ACT Form A09 Math Answer Explanation



Choice A

$$slope = \frac{3-(-6)}{5-(-2)}>0$$


2018 April(A09) Math Question 2

Number Types & Properties

- Complex Number

2. What is the sum of the complex numbers 3 – 4i and 5 + 3i?

Choice J

(3 - 4i) + (5 + 3i) = 8 - I


Choice C

$$\frac{630}{x}=\frac{2}{7} $$ $$ x=\frac{630\times7}{2}=2205$$


2018 April(A09) Math Question 4

Angles

- Collinear Points & Triangles

4. Three line segments intersect as shown in the figure below, forming angles with measures of 150°, 40°, and x°, respectively. What is the value of x ?

Choice K

a09

x = 40 + (180-150) = 70


Choice C

a09

$$p=\frac{4\times4}{6\times13}\approx 0.21$$


Choice H

$$9x-y-5=0 \\$$ $$\Rightarrow 9x-5=y \\$$ $$\Rightarrow y=9x-5$$


Choice A

$$XY=5k\\$$ $$YZ=9k \\$$ $$\Rightarrow XZ=14k\\$$ $$\Rightarrow \frac{XY}{XZ}=\frac{5k}{9k}=\frac{5}{9}$$


2018 April(A09) Math Question 8

Inequalities

- Inequality Expressions

8. The solution set of 2x + 4 —8 is the set of all real values of .v such that:

Choice G

$$2x+4\geq -8 \\$$ $$\Rightarrow 2x\geq -8-4 \\$$ $$\Rightarrow x\geq \frac{-8-4}{2}\\$$ $$\Rightarrow x\geq -6$$


Choice E

$$(a^3)^{21}=a^{3\times21}=a^{63}$$


Choice G

$$f(-3)=3(-3)^2+7(-3)-8=-2$$


Choice D

$$2a+(a+3)=93\\$$ $$a=30\\$$ $$a+3=33$$


2018 April(A09) Math Question 12

Sequences and Patterns

- Arithmetic Sequences

12. The 220 graduating seniors of Madison- High School will sit in the center section of the school auditorium at the graduation ceremony. How many rows of seats will be needed to seat all of the graduating seniors if the first row has 10 seats and each succeeding row has 2 more seats than the previous row?

Choice G

$$a_1=10 \\$$ $$d=2 \\$$ $$a_n=10+2(n-1)=2n+8 \\$$ $$S_n=\frac {a_1+a_n}{2}\cdot n=\frac {10+2n+8}{2}\cdot n=220 \\$$ $$n=11$$


2018 April(A09) Math Question 13

Equation of a Line

- Distance & Midpoint

13. In the standard (x,y) coordinate plane, the point (3,-7) is the midpoint of the line segment with endpoints (9, -11) and:

Choice B

$$\frac {9+x}{2}=3 \\$$ $$\frac {-11+y}{2}=-7 \\$$ $$\Rightarrow x=-3 ,\ y=-3$$


2018 April(A09) Math Question 14

Word Problems

- 2 Variable Addition

14. A community theater group performed at 5 local schools. For each school, the table below shows the total number of tickets sold and the total dollar amount collected from ticket sales. At School A, only 2 types of tickets were sold: premium tickets for $10 each and value tickets for $6 each. How many value tickets were sold at School A ?

Choice J

$$t+v=200\\$$ $$10t+6v=1400\\$$ $$\Rightarrow t=50,\ v=150$$


2018 April(A09) Math Question 15

Word Problems

- Fixed Cost + Variable Cost

15. The theater group had to pay each school a facility charge. For use of its facility, School C charged the theater group 10% of the ticket sales and a fixed fee of $200. How much money did School C charge the theater group for use of its facility?

Choice E

$$10\% \times 1800+200=380$$


2018 April(A09) Math Question 16

Statistics

- Mean, Median, Mode Combinations

16. What is the difference between the median and the mean number of tickets sold at the 5 schools?

Choice K

$$median=250 \\$$ $$mean=\frac {200+250+300+150+275}{5}=235 \\$$ $$\Rightarrow median - mean = 250-235=15$$


Choice E

$$Area = \frac{10\times24}{2}=120 \ dm^2$$


2018 April(A09) Math Question 18

Statistics

- Mean, Median

18. The mean of a list of 7 numbers is 85. The first 6 numbers on the list are 82, 93, 68, 93, 70, and 98. What is the 7th number on the list?

Choice J

$$Mean = \frac{82+93+68+93+70+x}{7}=85 \\$$ $$\Rightarrow x=91$$


2018 April(A09) Math Question 19

Trigonometry

- Pythagorean Theorem & sin cos tan

19. Which of the following is the sine of the smallest angle in a right triangle with side lengths 7, 24, and 25 inches, respectively?

Choice A

$$\sin A = \frac{7}{25}$$


Choice G

a09

The longer length of the smaller rectangle (S2) = 12-5=7

$$Total\ area = S1+S2=10 \times 5 + 3 \times 7=71$$


Choice A

x+(x+3)+(x+5)=26 => x=6


2018 April(A09) Math Question 22

Word Problems

- Distance, Speed, Time

22. A truck sprang a leak in its radiator, which held 480 ounces of fluid when it started to leak. Assuming the truck continues at 35 mph and its radiator leaks 4 ounces of fluid per minute, how many miles will the truck travel before the radiator is empty?

Choice J

$$t=\frac{480}{4}=120 \ minutes = 2 …


Choice A

v+c+s=42

c=v+11

v=s+2

=> c=s+13 …


Choice G

$$BC=\sqrt{AC^2-AB^2}=\sqrt{10^2-8^2}=6\\$$ $$BD=\frac{BC}{2}=3\\$$ $$\Rightarrow AD=\sqrt{BD^2+AB^2}=\sqrt{3^2+8^2}=\sqrt{73}$$


2018 April(A09) Math Question 25

Graphing Properties

- Interpret Graphs or Figures

25. Lyman has plotted 5 points in the standard (.v,y) coordinate plane below. He then plots a new point as follows: the x-coordinate of the new point is the mean of the .v-coordinates of the 5 points already plotted; the y-coordinate of the new point is the mean of the y-coordinates of the 5 points already plotted. Which of the following ordered pairs gives the coordinates of Lyman’s new point?

Choice C

$$x=\frac{(-2)+(-1)+(-2)+1+4}{5}=0 \\$$ $$y=\frac{4+1+(-2)+(-1)+(-2)}{5}=0 $$


2018 April(A09) Math Question 26

Sequences and Patterns

- Geometric Sequences

26. The 1st and 2nd terms of a certain geometric sequence are 10 and —5, respectively. What is the 5th term of the geometric sequence?

Choice H

$$10, -5, \frac{5}{2}, -\frac{5}{4}, \frac{5}{8}, …


2018 April(A09) Math Question 27

Word Problems

- 1 Variable Equations

27. Alani works 8 hours per day 5 days each week at a custom embroidery shop. Each day Alani is paid either $1.00 per shirt that she stitches or $10.00 per hour, whichever daily amount is higher. Alani stitched the following numbers of shirts: 60 on Monday, 52 on Tuesday, 85 on Wednesday, 80 on Thursday, and 90 on Friday. What is Alani’s total pay for these 5 days?

Choice C

Monday:

The daily amount for …


2018 April(A09) Math Question 28

Quadratic Equations

- Factoring

28. Which of the following most precisely describes the roots of the equation 5×2 + 7x + 2 = 0 ?

Choice H

To find the roots of …


2018 April(A09) Math Question 29

Trigonometry

- Pythagorean Theorem & sin cos tan

29. Paula is planning a course for a bike race. The course is in the shape of a right triangle, as shown below. Participants will begin at A, ride directly to B, then directly to C, and directly back to A. Paula wants to put a rest stop at the halfway point on the course. How many miles past B will the rest stop be?

Choice D

a09

$$BC=\sqrt{AB^2+AC^2}=13$$

Total distance of the …


Choice G

The number of students NOT …


Choice D

$$\sqrt{x}+3=6 \\$$ $$\Rightarrow \sqrt{x}=3 \\$$ …


2018 April(A09) Math Question 32

Trigonometry

- Pythagorean Theorem & sin cos tan

32. In the right triangle AABC shown below, the length of

Choice H

$$\sin A = \frac{3}{4}=\frac{BC}{AB}=\frac{12}{AB} \\$$ …


2018 April(A09) Math Question 33

Logarithms

- Logarithm Expressions

33. What integer does 3(log2 16) equal?

Choice A

$$3(\log_2 16)=3(\log_2 2^4)=3(4)=12$$


2018 April(A09) Math Question 34

Probability

- One Event

34. A bag contains several marbles. On 3 successive draws with replacement, a red marble is drawn from the bag each time. Which of the following statements must be: true about the marbles in the bag?

Choice F

F. is apparently true, because …


2018 April(A09) Math Question 35

Equation of a Line

- Distance & Midpoint

35. Students and adults from Western High School visited an amusement park on a field trip. The amusement park charged $25 for each adult ticket and $20 for each student ticket. Before the trip, the students were given this information about the Happy Dragon roller coaster: the average speed of the roller coaster is 50 miles per hour, and 1 ride on the roller coaster track is completed in 2.25 minutes. A graph showing the height above level ground, in feet, with respect to the time into the ride, in seconds, is given below. Which of the following values is closest to the total distance traveled, in miles, during 1. complete ride on the roller coaster track?

Choice B

Total distance travelled in 1 …


Choice G

$$\frac{5 \ seconds}{2.25 \ minutes …


Choice E

a09

$$The\ slope \approx \frac{200\ ft}{50\ …


Choice J

$$10^5(0.5\times10^2+2.5\times10+6)=10^5(50+25+6)=81\times10^5=8.1\times10^6 \\$$ $$\Rightarrow 8\times10^6<8.1\times10^6 …


2018 April(A09) Math Question 39

Statistics

- Mean, Median, Mode Combinations

39. For which of the following data sets is the difference between the mean and the median the greatest?

Choice D

A. { 10, 10, 10, …


2018 April(A09) Math Question 40

Perimeter, Area, Volume

- Mixed Shapes

40. Pablo has a deck and pool in his backyard. The deck’s shape is a rectangle with a semicircle removed and is shown shaded in the figure below. The lengths of the straight sides of the deck are given in feet. Pablo plans to cover the top of the deck with stain. To decide how much stain to purchase, he needs to find the area of the top of the deck. To the nearest square foot, what is the area of the top of the deck?

Choice K

a09

Diameter of the circle = …


Choice B

a09

a=30-3-3=24 feet

b=40-3-3=34 feet

Therefore, …


Choice F

$$g(f(x))=\frac{1}{4-f(x)}=\frac{1}{4-x^2}$$ Choice F.


Choice C

$$C=2 \pi r=2 \pi \sqrt{2} …


Choice F

a09

The graph of the function …


2018 April(A09) Math Question 45

Conics

- Parabola

45. Which of the following is an equation of a parabola that passes through the 3 points labeled in the standard (x,y) coordinate plane below?

Choice A

Only Choice A satisfies that …


2018 April(A09) Math Question 46

Sequences and Patterns

- Geometric Sequences

46. As x continually increases in value without bound, the

Choice H

When x become really large, …


2018 April(A09) Math Question 47

Probability

- More than One Event

47. Yulan will use a bag of 30 solid-colored marbles for a game in which each player randomly draws marbles from the bag. The number of marbles of each color is shown in the table below.

Choice B

The probability of drawing a …


Choice K

The expression is undefined whenever …


Choice E

$$\frac{\frac{2}{5}+\frac{8}{7}}{2}=\frac{27}{35}$$


Choice H

$$3^{x^2-x-2}=1=3^0\\$$ $$x^2-x-2=0\\$$ $$(x-2)(x+1)=0\\$$ $$\Rightarrow x=2,-1$$


2018 April(A09) Math Question 51

Conics

- Ellipse

51. The ellipse shown in the standard (x,y) coordinate plane below has equation 93)2 + `-Y 255,2 = 1. Which of the

Choice D

a09

The major radius p is …


2018 April(A09) Math Question 52

Probability

- One Event

52. A basket contains 10 solid-colored balls–2 blue, 3 red, and 5 green. Each ball has a single number printed on it. The blue balls are numbered 1 and 2 (each number is used once), the red balls are numbered 1-3 (each number is used once), and the green balls are numbered 1-5 (each number is used once). A ball will be drawn at random from the basket. What is the probability that the ball that is drawn will he red or have a 3 printed on it?

Choice J

a09

$$p=\frac{3+1}{10}=\frac{4}{10}$$


2018 April(A09) Math Question 53

Trigonometry

- Pythagorean Theorem & sin cos tan

53. In the figure below, the given side lengths of AABC are in inches. What is the area, in square inches, of /ABC ?

Choice B

a09

$$Area \ of \triangle ABC …


Choice J

$$\frac{59\%\times66}{15}\times5=12.98$$ Choice J.


2018 April(A09) Math Question 55

Perimeter, Area, Volume

- Mixed Shapes

55. In the diagram below, B, F, and H are on AC, AE, and BF, respectively, and GH I BE. The area of square ABGF is -12-t the area of square ACDE. What percent of the area of ACDE does the shaded portion represent?

Choice D

a09

Area of shaded portion = …


2018 April(A09) Math Question 56

Statistics

- Mean, Median, Mode Combinations

56. In a data set of 10 distinct values, the single largest value is replaced with a much greater value to form a new data set. Which of the following statements is true about the values of the mean and median for the new data set as compared to the mean and median of the original data set?

Choice F

The median will stay the …


2018 April(A09) Math Question 57

Matrices

- Multiplication

57. Valley High School and Mountain High School have decided that selected students will attend a daytime theatrical performance that costs $5 for each teacher and $3 for each student. One teacher and 10 students from Valley High will attend, and 2 teachers and 25 students from Mountain High will attend. Which of the following matrix products represents the ticket costs. in dollars, for each high school?

Choice A

The ticket cost is $$5\times1+3\times10$$ …


2018 April(A09) Math Question 58

Number Types & Properties

- Complex Number

58. Karie is inventing a new notation in math. She has decided to use n.L to denote the sum of the first a positive integers. For example, 51 means 5 + 4 + 3 + 2 + 1. Karie has written 3 statements that she is investigating as possible properties of

Choice F

$$I. n\downarrow+(n+1)=(n+1)\downarrow$$ $$(1+2+3+...+n)+(n+1)=1+2+3+...+n+(n+1)=(n+1)\downarrow\\$$ Therefore, I. …


Choice E

a09

$$(x-4)(x-4)\times2=72\\$$ $$x=10 \\$$ $$\Rightarrow 10 …


2018 April(A09) Math Question 60

Quadratic Equations

- Factoring

60. Which of the following quadratic equations has the complex number (1 + V-1) as a solution?

Choice K

Let's start with K. To …