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ACT Form A11 Math Answer Explanations

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Explain to me: 


1. Xuan sold 9 used books for $9.80 each. With the money from these sales, she bought 4 new books and had $37.80 left over. What was the average amount Xuan paid for each new book?

$$4x+37.8=9\times9.8 \\
x=12.6$$

Choice D.

1. Xuan sold 9 used books for $9.80 each. With the money from these sales, she bought 4 new books and had $37.80 left over. What was the average amount Xuan paid for each new book?

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2. A point at (-5,7) in the standard (x,v) coordinate plane is translated right 7 coordinate units and down 5 coordinate units. What are the coordinates of the point after the translation?

Choice H.

2. A point at (-5,7) in the standard (x,v) coordinate plane is translated right 7 coordinate units and down 5 coordinate units. What are the coordinates of the point after the translation?

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3. Shantiel left her home at 9:00 a.m. on Tuesday and traveled 648 miles. When she arrived at her destination it was 3:00 a.m. the next day. Given that her home and her destination are in the same time zone, which of the following is closest to her average speed, in miles per hour, for this trip?

Shantiel travelled for 18 hours from 9am to 3am the next day.

$$s=\frac{648}{18}=36$$

Choice C.

3. Shantiel left her home at 9:00 a.m. on Tuesday and traveled 648 miles. When she arrived at her destination it was 3:00 a.m. the next day. Given that her home and her destination are in the same time zone, which of the following is closest to her average speed, in miles per hour, for this trip?

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4. The text message component of each of Juan's monthly phone bills consists of $10.00 for the first 300 text messages sent that month, plus $0.10 for each additional text message sent that month. On Juan's most recent phone bill he was charged a total of $16.50 for text messages. For how many text messages in total was Juan charged on this bill?

$$n=300+\frac{16.5-10}{0.1}=365$$

Choice J.

4. The text message component of each of Juan’s monthly phone bills consists of $10.00 for the first 300 text messages sent that month, plus $0.10 for each additional text message sent that month. On Juan’s most recent phone bill he was charged a total of $16.50 for text messages. For how many text messages in total was Juan charged on this bill?

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5. Which of the following matrices is equal to

$$\begin{bmatrix}
9&8\\
-4&7\\
\end{bmatrix}+
\begin{bmatrix}
-6&6\\
5&4\\
\end{bmatrix}=
\begin{bmatrix}
9-6&8+6\\
-4+5&7+4\\
\end{bmatrix}=
\begin{bmatrix}
3&14\\
1&11\\
\end{bmatrix}$$

Choice A.

5. Which of the following matrices is equal to

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6. A function, f, is defined by f(x,y) = 3x2 — 4y. What is the value of f(4,3) ?

$$f(4,3)=3\cdot 4^2-4\cdot 3=36$$

Choice H.

6. A function, f, is defined by f(x,y) = 3×2 — 4y. What is the value of f(4,3) ?

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7. A certain group consists of 5 children, 3 of whom are age 10 and 2 of whom are age 5. What is the mean age of the children in the group?

$$mean \ age =\frac{10\times3+5\times2}{5}=8$$

Choice D.

7. A certain group consists of 5 children, 3 of whom are age 10 and 2 of whom are age 5. What is the mean age of the children in the group?

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8. In the figure shown below, AC II DE; BD = AD; D and E are on AB and BC, respectively; AC = 8 feet; and the height of AABC is 10 feet. What is DE, in feet?

$$AC \parallel DE , \ BD=AD\\
\Rightarrow DE=\frac{AC}{2}=\frac{8}{2}=4$$

Choice H.

8. In the figure shown below, AC II DE; BD = AD; D and E are on AB and BC, respectively; AC = 8 feet; and the height of AABC is 10 feet. What is DE, in feet?

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9. In a poll of 500 registered voters, 337 voters favored a proposal to increase funding for local schools. Suppose the poll is indicative of how the 22,000 registered voters will vote on the proposal. Which of the following values is closest to how many of the 22,000 registered voters will be expected to vote in favor of the proposal?

$$n=22000\times \frac{337}{500}=14828\approx 14830$$

Choice B.

9. In a poll of 500 registered voters, 337 voters favored a proposal to increase funding for local schools. Suppose the poll is indicative of how the 22,000 registered voters will vote on the proposal. Which of the following values is closest to how many of the 22,000 registered voters will be expected to vote in favor of the proposal?

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10. Diego purchased a car that had a purchase price of $13,400, which included all other costs and tax. He paid $400 as a down payment and got a loan for the rest of the purchase price. Diego paid off the loan by making 48 payments of $300 each. The total of all his payments, including the down payment, was how much more than the car's purchase price?

$$Total \ payment=400+300\times48=14800 \\
Difference=14800-13400=1400$$

Choice G.

10. Diego purchased a car that had a purchase price of $13,400, which included all other costs and tax. He paid $400 as a down payment and got a loan for the rest of the purchase price. Diego paid off the loan by making 48 payments of $300 each. The total of all his payments, including the down payment, was how much more than the car’s purchase price?

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11. In the standard (x,y) coordinate plane, what is the slope of the line 4x + 7y = 9 ?

$$4x+7y=9 \\
\Rightarrow 7y=-4x+9 \\
\Rightarrow y=-\frac {4}{7}x+\frac {9}{7} \\
\Rightarrow Slope =-\frac {4}{7}$$

Choice A.

11. In the standard (x,y) coordinate plane, what is the slope of the line 4x + 7y = 9 ?

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12. In the figure below, AD intersects BG at C and is perpendicular to DE. Line DE intersects BG at F. Given that the measure of ZEFG is 25°, what is the measure of ZBCD ?

$$\angle C= \angle D + \angle F = 90°+25°=115°$$

Choice G.

12. In the figure below, AD intersects BG at C and is perpendicular to DE. Line DE intersects BG at F. Given that the measure of ZEFG is 25°, what is the measure of ZBCD ?

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13. What is the sum of the 2 solutions of the equation X2+ x — 30 = 0 ?

$$x^2+x-30=0\\
\Rightarrow (x+6)(x-5)=0\\
\Rightarrow x=-6,\ 5\\
\Rightarrow sum \ of two \ solutions=-6+5=-1$$

Choice C.

13. What is the sum of the 2 solutions of the equation X2+ x — 30 = 0 ?

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14. The volume of a sphere is 41c3r	, where r is the radius
of the sphere. What is the volume, in cubic yards, of a sphere with a diameter of 4 yards?

$$d=4\\
\Rightarrow r=\frac{d}{2}=\frac{4}{2}=2\\
\Rightarrow v=\frac{4\pi r^3}{3}=\frac{4\pi 2^3}{3}=\frac{32}{3} \pi$$

Choice F.

14. The volume of a sphere is 41c3r , where r is the radius
of the sphere. What is the volume, in cubic yards, of a sphere with a diameter of 4 yards?

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15. What is the smallest integer greater than V85 ?

$$10=\sqrt{100} \ which\ is\ greater\ than\ \sqrt{85} \\
9=\sqrt{81} \ which\ is\ less\ than\ \sqrt{85} $$

Choice C.

15. What is the smallest integer greater than V85 ?

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16. The 3 statements below are true for the elements of sets A, B, C, and D.

Choice K.

16. The 3 statements below are true for the elements of sets A, B, C, and D.

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17. In the standard (x,y) coordinate plane, the midpoint of AB is at (2,1), and A is at (8,10). What is the x-coordinate of B ?

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

Choice A.

17. In the standard (x,y) coordinate plane, the midpoint of AB is at (2,1), and A is at (8,10). What is the x-coordinate of B ?

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18. Lena will pick 1 card at random from a pack of 25 baseball cards. Each card features the fielding position for 1 of 25 different baseball players. Each player in the pack has only 1 fielding position. The table below lists the frequency of fielding positions in the pack. What is the probability that the card Lena picks will feature an outfielder or a pitcher?

$$p=\frac{8+7}{25}=\frac{15}{25}=\frac{3}{5}=60\%$$

Choice K.

18. Lena will pick 1 card at random from a pack of 25 baseball cards. Each card features the fielding position for 1 of 25 different baseball players. Each player in the pack has only 1 fielding position. The table below lists the frequency of fielding positions in the pack. What is the probability that the card Lena picks will feature an outfielder or a pitcher?

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19. According to a soil analysis, a certain lawn requires an application of 40.0 kg of nitrogen phosphate when the average temperature is 75.0°F. To avoid burning the grass, the required application amount decreases 1.2 kg for each 1.0°F that the average temperature is above 75.0°F. To the nearest 0.1 kg, what is the required application amount of nitrogen phosphate when the average temperature is 83.0°F ?

$$x=40-(83-75)\times 1.2=30.4$$

Choice A.

19. According to a soil analysis, a certain lawn requires an application of 40.0 kg of nitrogen phosphate when the average temperature is 75.0°F. To avoid burning the grass, the required application amount decreases 1.2 kg for each 1.0°F that the average temperature is above 75.0°F. To the nearest 0.1 kg, what is the required application amount of nitrogen phosphate when the average temperature is 83.0°F ?

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20. In the figure below, all segments that meet do so at right angles. What is the area, in square units, of the shaded region?

$$x+2+x=5 \Rightarrow x=\frac{3}{2}\\
y+2+y=6 \Rightarrow y=2\\
\Rightarrow shaded \ area=xy=\frac{3}{2}\times 2=3$$

Choice G.

20. In the figure below, all segments that meet do so at right angles. What is the area, in square units, of the shaded region?

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21. The perimeter of a certain scalene triangle is 100 inches. The side lengths of the triangle are represented by 5x, 3x + 30, and 2x + 10, respectively. What is the length, in inches, of the longest side of the triangle?

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21. The perimeter of a certain scalene triangle is 100 inches. The side lengths of the triangle are represented by 5x, 3x + 30, and 2x + 10, respectively. What is the length, in inches, of the longest side of the triangle?

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22. The mayor of Westbrook is deciding how to assign the 6 council members to the row of seats below.

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22. The mayor of Westbrook is deciding how to assign the 6 council members to the row of seats below.

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23. The sum of 2 and 200% of 1 has the same value as which of the following calculations?

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23. The sum of 2 and 200% of 1 has the same value as which of the following calculations?

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24. The graph in the standard (x,y) coordinate plane below is represented by one of the following equations. Which equation?

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24. The graph in the standard (x,y) coordinate plane below is represented by one of the following equations. Which equation?

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25. Kamini is constructing the kite shown below. The kite includes 2 perpendicular supports, one of length 40 inches and the other of length 28 inches. The ends of the supports are connected with string to form a 4-sided figure that is symmetric with respect to the longer support. A layer of paper will cover the interior of the 4-sided figure. Which of the following is closest to the area, in square inches, that Kamini will cover with paper?

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25. Kamini is constructing the kite shown below. The kite includes 2 perpendicular supports, one of length 40 inches and the other of length 28 inches. The ends of the supports are connected with string to form a 4-sided figure that is symmetric with respect to the longer support. A layer of paper will cover the interior of the 4-sided figure. Which of the following is closest to the area, in square inches, that Kamini will cover with paper?

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26. The top view and side view of a 40-foot-long swimming pool are shown in the figure below. All dimensions given are in feet. The top view shows the top rectangular surface of the pool and the surrounding rectangular patio. All 4 walls of the pool are vertical and perpendicular to the top surface. The side view shows a cross section along the length of the pool. All cross sections parallel to the side view are congruent. The shallow end has a constant depth of 4 ft. The deep end has a constant depth of 9 ft. A rectangular surface connects the shallow and deep ends.

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26. The top view and side view of a 40-foot-long swimming pool are shown in the figure below. All dimensions given are in feet. The top view shows the top rectangular surface of the pool and the surrounding rectangular patio. All 4 walls of the pool are vertical and perpendicular to the top surface. The side view shows a cross section along the length of the pool. All cross sections parallel to the side view are congruent. The shallow end has a constant depth of 4 ft. The deep end has a constant depth of 9 ft. A rectangular surface connects the shallow and deep ends.

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27. Johann put up a fence along the outer edge of the patio. Given that the materials for the fence cost $12 per foot, what was the total cost of the materials for the fence?

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27. Johann put up a fence along the outer edge of the patio. Given that the materials for the fence cost $12 per foot, what was the total cost of the materials for the fence?

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28. A full lap is 2 times the length of the pool. Johann swam 5 full laps of the pool in 44- minutes. Which of the following values is closest to Johann's average swimming speed, in feet per minute?

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28. A full lap is 2 times the length of the pool. Johann swam 5 full laps of the pool in 44- minutes. Which of the following values is closest to Johann’s average swimming speed, in feet per minute?

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29. The side view of the pool is placed in the standard (x,y) coordinate plane, keeping the same orientation and scale, such that both vertical segments showing depth are parallel to the y-axis. Which of the following values is closest to the slope of the line segment connecting the shallow end to the deep end?

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29. The side view of the pool is placed in the standard (x,y) coordinate plane, keeping the same orientation and scale, such that both vertical segments showing depth are parallel to the y-axis. Which of the following values is closest to the slope of the line segment connecting the shallow end to the deep end?

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30. A construction company builds 3 different models of houses (A, B, and C). They order all the bathtubs, shower stalls, and sinks for the houses from a certain manufacturer. Each model of house contains different numbers of these bathroom fixtures. The tables below give the number of each kind of these fixtures required for each model and the cost to the company, in dollars, of each type of fixture.

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30. A construction company builds 3 different models of houses (A, B, and C). They order all the bathtubs, shower stalls, and sinks for the houses from a certain manufacturer. Each model of house contains different numbers of these bathroom fixtures. The tables below give the number of each kind of these fixtures required for each model and the cost to the company, in dollars, of each type of fixture.

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31. Shown below; a board 5 feet 6 inches long is cut into 2 equal parts. What is the length, to the nearest inch, of each part?

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31. Shown below; a board 5 feet 6 inches long is cut into 2 equal parts. What is the length, to the nearest inch, of each part?

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32. A company that builds bridges used a pile driver to drive a post into the ground. The post was driven 18 feet into the ground by the first hit of the pile driver. On each hit after the first hit, the post was driven into the ground an additional distance that was 3 the distance the post was driven in the previous hit. After a total of 4 hits, the post was driven how many feet into the ground?

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32. A company that builds bridges used a pile driver to drive a post into the ground. The post was driven 18 feet into the ground by the first hit of the pile driver. On each hit after the first hit, the post was driven into the ground an additional distance that was 3 the distance the post was driven in the previous hit. After a total of 4 hits, the post was driven how many feet into the ground?

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33. In the standard (x,y) coordinate plane, A' is the image resulting from the reflection of the point A(2,-3) across the y-axis. What are the coordinates of A' ?

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33. In the standard (x,y) coordinate plane, A’ is the image resulting from the reflection of the point A(2,-3) across the y-axis. What are the coordinates of A’ ?

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34. To increase the mean of 4 numbers by 3, by how much	would the sum of the 4 numbers have to increase?

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34. To increase the mean of 4 numbers by 3, by how much would the sum of the 4 numbers have to increase?

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35. Which of the following expressions is equivalent to (3 + x)^-100?

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35. Which of the following expressions is equivalent to (3 + x)^-100?

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36. Consider the graph of the equation y = 3x in the 2x — 6 standard (x,y) coordinate plane. Which of the following equations represents the vertical asymptote of the graph?

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36. Consider the graph of the equation y = 3x in the 2x — 6 standard (x,y) coordinate plane. Which of the following equations represents the vertical asymptote of the graph?

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37. For every pair of real numbers x and y such that xy = 0 and y = 0, which of the following statements is true?

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37. For every pair of real numbers x and y such that xy = 0 and y = 0, which of the following statements is true?

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38. Walter recently vacationed in Paris. While there, he visited the Louvre, a famous art museum. Afterward, he took a 3.7-kilometer cab ride from the Louvre to the Eiffel Tower. A tour guide named Amelie informed him that 2.5 million rivets were used to build the tower, which stands 320 meters tall. 38. Walter's cab ride lasted 15 minutes. Which of the following values is closest to the average speed, in miles per hour, of the cab? (Note: 1 mile 1.6 kilometers)

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38. Walter recently vacationed in Paris. While there, he visited the Louvre, a famous art museum. Afterward, he took a 3.7-kilometer cab ride from the Louvre to the Eiffel Tower. A tour guide named Amelie informed him that 2.5 million rivets were used to build the tower, which stands 320 meters tall. 38. Walter’s cab ride lasted 15 minutes. Which of the following values is closest to the average speed, in miles per hour, of the cab? (Note: 1 mile 1.6 kilometers)

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39. When written in scientific notation, the number of rivets used to build the Eiffel Tower is equal to which of the following expressions?

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39. When written in scientific notation, the number of rivets used to build the Eiffel Tower is equal to which of the following expressions?

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40. At a certain point, the angle of elevation formed by the level ground and the line from that point to the top of the Eiffel Tower is 70°. Which of the following expressions is equal to the distance, in meters, between that point and the center of the base of the tower?

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40. At a certain point, the angle of elevation formed by the level ground and the line from that point to the top of the Eiffel Tower is 70°. Which of the following expressions is equal to the distance, in meters, between that point and the center of the base of the tower?

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41. When the vector ai + 3j is added to the vector —2i + bj, the sum is 61 — 6j. What are the values of a and b ?

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41. When the vector ai + 3j is added to the vector —2i + bj, the sum is 61 — 6j. What are the values of a and b ?

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42. Given c = 10b^3 + 50, which of the following is an expression for b in terms of c ?

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42. Given c = 10b^3 + 50, which of the following is an expression for b in terms of c ?

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43. Given f(x) = x2 + 3x and g(x) = x + 1, what is f(g(x)) ?

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43. Given f(x) = x2 + 3x and g(x) = x + 1, what is f(g(x)) ?

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44. The diameter of one circle is 12 inches long. The diameter of a second circle is 25% longer than the diameter of the first circle. To the nearest square inch, how much larger is the area of the second circle than the area of the first circle?

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44. The diameter of one circle is 12 inches long. The diameter of a second circle is 25% longer than the diameter of the first circle. To the nearest square inch, how much larger is the area of the second circle than the area of the first circle?

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45. What is the product of the mean and the median of the first 6 prime numbers?(Note: 2 is the first prime number.)

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45. What is the product of the mean and the median of the first 6 prime numbers?(Note: 2 is the first prime number.)

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46. For all real values of x, which of the following equations is true?

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46. For all real values of x, which of the following equations is true?

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47. In the figure shown below, A, B, and D lie on a circle whose center is 0, a diameter is AB, CD is perpendicular to AB at C, the length of AD is 5 m, and the length of BD is 12 m. What is the length, in meters, of CD ?

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47. In the figure shown below, A, B, and D lie on a circle whose center is 0, a diameter is AB, CD is perpendicular to AB at C, the length of AD is 5 m, and the length of BD is 12 m. What is the length, in meters, of CD ?

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48. If a and h are real numbers such that a srcset= 0 and b< 0, then which of the following is equivalent to lal — I b1 ?" width="300" height="117">

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48. If a and h are real numbers such that a > 0 and b< 0, then which of the following is equivalent to lal — I b1 ?

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49. If x < y and y < 4, then what is the greatest possible integer value of x + y ?

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49. If x < y and y < 4, then what is the greatest possible integer value of x + y ?

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50. Given that y varies directly as the square of x, if y = 20 when x= 2, what is y when x = 3 ?

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50. Given that y varies directly as the square of x, if y = 20 when x= 2, what is y when x = 3 ?

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51. Shown below in the standard (x,y) coordinate plane are 2 circles and 1 ellipse, each centered at (0,0). The larger circle has equation x2 + y2 = 25 and intersects the ellipse at exactly 2 points, both on the x-axis. The smaller circle has equation x2 + y2 = 4 and intersects the ellipse at exactly 2 points, both on the y-axis. Which of the following equations represents the ellipse?

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51. Shown below in the standard (x,y) coordinate plane are 2 circles and 1 ellipse, each centered at (0,0). The larger circle has equation x2 + y2 = 25 and intersects the ellipse at exactly 2 points, both on the x-axis. The smaller circle has equation x2 + y2 = 4 and intersects the ellipse at exactly 2 points, both on the y-axis. Which of the following equations represents the ellipse?

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52. The mean of 5 integers is 52. The median of these 5 integers is 82. Three of the integers are 0, 12, and 82. Which of the following could be one of the other integers?

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52. The mean of 5 integers is 52. The median of these 5 integers is 82. Three of the integers are 0, 12, and 82. Which of the following could be one of the other integers?

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53. An integer is abundant if its positive integer factors, excluding the integer itself, have a sum that is greater than the integer. How many of the integers 6, 8, 10, and 12 are abundant?

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53. An integer is abundant if its positive integer factors, excluding the integer itself, have a sum that is greater than the integer. How many of the integers 6, 8, 10, and 12 are abundant?

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54. Vanna walked at a rate of 2 miles per hour for 10 minutes and then walked at a rate of 3 miles per hour for 5 minutes. Which of the following gives the average rate, in miles per hour, at which she walked over this 15-minute period?

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54. Vanna walked at a rate of 2 miles per hour for 10 minutes and then walked at a rate of 3 miles per hour for 5 minutes. Which of the following gives the average rate, in miles per hour, at which she walked over this 15-minute period?

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55. The ratio of Alani's height to Baahir's height is 5:7. The ratio of Baahir's height to Connor's height is 4:3. What is the ratio of Alani's height to Connor's height?

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55. The ratio of Alani’s height to Baahir’s height is 5:7. The ratio of Baahir’s height to Connor’s height is 4:3. What is the ratio of Alani’s height to Connor’s height?

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56. For all x srcset= 0, which of the following expressions is NOT equivalent to” width=”300″ height=”160″>

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56. For all x > 0, which of the following expressions is NOT equivalent to

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57. If the length of a rectangle is increased by 25% and the width is decreased by 10%, the area of the resulting rectangle is larger than the area of the original rectangle by what percent?

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57. If the length of a rectangle is increased by 25% and the width is decreased by 10%, the area of the resulting rectangle is larger than the area of the original rectangle by what percent?

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58. Five balls, numbered 1, 2, 3, 4, and 5, are placed in a bin. Two balls are drawn at random without replacement. What is the probability that the sum of the numbers on the balls drawn is 7 ?

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58. Five balls, numbered 1, 2, 3, 4, and 5, are placed in a bin. Two balls are drawn at random without replacement. What is the probability that the sum of the numbers on the balls drawn is 7 ?

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59. Consider the family of functions y = f(x)= sin x + c, where c is a real number. Which of the following number lines represents the graph of all and only the possible values of c for which the graph of y has no x-intercepts?

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59. Consider the family of functions y = f(x)= sin x + c, where c is a real number. Which of the following number lines represents the graph of all and only the possible values of c for which the graph of y has no x-intercepts?

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60. Tameka calculates that she needs 360 square feet of new carpet. But the type of carpet that she wants is priced by the square yard. How many square yards of carpet does she need?

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60. Tameka calculates that she needs 360 square feet of new carpet. But the type of carpet that she wants is priced by the square yard. How many square yards of carpet does she need?



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