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ACT Form 72G Math Answer Explanations

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1. Bella sells real estate. She is paid 3% of the sale price of every house she sells. What is Bella paid for selling a house at the sale price of $250,000 ?

$$$250,000\times 3\%=$7,500$$

Choice D.

1. Bella sells real estate. She is paid 3% of the sale price of every house she sells. What is Bella paid for selling a house at the sale price of $250,000 ?

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2. For what value(s) of x is (x + 3)(x — 4) = 0 true?

Solving for the equation, we have x = -3 and 4.

Choice F.

2. For what value(s) of x is (x + 3)(x — 4) = 0 true?

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3. Which of the following inequalities orders the numbers 0.1, 0.04, and 1/6 from least to greatest?

$$\frac{1}{6}\approx0.167$$

Choice B.

3. Which of the following inequalities orders the numbers 0.1, 0.04, and 1/6 from least to greatest?

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4. Given that 2x + 1 = 5, what is the value of (x + 3)^3 ?

$$2x+1=5\Rightarrow x=2 \\ \Rightarrow (x+3)^3=(2+3)^3=5^3=125$$

Choice H.

4. Given that 2x + 1 = 5, what is the value of (x + 3)^3 ?

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5. What vector is the result of adding the vectors (1,-2), (2,3), and (-4,2) ?

<1+2-4,-2+3+2>=<-1,3>

Choice B.

5. What vector is the result of adding the vectors (1,-2), (2,3), and (-4,2) ?

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6. What value of x makes (2\3)x + 5 = 11 true?

$$\frac{2}{3}x=6\\ \Rightarrow 2x=18 \\ \Rightarrow x=9$$

Choice H.

6. What value of x makes (2\3)x + 5 = 11 true?

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7. In Summit City, the daily low temperatures, in degrees Fahrenheit (°F), during the first week in December were —7, —11, 15, 23, 45, 23, and 2. To the nearest 1°F, what was the mean daily low temperature for that week?

$$\frac{-7-11+15+23+45+23+2}{7}=13°C$$

Choice B.

7. In Summit City, the daily low temperatures, in degrees Fahrenheit (°F), during the first week in December were —7, —11, 15, 23, 45, 23, and 2. To the nearest 1°F, what was the mean daily low temperature for that week?

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8. Assume that a, b, and c are real numbers. If a > b, and (a + c) > (b + c), then what is true of c ?” width=”300″></a></p>
<p>$$(a+c)>(b+c) \\<br />
\Rightarrow a>b$$</p>
<p>Choice K.</p>
<p><span style=8. Assume that a, b, and c are real numbers. If a > b, and (a + c) > (b + c), then what is true of c ?

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9. What is the slope of the line given by the equation 2x + 7y = —12 ?

$$7y=-2x-12 \\ \Rightarrow y=-\frac{2}{7}x-\frac{12}{7}$$

Choice C.

9. What is the slope of the line given by the equation 2x + 7y = —12 ?

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10. One leg of a right triangle is 4 meters long and the other leg is 8 meters long. The hypotenuse of the triangle is how many meters long?

$$h=\sqrt{4^2+8^2}=\sqrt{80}$$

Choice K.

10. One leg of a right triangle is 4 meters long and the other leg is 8 meters long. The hypotenuse of the triangle is how many meters long?

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11. 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.

Only the equation in C fulfils all  sets of d and t in the table.

Choice C.

11. 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.

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12. The ratio of 30 to 24 is the same as the ratio of 40 to what number?

$$\frac{30}{24}=\frac{40}{x} \\ \Rightarrow x=32$$

Choice G.

12. The ratio of 30 to 24 is the same as the ratio of 40 to what number?

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13. A box contains 3 red, 2 white, and 5 blue beads, each of which is a solid color. Miguel will randomly remove 1 bead from the box, record the color of the bead, and return the bead to the box. Miguel will complete this process 50 times. How many times should Miguel expect to record a bead color that is NOT white?

$$50\times \frac{3+5}{3+2+5}=40$$

Choice D.

13. A box contains 3 red, 2 white, and 5 blue beads, each of which is a solid color. Miguel will randomly remove 1 bead from the box, record the color of the bead, and return the bead to the box. Miguel will complete this process 50 times. How many times should Miguel expect to record a bead color that is NOT white?

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14. Whenever 3(a + 6b) — c = 0, which of the following expressions is equal to a ?

$$3(a+6b)=c \\ \Rightarrow a+6b=\frac{c}{3} \\ \Rightarrow a=\frac{c}{3}-6b$$

Choice F.

14. Whenever 3(a + 6b) — c = 0, which of the following expressions is equal to a ?

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15. Given that /2x —9 = 1, x = ?

$$\sqrt{2x}=10 \\ \Rightarrow 2x=100 \\ \Rightarrow x=50$$

Choice E.

15. Given that /2x —9 = 1, x = ?

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16. A high school ice hockey conference that consists of 10 teams requires that every conference team must play each of the other conference teams at least once in a season. For n teams in the conference, the number of conference games played in a season must then be at least n 2— n . What is the minimum number of conference games that must be played in a season for this conference?

$$\frac{10^2-10}{2}=45$$

Choice H.

16. A high school ice hockey conference that consists of 10 teams requires that every conference team must play each of the other conference teams at least once in a season. For n teams in the conference, the number of conference games played in a season must then be at least n 2— n . What is the minimum number of conference games that must be played in a season for this conference?

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17. For an angle with measure a in a right triangle, sin a = 113	15 and tan a = 112/15. What is the value of cos a ?

$$\tan\alpha=\frac{\sin\alpha}{\cos\alpha} \\ \Rightarrow \cos\alpha=\frac{\sin\alpha}{\tan\alpha}=\frac{\frac{112}{113}}{\frac{112}{15}}=\frac{15}{113}$$

Choice A.

17. For an angle with measure a in a right triangle, sin a = 113 15 and tan a = 112/15. What is the value of cos a ?

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18. In isosceles triangle ABC, AB is congruent to BC and the measure of the vertex angle, B, is 75. What is the measure of A?

$$\angle A = \frac{180°-75°}{2}=52.5°$$

Choice H.

18. In isosceles triangle ABC, AB is congruent to BC and the measure of the vertex angle, B, is 75. What is the measure of A?

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19. Cowling's rule is a method of calculating a medication dosage for a child by using the child's age and the adult dosage. The rule uses the formula d = a( t + 1 / 24 ), where d is the child's dosage in milligrams, a is the adult dosage in milligrams, and t is the child's age in years. Dato is taking a 150-milligram dosage of medication that has an adult dosage of 300 milligrams. According to this rule, which of the following values is closest to Dato's age in years?

$$150=300(\frac{t+1}{24}) \\ \Rightarrow t\approx 11$$

Choice B.

19. Cowling’s rule is a method of calculating a medication dosage for a child by using the child’s age and the adult dosage. The rule uses the formula d = a( t + 1 / 24 ), where d is the child’s dosage in milligrams, a is the adult dosage in milligrams, and t is the child’s age in years. Dato is taking a 150-milligram dosage of medication that has an adult dosage of 300 milligrams. According to this rule, which of the following values is closest to Dato’s age in years?

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20. Each side of a square is 3 cm long. One vertex of the square is at (4,2) on a square coordinate grid marked in centimeter units. Which of the following points on the grid could be another vertex of the square?

20. Each side of a square is 3 cm long. One vertex of the square is at (4,2) on a square coordinate grid marked in centimeter units. Which of the following points on the grid could be another vertex of the square?

As can be seen from the graph, only (7,2) is 3 cm from (4,2).

Choice F.

20. Each side of a square is 3 cm long. One vertex of the square is at (4,2) on a square coordinate grid marked in centimeter units. Which of the following points on the grid could be another vertex of the square?

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21. The Ferris wheel at a state fair has a radius of 60 feet, rotates at a constant speed, and completes 1 rotation in 4 minutes. How many degrees does the Ferris wheel rotate in 30 seconds?

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21. The Ferris wheel at a state fair has a radius of 60 feet, rotates at a constant speed, and completes 1 rotation in 4 minutes. How many degrees does the Ferris wheel rotate in 30 seconds?

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22. At Gussie's Pizzeria, Odetta spends $8.75 (before tax) on the purchase of 2 appetizers and 3 slices of pizza. The price of each slice of pizza is p dollars. The price of each appetizer is twice the price of a slice of pizza. Which of the following systems of equations, when solved, gives the price, a dollars, of an appetizer and the price, p dollars, of a slice of pizza at Gussie's?

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22. At Gussie’s Pizzeria, Odetta spends $8.75 (before tax) on the purchase of 2 appetizers and 3 slices of pizza. The price of each slice of pizza is p dollars. The price of each appetizer is twice the price of a slice of pizza. Which of the following systems of equations, when solved, gives the price, a dollars, of an appetizer and the price, p dollars, of a slice of pizza at Gussie’s?

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23. The perimeter of a parallelogram is 76 inches, and the length of 1 side is 10 inches. If it can be determined, what are the lengths, in inches, of the other 3 sides?

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23. The perimeter of a parallelogram is 76 inches, and the length of 1 side is 10 inches. If it can be determined, what are the lengths, in inches, of the other 3 sides?

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24. John told Raquel that if he spent up to $30 from his savings account, his savings account would have at least 3/4 as much in it as it has now. From John's statement, Raquel can deduce that the least amount of money that John could have in his savings account now is:

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24. John told Raquel that if he spent up to $30 from his savings account, his savings account would have at least 3/4 as much in it as it has now. From John’s statement, Raquel can deduce that the least amount of money that John could have in his savings account now is:

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25. At East High School, the 20 cast members of the spring musical sold 2 types of tickets—student and adult—for the Friday night and Saturday night performances. The price of each student ticket was $2 and the price of each adult ticket was $3. The table below gives the number of tickets sold, by type and by night. The stem-and-leaf plot below shows the number of tickets, regardless of type, sold by each of the 20 cast members. The auditorium where the spring musical will be performed has 10 seats in the 1st (front) row. Each row behind the 1st row has 4 more seats than does the row in front of it. Suppose 1 cast member will be picked at random from the 20 cast members who sold tickets to receive a prize. What is the probability of picking a cast member who sold more than 30 tickets?

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25. At East High School, the 20 cast members of the spring musical sold 2 types of tickets—student and adult—for the Friday night and Saturday night performances. The price of each student ticket was $2 and the price of each adult ticket was $3. The table below gives the number of tickets sold, by type and by night. The stem-and-leaf plot below shows the number of tickets, regardless of type, sold by each of the 20 cast members. The auditorium where the spring musical will be performed has 10 seats in the 1st (front) row. Each row behind the 1st row has 4 more seats than does the row in front of it. Suppose 1 cast member will be picked at random from the 20 cast members who sold tickets to receive a prize. What is the probability of picking a cast member who sold more than 30 tickets?

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26. For which night was the total amount collected for the tickets greater, and by how many dollars was it greater?

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26. For which night was the total amount collected for the tickets greater, and by how many dollars was it greater?

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27. How many seats are in the 8th row of the auditorium?

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27. How many seats are in the 8th row of the auditorium?

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28. In rectangle ABDE shown below, C lies on BD and F lies on AE such that rectangles ABCF and CDEF are similar (ABCF — CDEF), the length of AB is 4 cm, and the length of BC is 8 cm. What is the area, in square centimeters, of ABDE ?

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28. In rectangle ABDE shown below, C lies on BD and F lies on AE such that rectangles ABCF and CDEF are similar (ABCF — CDEF), the length of AB is 4 cm, and the length of BC is 8 cm. What is the area, in square centimeters, of ABDE ?

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29. Points A, B, C, and D lie on the real number line as shown below. The coordinate of B is 0, AC is 15 units long, BD is 20 units long, and AD is 32 units long. What is the coordinate of C ?

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29. Points A, B, C, and D lie on the real number line as shown below. The coordinate of B is 0, AC is 15 units long, BD is 20 units long, and AD is 32 units long. What is the coordinate of C ?

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30. In a certain country, last year's population was estimated to be 3.6 x 10^6 people, and last year's public debt was estimated to be 1.8 x 10^9 dollars. Based on these estimates, what was last year's public debt per person in this country?

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30. In a certain country, last year’s population was estimated to be 3.6 x 10^6 people, and last year’s public debt was estimated to be 1.8 x 10^9 dollars. Based on these estimates, what was last year’s public debt per person in this country?

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31. A landscaper will install a triangular garden near the intersection of 2 streets. In the plan of the garden shown below, the angle nearest the intersection is a right angle. The landscaper has purchased the following materials for the garden: 1 bag of fertilizer at a price of $15.50, 50 plants at a price of $0.70 each, and 3 bags of mulch at a price of $4.00 each. An 8% sales tax was added to the total price of these materials. The landscaper will also install a water sprinkler that will be located an equal distance from the 3 vertices of the garden. The landscaper calculated the area of the garden before purchasing the fertilizer, plants, and mulch. What is the area, in square feet, of the garden?

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31. A landscaper will install a triangular garden near the intersection of 2 streets. In the plan of the garden shown below, the angle nearest the intersection is a right angle. The landscaper has purchased the following materials for the garden: 1 bag of fertilizer at a price of $15.50, 50 plants at a price of $0.70 each, and 3 bags of mulch at a price of $4.00 each. An 8% sales tax was added to the total price of these materials. The landscaper will also install a water sprinkler that will be located an equal distance from the 3 vertices of the garden. The landscaper calculated the area of the garden before purchasing the fertilizer, plants, and mulch. What is the area, in square feet, of the garden?

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32. What was the total cost, including sales tax, of the fertilizer, plants, and mulch that the landscaper purchased?

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32. What was the total cost, including sales tax, of the fertilizer, plants, and mulch that the landscaper purchased?

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33. The plan of the garden will be aligned in the standard (x,y) cdordinate plane so that the right angle is at (0,0) and the other 2 vertices are at (12,0) and (0,-9). What coordinates give the location of the water sprinkler?

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33. The plan of the garden will be aligned in the standard (x,y) cdordinate plane so that the right angle is at (0,0) and the other 2 vertices are at (12,0) and (0,-9). What coordinates give the location of the water sprinkler?

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34. The landscaper has decided to enclose the garden by installing a decorative fence of uniform height along the garden's perimeter. What is the minimum length, in feet, of fencing required to enclose the garden?

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34. The landscaper has decided to enclose the garden by installing a decorative fence of uniform height along the garden’s perimeter. What is the minimum length, in feet, of fencing required to enclose the garden?

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35. Shown below are right triangles ACD and ABE with lengths given in inches. What is the length, in inches, of AB ?

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35. Shown below are right triangles ACD and ABE with lengths given in inches. What is the length, in inches, of AB ?

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36. The circumference of a circle is 50 cm. What is the length, in centimeters, of the diameter of the circle?

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36. The circumference of a circle is 50 cm. What is the length, in centimeters, of the diameter of the circle?

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37. Which of the following values is closest to the distance, in inches, between point A and point B on the ruler shown below? (Note: The ruler shown is NOT actual size.)

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37. Which of the following values is closest to the distance, in inches, between point A and point B on the ruler shown below? (Note: The ruler shown is NOT actual size.)

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38. If r + 5 = m and r + 6 = n, what is the value of n — m ?

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38. If r + 5 = m and r + 6 = n, what is the value of n — m ?

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39. In the figure below, lines l and m are parallel. Lines a and b intersect l at the same point. What is the value of x?

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39. In the figure below, lines l and m are parallel. Lines a and b intersect l at the same point. What is the value of x?

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40. The line y = 1/2x + 3 and the line y = 3x — 12 intersect at (6,6), as shown in the standard (x,y) coordinate plane below. The shaded region is bounded by these 2 lines, the x-axis, and the y-axis. What is the area of the shaded region, in square coordinate units?

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40. The line y = 1/2x + 3 and the line y = 3x — 12 intersect at (6,6), as shown in the standard (x,y) coordinate plane below. The shaded region is bounded by these 2 lines, the x-axis, and the y-axis. What is the area of the shaded region, in square coordinate units?

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41. In the figure below, AD, BE, and CF all intersect at point Z. The measures of 2 angles are given. What is the measure of CZE ?

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41. In the figure below, AD, BE, and CF all intersect at point Z. The measures of 2 angles are given. What is the measure of CZE ?

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42. Emma and Kristin each ran 10 laps (4,000 meters) around a 400-meter track without stopping. Emma ran at a constant speed of 250 meters per minute. Kristin ran at a constant speed of 200 meters per minute. Both Emma and Kristin began running at the same instant. How many laps did Kristin have left to run when Emma had completed her run?

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42. Emma and Kristin each ran 10 laps (4,000 meters) around a 400-meter track without stopping. Emma ran at a constant speed of 250 meters per minute. Kristin ran at a constant speed of 200 meters per minute. Both Emma and Kristin began running at the same instant. How many laps did Kristin have left to run when Emma had completed her run?

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43. Three linear equations are given below. y= x y= x + 1 y = —x + 1 In terms of being parallel, being perpendicular, or intersecting, how are the graphs of these equations related in the standard (x,y) coordinate plane?

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43. Three linear equations are given below. y= x y= x + 1 y = —x + 1 In terms of being parallel, being perpendicular, or intersecting, how are the graphs of these equations related in the standard (x,y) coordinate plane?

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44. For what value of a would the following system of equations have an infinite number of solutions?

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44. For what value of a would the following system of equations have an infinite number of solutions?

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45. Given that x = 2 when y = 12 for the proportion x/6 = k/y, what is x when y = 8 ?

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45. Given that x = 2 when y = 12 for the proportion x/6 = k/y, what is x when y = 8 ?

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46. Two events are independent if the outcome of one event does not affect the outcome of the other event. One of the following statements does NOT describe independent events. Which one?

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46. Two events are independent if the outcome of one event does not affect the outcome of the other event. One of the following statements does NOT describe independent events. Which one?

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47. Which of the following binomials is a factor of 3x^2 + 11x — 4 ?

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47. Which of the following binomials is a factor of 3x^2 + 11x — 4 ?

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48. The figure below shows a lighthouse keeper looking down at a rowboat on the sea through a navigational instrument. The instrument is 60 feet above sea level and indicates an angle of depression of 53° to the rowboat. Which of the following is closest to the horizontal distance, in feet, between the navigational instrument and the rowboat? (Note: sin 53°=0.80, cos 53°=0.60, tan 53°=1.33)

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48. The figure below shows a lighthouse keeper looking down at a rowboat on the sea through a navigational instrument. The instrument is 60 feet above sea level and indicates an angle of depression of 53° to the rowboat. Which of the following is closest to the horizontal distance, in feet, between the navigational instrument and the rowboat? (Note: sin 53°=0.80, cos 53°=0.60, tan 53°=1.33)

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49. Each of 20 students in a class took a test and received a whole number score. The median of the scores was 87. None of the students received a score of 87, and 30% of the students received scores of 90 or above. How many students received scores of 88 or 89 ?

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49. Each of 20 students in a class took a test and received a whole number score. The median of the scores was 87. None of the students received a score of 87, and 30% of the students received scores of 90 or above. How many students received scores of 88 or 89 ?

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50. The radius and height of the right circular cylinder shown below are given in centimeters. Which of the following is closest to the volume, in cubic centimeters, of the cylinder?

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50. The radius and height of the right circular cylinder shown below are given in centimeters. Which of the following is closest to the volume, in cubic centimeters, of the cylinder?

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51. What value of x satisfies the equation below?

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51. What value of x satisfies the equation below?

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52. Let x be a negative odd integer. The expression xy3 is a positive even integer whenever y is any member of which of the following sets?

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52. Let x be a negative odd integer. The expression xy3 is a positive even integer whenever y is any member of which of the following sets?

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53. Point I(3,4) is shown in the standard (x,y) coordinate plane below. Point I is reflected across the line x = 7 and, after the reflection, is labeled J. What are the coordinates of J ?

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53. Point I(3,4) is shown in the standard (x,y) coordinate plane below. Point I is reflected across the line x = 7 and, after the reflection, is labeled J. What are the coordinates of J ?

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54. The angle measures of CDE are shown below. Point B of ABC lies on CD, and point C lies on AE. What is the value of cos(a + b) ?

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54. The angle measures of CDE are shown below. Point B of ABC lies on CD, and point C lies on AE. What is the value of cos(a + b) ?

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55. The determinant of any 2 x 2 matrix is ad - bc. The determinant of is equal to 0. What are all possible values of x?

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55. The determinant of any 2 x 2 matrix is ad – bc. The determinant of is equal to 0. What are all possible values of x?

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56. Which of the following inequalities is equivalent to (|x|+1)2<=4 ?

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56. Which of the following inequalities is equivalent to (|x|+1)2<=4 ?

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57. Given that 2 sin a = 2 and 2 cos(- b) = 2, which of the following could be a value, in radians, of a + b ?

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57. Given that 2 sin a = 2 and 2 cos(- b) = 2, which of the following could be a value, in radians, of a + b ?

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58. One of the following equations determines the graph in the standard (x,y) coordinate plane below. Which one?

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58. One of the following equations determines the graph in the standard (x,y) coordinate plane below. Which one?

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59. Cubes each having a side length of 0.5 cm are put together to form a rectangular solid with 8 layers. Each layer has 6 cubes. What is the volume, in cubic centimeters, of the rectangular solid?

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59. Cubes each having a side length of 0.5 cm are put together to form a rectangular solid with 8 layers. Each layer has 6 cubes. What is the volume, in cubic centimeters, of the rectangular solid?

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60. Consider the 2 functions f(x) = 2x + 5 and g(x) = 3x + b,

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60. Consider the 2 functions f(x) = 2x + 5 and g(x) = 3x + b,

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