of the cube = π(6)(12)²
–> Volume of packing material = 18³ – π(6)(12)²

46. A box in the shape of a cube has an interior side length of 18 inches and is used to ship a right circular cylinder with a radius of 6 inches and a height of 12 inches. The interior of the box not occupied by the cylinder is filled with packing material. Which of the following numerical expressions gives the number of cubic inches of the box filled with packing material?

B.

1 yard = 3 feet
1 feet = 12 inches

Area of the floor = 15 feet x 21 feet = 5 yard x 7 yard = 35 square yards

47. A room has a rectangular floor that is 15 feet by 21 feet. What is the area of the floor in square yards?

Choice G.

48. ABC Cabs and Tary Taxicabs both have an initial fare of a whole number of dollars for 1 passenger. The fare increases a whole number of dollars at each whole number of miles traveled. The graphs below show the 1-passenger fares, in dollars, for both cab companies for trips up to 6 miles. When the fares of the 2 cab companies are compared, what is the cheaper fare for a 5-mile trip?

B.

The area is simply the area of the trapezoid spanned from x = 0 to 2 plus the area of the rectangle from x = 2 to 3 plus the area of the triangle from x = 3 to 5. Thus, the area = .5(4+3)(2)+(1)(3)+.5(3)(2) = 13

49. The graph of a function y = f(x) consists of 3 line segments. The graph and the coordinates of the endpoints of the 3 line segments are shown in the standard (x,y) coordinate plane below. What is the area, in square coordinate units, of the region bounded by the graph of y .f(x), the positive y-axis, and the positive x-axis?

J.

x + y = 151
x=19+√y
–>  x²-37x+210=0
–>  (x-7)(x-30)=0
–> x = 7 or 30

If x = 7, √y<0, therefore x cannot be 7
–> x = 30, y = 121
–>y-x = 91

50. The sum of 2 positive numbers is 151. The lesser number is 19 more than the square root of the greater nunthcr. What is the value of the greater number minus the lesser number?

C.

The list of number is ordered from the highest to the lowest.

The median of the list is 25, therefore (30+X) / 2 = 25 –> X = 20
The mode of the list is 15, therefore Y = 15

The mean of the list = (41+35+30+20+15+15) / 6 = 26

51. The list of numbers 41, 35, 30, X, Y, 15 has a median of 25. The mode of the list of numbers is 15. To the nearest whole number, what is the mean of the list?

F.

Substituting y for x², we have

s·x² + r·x – t = 0

For the equation to have two different solutions,

Δ = r² + 4st › 0

52. You are given the following system of equations: y = x2 rx + sy t where r, s, and t are integers. For which of the following will there be more than one (x,y) solution, with real-number coordinates, for the system?

$$a_{3}=13 \\ a_{4}=18 \\ \Rightarrow d=a_{4}-a_{3}=5 \\ \Rightarrow a_{50}= a_{4} +(50-4)\times d =248$$

Choice A.

53. The 3rd and 4th terms of an arithmetic sequence are 13 and 18, respectively. What is the 50th term of the sequence?

H.

The Pythagorean trigonometric identity is expressed as:

sin² x + cos² x = 1

Therefore, y = sin² x + cos² x = 1

54. One of the following graphs in the standard (x.y) coordinate plane k the graph of v = sin2x + cos2x over the domain —It– < x < . Which one?

E.

The regular period of csc(x) is 2π. Thus, csc(4x) represents a compression of csc(x) towards the y axis by a factor of 4. Thus, the period will be 2π/4 = π/2

55. What is the period of the function f(x) = esc(4x) ?

H.

For the toss of a penny, the probability of it landing with its head faceup is 50%. If it lands with its head up, the awarded value is 3 points. Otherwise the point is 0. Therefore the expected value of the point awarded for the toss of a penny is 3 x 50% + 0 x 50% = 3/2.

The same can also be said of the toss of a nickel and a dime. Therefore the expected value of the total points awarded is 3/2 +3/2 +3/2 = 9/2.

56. At the school carnival, Mike will play a game in which he will toss a penny, a nickel, and dime at the same time. He will be awarded 3 points for each coin that lands with heads faceup. Let the random variable x represent the total number of points awarded on any toss of the coins. What is the expected value of x

B.

We wish to find k for which k2-12 = k. Rearranging yields k2-k-12 = (k-4)(k+3) = 0. Thus, k = 4

57. For what positive real value of k, if any, is the determinant of the matrix [ k 3 k 4

F.

If i² = -1, then i^4 = (-1)² = 1. And notice that (i^4)k = i^4k = 1k = 1, k ∈ ℤ. In other words, i^4 = i^8 = i^12 = … = 1. Thus, we may conclude that 4|n (4 divides n; you do not need to know this notation).

58. Given a positive integer n such that i^n = 1, which of the following statements about n must be true? (Note: i² = – 1)

A.

Recall that |sinθ| ≤ 1 for all θ ∈ R. Thus, we need only to find θ where sinθ = ±1. Going back to your unit circle, those values are θ = ±π/2

59. For 5 0 5 1, ‘sin 01 1 is true for all and only the values of B in which of the following sets?

K.

∠LPK · 2 =∠LPM
->(4x + 18°) · 2 = 11x
-> x = 12°
-> ∠KPM = 4· 12° + 18°=66°

60. Ray PK bisects LLPM, the measure of ZLPM is I lx°, and the measure of LLPK is (4x + 18)°. What is the measure of ZKPM ?