ACT Form A11 Math Answer Explanation



2019 June(A11) Math Question 1

Statistics

- Mean, Median

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?

Choice D

\begin{align*} & 4x+37.8=9\times9.8 \\ & x=12.6 \end{align*}


2018 June(A11) Math Question 3

Word Problems

- Distance, Speed, Time

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?

Choice C

Shantiel travelled for 18 hours from 9am to 3am the next day. \begin{align*} s=\frac{648}{18}=36 \end{align*}


2018 June(A11) Math Question 4

Word Problems

- Fixed Cost + Variable Cost

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?

Choice J

\begin{align*} n=300+\frac{16.5-10}{0.1}=365 \end{align*}


2018 June(A11) Math Question 5

Matrices

- Addition & Subtraction

5. Which of the following matrices is equal to

Choice A

\begin{align*} \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} \end{align*}


Choice H

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


2018 June(A11) Math Question 7

Statistics

- Mean, Median

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?

Choice D

\begin{align*} mean \ age =\frac{10\times3+5\times2}{5}=8 \end{align*}


Choice H

\begin{align*} AC \parallel DE , \ BD=AD\\ \Rightarrow DE=\frac{AC}{2}=\frac{8}{2}=4 \end{align*}


Choice B

\begin{align*} n=22000\times \frac{337}{500}=14828\approx 14830 \end{align*}


2018 June(A11) Math Question 10

Word Problems

- Fixed Cost + Variable Cost

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?

Choice G

\begin{align*} Total \ payment=400+300\times48=14800 \\ Difference=14800-13400=1400 \end{align*}


Choice A

\begin{align*} 4x+7y=9 \\ \Rightarrow 7y=-4x+9 \\ \Rightarrow y=-\frac {4}{7}x+\frac {9}{7} \\ \Rightarrow Slope =-\frac {4}{7} \end{align*}


2018 June(A11) Math Question 12

Angles

- Vertex Angles & Bisectors

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 ?

Choice G

a11

\begin{align*} \angle C= \angle D + \angle F = 90°+25°=115° \end{align*}


2018 June(A11) Math Question 13

Quadratic Equations

- Factoring

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

Choice C

\begin{align*} x^2+x-30=0\\ \Rightarrow (x+6)(x-5)=0\\ \Rightarrow x=-6,\ 5\\ \Rightarrow sum \ of two \ solutions=-6+5=-1 \end{align*}


2018 June(A11) Math Question 14

Perimeter, Area, Volume

- Other Shapes

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?

Choice F

\begin{align*} 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 \end{align*}


Choice C

\begin{align*} 10=\sqrt{100} \ which\ is\ greater\ than\ \sqrt{85} \\ 9=\sqrt{81} \ which\ is\ less\ than\ \sqrt{85} \end{align*}


2018 June(A11) Math Question 16

Sequences and Patterns

- Logical Expressions

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

Choice K

a11

\begin{align*} \end{align*}


2018 June(A11) Math Question 17

Equation of a Line

- Distance & Midpoint

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 ?

Choice A

\begin{align*} 2=\frac {8+x}{2}\Rightarrow x=-4 \end{align*}


2018 June(A11) Math Question 18

Probability

- One Event

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?

Choice K

\begin{align*} p=\frac{8+7}{25}=\frac{15}{25}=\frac{3}{5}=60\% \end{align*}


2018 June(A11) Math Question 19

Word Problems

- Temperature

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 ?

Choice A

\begin{align*} x=40-(83-75)\times 1.2=30.4 \end{align*}


2018 June(A11) Math Question 20

Perimeter, Area, Volume

- Mixed Shapes

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?

Choice G

\begin{align*} 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 \end{align*}


Choice D

\begin{align*} 5x+(3x+30)+(2x+10)=100\Rightarrow x=6\\ 3x+30=3\times 6+30=48 …


2018 June(A11) Math Question 22

Probability

- Combinations

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

Choice J

\begin{align*} ^nP_k=\frac{n!}{(n-k)!} \end{align*} \begin{align*} ^6P_6=\frac{6!}{(6-6)!} …


Choice E

\begin{align*} 2+200\%\cdot1=2+2=4=400\%\cdot1 \end{align*}


Choice H

\begin{align*} Choice \ H: y=f(x)=-\frac{2}{3}\cdot …


2018 June(A11) Math Question 25

Perimeter, Area, Volume

- Mixed Shapes

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?

Choice C

a11

\begin{align*} & Let\ a\ be\ …


Choice H

a11

\begin{align*} Area \ of \ …


Choice D

\begin{align*} cost=12\times(60+25+60+25)=2040 \end{align*}


2018 June(A11) Math Question 28

Word Problems

- Distance, Speed, Time

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?

Choice J

\begin{align*} 1 \ full \ …


Choice B

\begin{align*} k=\frac{0-5}{0-(-8)}=-\frac{5}{8}=-0.625\approx-0.63 \end{align*}


Choice K

\begin{align*} cost \ of \ …


Choice C

\begin{align*} \frac{5 \ feet \ …


Choice H

\begin{align*} 18+18\times \frac{2}{3}+18\times \frac{2}{3}\times \frac{2}{3}+18\times …


Choice B

a11

\begin{align*} \end{align*}


2018 June(A11) Math Question 34

Statistics

- Mean, Median

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

Choice K

\begin{align*} increase \ in \ …


Choice E

One need to know that …


Choice G

\begin{align*} 2x-6=0\\ \Rightarrow x=3 \end{align*}


Choice C

\begin{align*} xy=0 \Rightarrow x=0 \ …


2018 June(A11) Math Question 38

Word Problems

- Distance, Speed, Time

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)

Choice F

\begin{align*} v=\frac{3.7 \ km}{15 \ …


Choice A

\begin{align*} 2.5 \ million = …


2018 June(A11) Math Question 40

Trigonometry

- Pythagorean Theorem & sin cos tan

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?

Choice K

a11

\begin{align*} \tan 70°=\frac{320}{x}\Rightarrow x=\frac{320}{\tan 70°} …


2018 June(A11) Math Question 41

Graphing Properties

- Vectors

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 ?

Choice E

\begin{align*} a-2=6 \Rightarrow a=9\\ 3+b=-6\Rightarrow …


Choice F

\begin{align*} c=10b^3+50 \\ \Rightarrow c-50=10b^3\\ …


Choice A

\begin{align*} f(g(x))=(g(x))^2+3g(x)=(x+1)^2+3(x+1)\\ =x^2+2x+1+3x+3 \\ =x^2+5x+4 …


Choice J

\begin{align*} d_1=12 \Rightarrow 2\pi r_1=12\Rightarrow …


2018 June(A11) Math Question 45

Statistics

- Mean, Median, Mode Combinations

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

Choice D

The first 6 prime numbers …


2018 June(A11) Math Question 46

Trigonometry

- Advanced sin, cos, tan functions

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

Choice K

One need to know that …


2018 June(A11) Math Question 47

Trigonometry

- Pythagorean Theorem & sin cos tan

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 ?

Choice A

a11

\begin{align*} 13x=5\times12 \Rightarrow x=\frac{60}{13} \end{align*}


2018 June(A11) Math Question 48

Absolute Values

- Absolute Value Expressions

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 ?

Choice K

\begin{align*} a>0 \Rightarrow |a|=a …


Choice D

\begin{align*} x < y \Rightarrow x+y<2y \ (1)\\ y<4 \Rightarrow 2y<8 \ (2)\\ (1)+(2) \Rightarrow x+y<8 \\ (x+y)_{greatest \ possible \ integer}=7 \end{align*}


Choice G

\begin{align*} y=kx^2\\ 20=4k\Rightarrow k=5 \\ …


2018 June(A11) Math Question 51

Conics

- Ellipse

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?

Choice D

a11

\begin{align*} \frac{(x-h)^2}{a^2}+\frac{(y-h)^2}{b^2}=1 \end{align*}

This is …


2018 June(A11) Math Question 52

Statistics

- Mean, Median, Mode Combinations

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?

Choice H

The sum of the other …


Choice B

The sum positive integers of …


2018 June(A11) Math Question 54

Word Problems

- Distance, Speed, Time

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?

Choice G

\begin{align*} average \ rate =\frac{2\times\frac{10}{60}+3\times\frac{5}{60}}{\frac{10}{60}+\frac{5}{60}}=\frac{7}{3} …


Choice D

\begin{align*} A=5n\\ B=7n=4m\Rightarrow n=\frac{4m}{7}\\ C=3m\Rightarrow …


Choice K

\begin{align*} x^\frac{2}{3}=\sqrt[3]{x^2} \end{align*}


Choice B

\begin{align*} 1.25a\times0.9b=1.125ab \end{align*}


2018 June(A11) Math Question 58

Probability

- More than One Event

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 ?

Choice F

When 3 and 4, or, …


2018 June(A11) Math Question 59

Trigonometry

- Graphing Trig Functions

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?

Choice D

a11

\begin{align*} \end{align*}


Choice G

\begin{align*} 1 \ yard = …