ACT Form B02 Math Answer Explanation



2019 June(B02) Math Question 1

Word Problems

- Cost

1. A calculator has a regular price of $59.95 before taxes It goes on sale at 20% below the regular price. Before taxes are added, what is the sale price of the calculator? calculator?

Choice D

$$$59.95\approx$60$$ $$\frac{20}{100}\times$60=$12$$ $$$60 - $12 = $48$$ $$$48\approx.$47.96$$


Choice F

$$(r + b - g)(b + g) = [6 + 4 -(-9)][(4+(-9)] = 19\times(-5)= -95$$


2019 June(B02) Math Question 3

Angles

- Collinear Points & Triangles

3.In the figure below, C is on BD, LBAC measures 42~,
and LABC measures 108°. What is the measure of
LACD?

Choice E

$$\angle ACB = 180°- \angle ABC - \angle BAC = 180° - 108° - 42° = 30°$$ $$\angle ACD = 180° - \angle ACB = 180° - 30° = 150°$$


Choice H

$$X=(17-10)\cdot\frac{5}{3}=\frac{35}{3}$$


Choice C

b02

$$9^{2}+2^{2}=C^{2}$$ $$81+4=85=C^{2}$$ $$C=\sqrt{85}$$


2019 June(B02) Math Question 6

Probability

- One Event

6. A bag contains exactly 18 solid-colored buttons: 3 red,
5 blue; and 10 white. What is the probability of
randomly selecting 1 button that is NOT white?

Choice H

White : 10

Not-white : 3 red + 5 blue = 8

The probability of randomly selecting 1 button that is NOT white

$$= \frac{8}{10+8}=\frac{8}{18}=\frac{4}{9}$$


2019 June(B02) Math Question 7

Statistics

- Mean, Median

7. What is the sum of 3 consecutive odd integers whose
mean is 27?

Choice C

Sum of 3 consecutive odd integers

$$= 27\times3 = 81$$


Choice K

1: 1+0=1

4: 2+2=4

6: 4+2=6

7: 5+2=7

8 is NOT a possible sum of those 2 numbers


2019 June(B02) Math Question 9

Equation of a Line

- Distance & Midpoint

9. On a bike trail there 'are 5 2heckpoints numbered in
order, Checkpoint 1 through Checkpoint 5,.as shown in
the figure below. Some di!ltances along the trail
between 2 checkpoints are given: 6.6 miles between
1 and 3; 4.5 miles between 2 and 3; and 9.7 miles
betwe~ri 2 and 5. Which .of the following values is
closest to the distance, in miles, along the tr.ail between
Checkpoint 1 and Checkpoint 5 ?

Choice B

$$\widetilde{13}=6.6 miles$$ $$\widetilde{23}=4.5 miles$$ $$\widetilde{25}=9.7 miles$$ $$\widetilde{12}=\widetilde{13}-\widetilde{23}=6.6-4.5=2.1 miles$$ $$\widetilde{15}=\widetilde{12}+\widetilde{25}=2.1+9.7=11.8 miles$$


2019 June(B02) Math Question 10

Perimeter, Area, Volume

- Mixed Shapes

10; Iv.th e figure below, a circle with a radius of i O meters
circumscribes a regular hexagon. What is the perimeter,
in meters, of the hexagon?

Choice H

b02

The perimeter of the hexagon:

$$6\times10=60$$


2019 June(B02) Math Question 11

Word Problems

- Fixed Cost + Variable Cost

11. To produce aluminum softball bats, it costs the
Recreation Equipment Supply Company $3,500 for
overhead, plus $2 per softball bat produced. What is
the maximum number of bats that can be produced by
the company for $15,000 ?

Choice D

N: the maximum number of bats that can be produced by the company

$$ $3,500+$2\times N =$15000 $$ $$ $2\times N = $11500 $$ $$N=5750 $$


Choice G

$$3x+2=4$$ $$\Rightarrow 3x=2$$ $$\Rightarrow x=\frac{2}{3}$$ $$2y+6=5$$ $$\Rightarrow 2y=-1$$ $$\Rightarrow y=\frac{-1}{2}$$ $$x+y=\frac{2}{3}+\frac{-1}{2}=\frac{4}{6}+\frac{-3}{6}=\frac{1}{6}$$


Choice C

$$\require{cancel} $$ $$\frac{ 15x ^{2}+25x}{5x}=\frac{\cancel{5}(3x^{2}+5x)}{\cancel{5}x}=\frac{3x^\cancel{2}+5\cancel{x}}{\cancel{x}}=3x+5$$


2019 June(B02) Math Question 14

Absolute Values

- Absolute Value Expressions

14. What is th~ valu~ of the expressibri . 16 + 4 x 2 _ 5 •?

Choice K

$$ {\displaystyle \left\vert -5\right\vert^{2} =25}$$ $$\Rightarrow\frac{\displaystyle \left\vert -5\right\vert^{2}+(-1)}{(16\div4)\times2-5}=\frac{25-1}{4\times2-5}=\frac{24}{8-5}=\frac{24}{3}=8$$


Choice D

This special savings account will double every 5 years.

$$ \because 40\ years = 5\ years\times 8 $$

Therefore, this account will double 8 times.

$$ $2000\times 2^{8}=$2000\times256=$512000 $$


Choice F

2 hours ~ 4 hours:

the number of hours the amount of water in the pond remained constant.


2019 June(B02) Math Question 17

Sequences and Patterns

- Logical Expressions

17. lf it rains in Franklin City on a particular day, the
probability that it will rain there the following day is
0.70, If it does not rain Franklin City on a particular
 day, the probability that it will rain there. the following
day is 0.1. Given that it rained in Franklin City on
Monday, what is the probability that it will NOT rain
in Franklin City on Tuesday of the same week? ·

Choice B

lf it rains in Franklin City on a particular day, the probability that it will rain there the following day is 0.70.

Therefore,

if it rains on a Monday,

the probability that it will NOT rain there the following day is

1 - 0.70 = 0.30


Choice G

$$5x=9y+18$$ $$\Rightarrow 9y=5x-18$$ $$\Rightarrow y=\frac{5}{9}-2$$


Choice B

b02

$$Slope=\frac{y-y_{0}}{x-x_{0}}$$ $$\Rightarrow \frac{4-0}{0-2}=-2$$ $$\Rightarrow y=-2x+4$$


2019 June(B02) Math Question 20

Angles

- Parallel Lines

20. In the figure below, line m is perpendicular to line n, and
line p is parallel to line q. Lines m, n, and p intersect at a
single point Some angle measures are given. What is
the value of x ?


2019 June(B02) Math Question 21

Probability

- More than One Event

21. A bag contains 10 solid-colored marbles of the same
size: 3 red, 2 green,.1 yellow, and 4 blue. Which of the
following expressions gives the probability of drawing,
at random and without replacement, a blue marble on
the 1st draw, a green marble on the 2nd draw, and a
blue marble on the 3rd draw?

Choice D

b02

The 1st draw: a blue …


2019 June(B02) Math Question 22

Probability

- One Event

22. A physical education teacher recorded the distances, in
inches, that her students jumped during a long jump
lesson. The distances of 1 jump by each of the students
are represented in the stem-and-leaf plot below.

Choice K

b02

At least 60 inches ⇒ …


Choice C

$$f(x)=5-3x $$ $$\left\{-1,0,2\right\} $$ $$f(-1)=5-3(-1)=8$$ …


2019 June(B02) Math Question 24

Perimeter, Area, Volume

- Other Shapes

24. To the nearest 1 foot, what is the heig~t of a rectangular
prism with a base length of 15 feet, a base width of
1 ½ feet, and a volume of 100 cubic feet?

Choice F

$$100 cubic feet =15\times 1\frac{1}{3}\times …


Choice C

b02


2019 June(B02) Math Question 26

Trigonometry

- Pythagorean Theorem & sin cos tan

26. In 'the figure shown below, a ladder 15 feet long formi
an angle of 56° with the level ground as it leans against
the vertical side of a building. The distance along the
building, fa feet, between the ground and the top of the
ladder is equal to which of the following expressions?

Choice H

b02

$$\frac{h}{15}=\sin56°$$ $$h=15 \sin56°$$


Choice C

The amount of iodine-131 remaining …


Choice F

$$\sqrt[4]{256x^{16}}=(256x^{16})^{\frac{1}{4}}$$ $$=256^{\frac{1}{4}}\times (x^{16})^{\frac{1}{4}}$$ $$=(16^{2})^{\frac{1}{4}}\times x^{4}$$ …


2019 June(B02) Math Question 29

Perimeter, Area, Volume

- Mixed Shapes

29. Two concentric circles have radii of 5 centimeters. and
6 centimeters, respectively. How many centimeters
longer is .the circumference of the larger circle than
that of the smaller circle?

Choice C

b02

The formula for the circumference …


Choice G

b02

$$Area= 8\times18-4\times x^{2} =144-x^{2}$$


Choice E

b02

A(-5,1-10)=A(-5,-9)


2019 June(B02) Math Question 32

Trigonometry

- Pythagorean Theorem & sin cos tan

32. Olivia, Ashton, and Jane are standing on a soccer field
such that Olivia is 20 meters due west of Ashton and
Jane is 40 meters due north of Ashton. Their positions
are at the vertices of a triangle. Which of the following
expressions gives the degree measure of the angle of
the triangle at the vertex where Olivia is standing?

Choice J

b02

$$\tan\theta=\frac{20}{40}$$ $$\Rightarrow\theta=\tan^{-1}\frac{20}{40}$$


Choice D

b02

$$ Area = \frac{1}{2}\times6\times6=18 $$


2019 June(B02) Math Question 34

Equation of a Line

- Distance & Midpoint

34. Whatis the length of AB in coordinate units?

Choice G

b02


Choice A

$$\because y=mx+b$$ $$A(0,6)\Rightarrow6=0+b\Rightarrow b=6$$ $$B(6,0)\Rightarrow0=m\cdot6+6\Rightarrow …


2019 June(B02) Math Question 36

Number Types & Properties

- Order of Values

36. Which of the following arranges the numbers f, 1.8,
1.08, and 1.08 into ascending order? (Note: The
overbar notation shows that the digits under the bar
will repeat. For example, 1.73 == 1.737373 ... )

Choice K

$$\frac{9}{5}=1.8 $$ $$1.\overline{8}=1.88888... $$ $$1.08=1.08$$ …


2019 June(B02) Math Question 37

Statistics

- Mean, Median

37. Andre's Floral Shop asked each of 20 customers to
give a rating of the shop's service. The table below
, summarizes the 20 customer ratings.
Which of the following values is closest to the mean of
the 20 customer ratings?

Choice A

$$\frac{6\times3+8\times2+2\times1+4\times0}{20}$$ $$=\frac{18+16+2+0}{20}$$ $$=\frac{36}{20}$$ $$=\frac{9}{5}=1.8$$


Choice G

$$(F)14+(\frac{-1}{40})=14-(\frac{1}{40})$$ $$(G)14-(\frac{-1}{40})=14+\frac{1}{40}$$ $$(H)14\div(\frac{-1}{40})=14\times(-40)=-560$$ $$(J)14\times(\frac{-1}{40})=\frac{-14}{40}$$ $$(K)\frac{14+\frac{-1}{40}}{2}=7+\frac{-1}{80}$$


Choice E

$$H=0.75\times(200-A)$$


2019 June(B02) Math Question 40

Word Problems

- Temperature

40. The equation t = -0.0066a + 15 models the noon
temperature, t degrees Celsius,  a meters above sea
level, on a certain day on Laurel Mountain. According
to. this equation, what would be the noon temperature
for that certain day on Laurel Mountain at sea level?

Choice J

t=-0.0066a+15

t : degrees Celsius …


Choice C

b02

$$The\ formula\ for\ the\ area\ …


2019 June(B02) Math Question 42

Conics

- Circles

42. Which of the following equations is that of a circle that
is in the standard (x,y) coordinate plane, has center
(1,--4), and has a radius of 5 coordinate units?

Choice J

$$(x-h)^{2}+(y-k)^{2}=r^{2}$$ $$(h,k)=(1,-4)$$ $$r=5$$ $$\Rightarrow (x-1)^{2}+(y+4)^{2}=5^{2}$$


Choice D

$$5= 1\times5 \Rightarrow1,5$$ $$6=1\times6, 2\times3 …


Choice K

$$49^{a}= (7^{2})^{a}=7 $$ $$\Rightarrow a=\frac{1}{2}$$ …


Choice A

b02

$$(A)16\times16=256,\ 16\times4=64$$ $$(B)17\times17=289,\ 17\times4=68\gt64$$ $$(C)14\times16=224\lt256,\ …


2019 June(B02) Math Question 46

Graphing Properties

- Create Graphs or Figures

46. The difference  3/5 - (-1)/3 lies in which of the following
intervals graphed on the real number line?

Choice J

$$\frac{3}{5}-(\frac{-1}{3})$$ $$=\frac{9}{15}+\frac{5}{15}=\frac{14}{15}$$ $$\frac{14}{15}=1-\frac{1}{15} \Rightarrow \frac{14}{15} …


2019 June(B02) Math Question 47

Sequences and Patterns

- General Sequences

47. The recursive formula for a sequence is given below,
where an is the value of the nth term.
a1 = 10
an= a n-1 + 5
Which of the following equations is an explicit
formula for this sequence?

Choice B

$$a_{1}=10 $$ $$a_{n}=a_{n-1}+5 $$ $$\Rightarrow …


2019 June(B02) Math Question 48

Probability

- More than One Event

48. The probabilities that each of 2 independent events
will occur are given in the table below.

Choice F

$$P(A\ and\ B)=P(A)\times P(B) $$ …


2019 June(B02) Math Question 49

Quadratic Equations

- Factoring

49. What is the solution set of the equation
x4 +21x 2 -100=0?

Choice E

$$x^{4}+21x^{2}-100=0 $$ $$Let\ x^{2}=a, $$ …


2019 June(B02) Math Question 50

Trigonometry

- Pythagorean Theorem & sin cos tan

50. Jarrod is looking up at a hot-air balloon. The balloon is
currently at the maximum altitude during Tour C. The
angle of elevation from the horizon is 37°, as shown in
the figure below. Which of the following expressions
is closest to the distance, d feet, from Jarrod to the
basket?

Choice F

b02

$$\sin37°=\frac{1000}{d}$$ $$\Rightarrow d=\frac{1000}{\sin37°}$$


Choice E

$5000 in 1 day

Tickets …


2019 June(B02) Math Question 52

Word Problems

- Distance, Speed, Time

52. Jarrod went on Tour A, and his trip. covered a distance
of 6 miles. Bhumi went on Tour C, and her trip covered
a distance of 9 miles. Which of the following values is
the difference, in miles per hour, of the average speeds
of their balloons during their tours?

Choice G

$$Speed=\frac{Distance}{Time}$$ $$6\ miles\div \frac{45}{60}\ hours …


2019 June(B02) Math Question 53

Trigonometry

- Advanced sin, cos, tan functions

53. A forest fire is contained within a triangular region,
 which is shown below. The supervising firefighter plans
to fight the fire by positioning a firefighter about every
4 meters along the perimeter of the triangle. Among the
following, which expression best estimates the planned
number of firefighters along the perimeter? ·
(Note: The law of sines states that in every triangle, the
3 ratios of length of a side to the sine of the angle
opposite that side are equal.)

Choice A

b02

$$\frac{perimeter}{4}=\frac{130+x+y}{4}$$ $$\frac{x}{\sin47°}=\frac{130}{\sin91°}=\frac{y}{\sin42°}$$ $$x=\frac{130\sin47°}{\sin91°}$$ $$y=\frac{130\sin42°}{\sin91°}$$ $$\Rightarrow\frac{perimeter}{4}=\frac{130+\frac{130\sin47°}{\sin91°}+\frac{130\sin42°}{\sin91°}}{4}$$


Choice G

$$21=3\times7 $$ $$22=2\times11 $$ $$23=23 …


2019 June(B02) Math Question 55

Inequalities

- Intersection of Lines

55. ·The graph of y= x+1, y= x-3 and x2 + y2= 9 are
shown in the standard (x,y) coordinate plane below.
The, shaded region is the solution set to one o{ the
following systems of inequalities. Which system is it?

Choice A

b02


Choice J

b02

$$g(x)=-[f(x)-3]$$ $$g(4)=-[f(4)-3]$$ $$f(4)=-1$$ $$\Rightarrow g(4)=-[(-1)-3]=4$$


Choice D

$$\require{cancel} $$ $$\frac{a}{b}=\frac{6}{1}\Rightarrow a=6b $$ …


2019 June(B02) Math Question 58

Statistics

- Mean, Median

58. The frequency histogram below shows the distribution '
of the heights, in inches, of 11 basketball players.

Using the data from the frequency histogram, what is
the sum of the mean the median of this distribution?

Choice F

b02

$$Median=70 $$ $$Mean=\frac{66\times3+68\times2+70+73+74+76+77\times2}{11}=71 $$ $$Median+Mean=141$$


Choice A

Definition of y-intercept: the y-coordinate …


2019 June(B02) Math Question 60

Matrices

- Multiplication

60. What is the matrix product [2 4 ] [a b] ? _

Choice K

b02