ACT Form A11 Math Answer Explanation


2019 June(A11) Math Question 1

Statistics Practice

- 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*}

Choice C

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

Choice J

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

2018 June(A11) Math Question 5

Matrices Practice

- 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 Practice

- 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*}

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 Practice

- 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

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

2018 June(A11) Math Question 13

Quadratic Equations Practice

- 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*}

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*}

Choice K

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

Choice A

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

2018 June(A11) Math Question 18

Probability Practice

- 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 Practice

- 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*}

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 Practice

- 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 …

Choice C

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\begin{align*} & Let\ a\ be\ …

Choice H

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\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 Practice

- 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

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

2018 June(A11) Math Question 34

Statistics Practice

- 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 Practice

- 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 = …

Choice K

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

2018 June(A11) Math Question 41

Graphing Properties Practice

- 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 …

Choice D

The first 6 prime numbers …

Choice K

One need to know that …

Choice A

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

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 Practice

- 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

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

This is …

Choice H

The sum of the other …

Choice B

The sum positive integers of …

2018 June(A11) Math Question 54

Word Problems Practice

- 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 Practice

- 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, …

Choice D

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

Choice G

\begin{align*} 1 \ yard = …