ACT Form C01 Math Answer Explanation



2020 July(C01) Math Question 1

Angles

- Other Shapes

1. The parallelogram below has consecutive angles with measures x and 25. What is the value of x?

Choice E

c01

$$x+25°=180°$$ $$x=155°$$


2020 July(C01) Math Question 2

Statistics

- Mean, Median

2. A retail sales associate's daily commission during I week was $30 on Monday and Tuesday and $70 on Wednesday, Thursday, and Friday. What was the associate's average daily commission for these 5 days?

Choice H

$$\frac{(30+30+70+70+70)}{5}=\frac{270}{5}=54$$


Choice B

$$45=1\times3\times3\times5$$ $$50=1\times2\times5\times5$$ $$84=1\times2\times2\times3\times7$$ $$The\ greastest\ common\ factor=1 $$


Choice J

$$\Rightarrow 2x-24+x=36$$ $$\Rightarrow 3x=36+24$$ $$\Rightarrow 3x=60$$ $$\Rightarrow x=20$$


2020 July(C01) Math Question 5

Probability

- One Event

A bag contains exactly 22 solid-colored buttons:4 red
6 blue, and 12 white. What is the probability of randomly selecting I button that is NOT white?

Choice A

$$Total=22$$ $$Not\ white=22-12=10$$ $$P(Not\ white)=\frac{Not\ white}{Total}=\frac{10}{22}=\frac{5}{11}$$


Choice F

$$\frac{1}{2}:12=5:x$$ $$12\times5=\frac{1}{2}x$$ $$60=\frac{1}{2}x$$ $$x=120$$


2020 July(C01) Math Question 7

Sequences and Patterns

- General Sequences

7.Caden had exactly 45 plants to sell. After Day I of his sale, he had exactly 42 plants left. After Day 2, Caden had exactly 39 plants left. After Day 3. he has exactl
36 plants lcft. Assuming Caden will continue to sell plants at that daily rate, how many of these plants will he have left at the end of Day 6?

Choice B

$$45-42=3$$ $$42-39=3$$ $$39-36=3$$ $$45-3\times6=45-18=27$$


Choice H

$$\Rightarrow T=5+2M$$


2020 July(C01) Math Question 9

Quadratic Equations

- Factoring

9. What are the solutions to the quadratic equation
(2x+5)(3x-4)=0?

Choice C

$$\Rightarrow 2x+5=0\ or\ 3x-4=0$$ $$\Rightarrow x=-\frac{5}{2}\ or\ x=\frac{4}{3}$$


Choice J

$$8\times6=48$$ $$48\div2=24$$


Choice C

$$\frac{2}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{60}$$ $$=\frac{40}{60}+\frac{10}{60}+\frac{6}{60}+\frac{1}{60}$$ $$=\frac{40+10+6+1}{60}$$ $$=\frac{57}{60}$$ $$1-\frac{57}{60}=\frac{3}{60}=\frac{1}{20}$$


Choice H

$$7^{2}=49$$ $$8^{2}=64$$ $$\sqrt{61}\lt\sqrt{64}$$ $$\Rightarrow \sqrt{61}\lt8$$


2020 July(C01) Math Question 13

Trigonometry

- Pythagorean Theorem & sin cos tan

13. In AABC shown below. sin C== and the length of AB is 6 inches. What is the length in inches, of AC

Choice E

$$\sin C=\frac{2}{3}=\frac{6}{?}$$ $$?=9$$


2020 July(C01) Math Question 14

Sequences and Patterns

- Arithmetic Sequences

14. The table below shows the first 5 terms of an arithmetic sequence. Which of the following is a general expression for the nth term?

Choice H

$$n=1,\ 1=2n-1,\ 1=3n-2,\ 1=4n-3,\ 1=5n-4,\ 1=6n-5$$ $$n=2,\ 5\neq4-1,\ 5\neq6-2,\ 5=8-3,\ 5\neq10-4,\ 5\neq12-5$$ $$Only\ 4n-3\ is correct.$$


Choice D

c01

$$8+6+12+14+20+20=80$$


2020 July(C01) Math Question 16

Statistics

- Mean, Median

16. Manuel estimates that s of a pizza is left. Stephen estimates that, is left. They are going to compromise for a joint estimate by using the number halfway between their 2 estimates, What is their joint estimate?

Choice F

$$(\frac{2}{3}+\frac{3}{4})\div2$$ $$=(\frac{8}{12}+\frac{9}{12})\div2$$ $$=\frac{17}{12}\div2$$ $$=\frac{17}{24}$$


2020 July(C01) Math Question 17

Statistics

- Mean, Median

17. So far this basketball season. Sherita made 46 of her first 60 free throws, giving her a free-throw average of about 76.7%. What is the minimum number of free throws she would need to make from now on in order to have a free-throw average of at least 80?

Choice C

$$\frac{46+x}{60+x}=0.8$$ $$\Rightarrow 0.8\times(60+x)=46+x$$ $$\Rightarrow 48+0.8x=46+x$$ $$\Rightarrow 0.2x=2$$ $$\Rightarrow x=10$$


Choice G

$$f(g(x))=2(x^{2}+1)-1=2x^{2}+2-1=2x^{2}+1$$


2020 July(C01) Math Question 19

Statistics

- Mean, Median, Mode Combinations

19. Data Set A consists of the 8 numbers listed below Data Set B consists of the 8 numbers in A and a gth number, which is greater than 90. How will the mean and the median of B compare to the mean and the median of A?
62,76,76,80.82.87.9496

Choice A

Let the 9th number be 91 which is greater than 90.

$$The\ mean\ of\ Set\ A=(62+76+76+80+82+87+94+96)\div8=\frac{653}{8}=81.625$$ $$The\ median\ of\ Set\ A=(80+82)\div2=81$$ $$The\ mean\ of\ Set\ B=\frac{653+91}{9}\approx82.66$$ $$The\ median\ of\ Set\ B=82$$ $$\Rightarrow 82.66\gt 81.625$$ $$\Rightarrow 82\gt 81$$

The mean and the median of Set B will each be greater than the mean and the median of Set A.


2020 July(C01) Math Question 20

Word Problems

- Distance, Speed, Time

20. A truck traveling at 35 mph has a leaky radiator that is losing 4 fluid ounces per minute. How many miles will the truck travel before the radiator, which held
4so nuid ounces when it began to leak is empty?

Choice J

$$\frac{480}{4}=120\ mins=2\ hrs$$ $$\Rightarrow 35\times2=70 $$


2020 July(C01) Math Question 21

Equation of a Line

- Distance & Midpoint

21. In the standard (r, v) coordinate plane. what is the midpoint of the line segment that has endpoints(-5.8)
and(3.-1)?

Choice B

$$\Rightarrow \frac{((-5)+3),(8+(-1))}{2}=(-1,\frac{7}{2})$$


Choice H

When the X value is …


2020 July(C01) Math Question 23

Angles

- Other Shapes

23.In△ ABC shown below,m∠A=x°,m∠B=(2x)°, m∠C=(3x)°,AB= c inches,AC= b inches, and BC a inches. Which of the following inequalities correctly relates the side lengths of AABC Note:m∠ A denotes the measure of∠A. and AB denotes the length of AB. The triangle is NOT drawn to scale.)

Choice A

The larger the angle in …


Choice F

$$y=mx+b$$ $$5=m+b---(1)$$ $$7=17m+b---(2)$$ $$2=16m$$ $$\Rightarrow …


Choice C

$$The\ longer\ side=x+2$$ $$The\ Shorter\ …


2020 July(C01) Math Question 26

Word Problems

- 2 Variable Addition

Use the following information to answer questions 26-28
winter Fun Ski Resort sells only 2 types of tickets--adult and student. On Monday. the resort sold 200 tickets I ticket to cach skier. Of those tickets. 25 were sold to first-time skiers. When Alyssa skis the resort's main, run her elevation, E feet, at any point on the run is modeled by the equation E= 300,000 where t is the number of seconds after she begins skiing at the start of the main run.

Choice F

We assume that x adult …


Choice E

NOT first-time skiers=200-25=175

$$\frac{175}{200}=\frac{x}{8}$$ $$\Rightarrow …


Choice H

$$E=\frac{300,000}{t+100}$$ $$t=0 \Rightarrow E=\frac{300,000}{100}=3,000$$


2020 July(C01) Math Question 29

Perimeter, Area, Volume

- Mixed Shapes

29. One side of square ABCD has a length of 18 meters A certain rectangle whose area is equal to the area of ABCD has a width of 6 meters. What is the length. in meters, of the certain rectangle?

Choice E

Let another side of the …


2020 July(C01) Math Question 30

Matrices

- Addition & Subtraction

30. The 2 x 2 matrices A and B below are related to matrix C by the equation C=2A-3B. What is matrix C?

Choice F

$$C=2A-3B$$ $$\Rightarrow C=2\begin{bmatrix}3 & 5 …


2020 July(C01) Math Question 31

Statistics

- Mean, Median

31. Jen is doing an experiment to determine whether a high-protein food affects the ability of white mice to ind their way through a maze. The mice in the fi experimental group were given the high-protein food the mice in the control group were given regular food Jen then timed the mice as they found their way through the maze. The table below shows the results Mouse ExperimentalControl number group group I I min 46 see 2 min 13 sec
23452 min 2 sec I min 49 see
2 min 20 sec 2 min 28 sce I min SI
2 min 7 sec I min 4l sec I min 58 sec The average time the mice in the experimental group took to find their way through the maze was how many seconds less than the average time taken by the mice in the control group

Choice B

Experimental group:

1 min 46 …


2020 July(C01) Math Question 32

Probability

- Combinations

32. In the United States, phone numbers begin with a
3-digit area code. Now, there are restrictions on some of the digits, but in the future as more and more area codes are needed. the restrictions may need to be lifted. If, and when. there are no restrictions and eacl of the 3 digits can be any integer from 0 through 9.
how many area codes will be possible?

Choice K

$$0,1,2,3,4,5,6,7,8,9\rightarrow 10\ numbers$$ $$10\times10\times10=1000$$


2020 July(C01) Math Question 33

Trigonometry

- Graphing Trig Functions

33. The function y= 2 sin(&rx) is graphed in the standard
(r, y) coordinate plane below. How many x-intercepts does the graph of this function have on the interval
0<x<1

Choice C

c01


Choice J

$$3=1\times3$$ $$7=1\times7$$ $$9=1\times3\times3$$ $$\Rightarrow[3,7,9]=1\times3\times3\times7=63$$


2020 July(C01) Math Question 35

Angles

- Other Shapes

Use the following information to answer questions 35-38
In parallelogram ABCD shown below, the diagonals intersect at E,m∠BDC=42°,m∠BDA=71°,and AB=l4n-5 inches
(Note:The figure is not drawn to scale; mZPQR denotes the measure of∠PQR.)

Choice D

$$\because\ ABCD\ is\ a\ parallelogram.$$ …


2020 July(C01) Math Question 36

Perimeter, Area, Volume

- Other Shapes

36. Given that DC=【in+7)inches, what is the value of n

Choice K

$$\because\ ABCD\ is\ a\ parallelogram.$$ …


2020 July(C01) Math Question 37

Equation of a Line

- Distance & Midpoint

37. Suppose A and C are located at (2, 10) and(30.3)。
What are the coordinates of F2'-t, y) coordinate plane respectively, in the standard

Choice E

$$\frac{2+30}{2}=16$$ $$\frac{10+3}{2}=\frac{13}{2}$$ $$\Rightarrow\ E(16,\frac{13}{2})$$


Choice K

$$\because\ All\ three\ sides\ are\ …


2020 July(C01) Math Question 39

Quadratic Equations

- FOIL

39.Which of the following expressions is equal to
(a+Vbla-2Vb)for all positive real numbers a and b

Choice B

$$=(a+\sqrt{b})a-(a+\sqrt{b})(2\sqrt{b})$$ $$=a^{2}+a\sqrt{b}-2a\sqrt{b}-2b$$ $$=a^{2}-a\sqrt{b}-2b$$


Choice H

$$\pi(30)+\pi(50)=80\pi$$ $$\frac{80\pi}{75}=\frac{80(3.14)}{75}=3.3493$$


Choice B

$$(\frac{4}{5})^{-\frac{3}{2}}=(\frac{5}{4})^{\frac{3}{2}}=\frac{5^{\frac{3}{2}}}{4^{\frac{3}{2}}}$$ $$=\frac{5^{1}\times5^{\frac{1}{2}}}{(2^{2})^{\frac{3}{2}}}$$ $$=\frac{5^{1}\times5^{\frac{1}{2}}}{8}=\frac{5\sqrt{5}}{8}$$


2020 July(C01) Math Question 42

Logarithms

- Logarithm Expressions

42. What is the value of the positive real number x such that log,
=-2?

Choice F

$$\frac{1}{25}=5^{-2}$$ $$\log_{x}{5^{-2}}=-2$$ $$\Rightarrow x^{2}=5^{-2}$$ $$\Rightarrow …


Choice D

$$y=mx+b$$ $$(-4,-5)\rightarrow -5=-4m+b$$ $$(0,-3)\rightarrow -3=0m+b …


2020 July(C01) Math Question 44

Word Problems

- Distance, Speed, Time

44. Rya and Sampath start running laps from the same starting line at the same time and in the same direction on a certain indoor track Rya completes one lap in 16 seconds, and Sampath completes the same lap in 28 seconds. Both continue running at their same respective rates and in the same direction for 10 minutes. What is the fewest number of seconds after starting that Rya and Sampath will again be at their starting line at the same time?

Choice G

$$16=4\times4$$ $$28=4\times7$$ $$[16,28]=4\times4\times7=112$$


Choice D

c01

$$p(c)=1600c-4c^{2}=-4c^{2}+1600c$$ $$=-4c(c-400)$$ $$\Rightarrow c=0,\ c=400$$ …


Choice H

$$a\lt b\lt c\lt d,$$ $$Let\ …


2020 July(C01) Math Question 47

Perimeter, Area, Volume

- Mixed Shapes

47. The ratio of the perimeters of two squares is 2:3. If the area of the larger square is 324 square feet, what is the length, in feet, of the side of the smaller square?

Choice A

$$The\ ratio\ of\ the\ area\ …


2020 July(C01) Math Question 48

Inequalities

- Inequality Expressions

48. What is the set of all integer solutions for the inequality-l sx-V5<4?

Choice G

$$\sqrt{5}\approx2.23$$ $$-1\leq x-\sqrt{5}\leq 4$$ $$\Rightarrow …


Choice D

$$The\ maximum\ force=kSv^{2}$$ $$50=k(1)(40)^{2}=1600k$$ $$\Rightarrow …


Choice J

c01

$$4(10)+16(4)=40+64=104$$ $$104\times4=416$$ $$\frac{416}{12}=34.6$$ $$34.6\div0.6=57.7$$


2020 July(C01) Math Question 51

Trigonometry

- Pythagorean Theorem & sin cos tan

51. Radius OA of the circle shown below is perpendicular to AP. The circle intersects OP at B. The length of AP is 12 centimeters, and the measure of ZAPO is 20..
Which of the following values is closest to the length in centimeters, of Bp

Choice E

c01

$$\sin 20°=\frac{\overline{OA}}{\overline{OP}}=\frac{r}{r+x}=0.342$$ $$\Rightarrow(r+x)0.342=r$$ $$\Rightarrow(4.368+x)=\frac{4.368}{0.342}=12.77$$ $$\Rightarrow …


2020 July(C01) Math Question 52

Statistics

- Mean, Median

52. The average of 10 test scores is r. When the highest score and lowest score are removed from the 10 scores the average is y Which of the following is an expression for the average of the highest score and lowest score

Choice J

The average of 10 test …


Choice E

$$27^{n^{2}}=9^{5n-4}$$ $$\Rightarrow (3^{3})^{n^{2}}=(3^{2})^{5n-4}$$ $$\Rightarrow 3^{3(n^{2})}=3^{2(5n-4)}$$ …


2020 July(C01) Math Question 54

Probability

- One Event

54. Each face of 2 cubes with faces numbered from I through 6 has a I chance of landing up when the
2 cubes are tossed. What is the probability that the sum of the numbers on the faces landing up will be less than 6?

Choice G

$$Less\ than\ 6: $$ $$(1,1),(1,2),(1,3),(1,4),$$ …


2020 July(C01) Math Question 55

Word Problems

- Distance, Speed, Time

55. At 2:00 P.m., Louisa leaves Kansas City in her car traveling east on I-70 toward St Louis at an average spced of 68 mph. At precisely the same time, Antonio leaves St. Louis in his car traveling west on I-70
toward Kansas Cit y at an average speed of 57 mph The driving distance from St Louis to Kansas City is
10 miles. At what time, to the nearest minute, will they drive past each other on 1-70?

Choice D

$$68\times x=240-57\times x$$ $$68x=240-57x$$ $$125x=240$$ …


2020 July(C01) Math Question 56

Angles

- Collinear Points & Triangles

56. There are 10 points in a plane, and no 3 of the points are collinear. These 10 points, taken 2 points at a time determine how many distinct lines?

Choice J

c01


2020 July(C01) Math Question 57

Probability

- Factorials

57. The expression n! (read as n factorial) is defined as the product of all positive integers up to and including n, whenever n is a positive integer. For example.
4!=1·2·3:4. Whenever n is a positive integer which of the following is equivalent to-m31?

Choice A

$$\frac{(n+1)!6!}{n!3!}$$ $$=\frac{(n+1)!(6\cdot5\cdot4\cdot3\cdot2\cdot1)}{n!(3\cdot2\cdot1)}$$ $$=\frac{(n+1)!(6\cdot5\cdot4)}{n!}$$ $$=\frac{n!(n+1)(6\cdot5\cdot4)}{n!}$$ $$=(n+1)(6\cdot5\cdot4)$$ …


Choice G

I.

1,2,3,4

24,25,26,27

24, 25, …


2020 July(C01) Math Question 59

Quadratic Equations

- FOIL

59.When (x+ I) is expanded and like terms ar combined, what is the coefficient of r.

Choice E

c01

$$(x+1)^{4}$$ $$=(x+1)^{2}(x+1)^{2}$$ $$=(x^{2}+2x+1)(x^{2}+2x+1)$$ $$x^{2}(1)+2x(2x)+1(x^{2})$$ $$=2x^{2}+4x^{2}$$ …


2020 July(C01) Math Question 60

Trigonometry

- Advanced sin, cos, tan functions

60. A hill makes an angle of 20 with the horizontal AD as shown below. A taut guy wire, AB, extends from the base of the hill, point A, to point B on a vertical pole.
Point B is 25 ft directly above where the pole is inserted into the ground at point C. Given that the length of AC is 60 ft, which of the following expressions represents the length, in feet. of the guy wire?
(Note:For a triangle with sides of length a, b and c that are opposite angles∠A,∠B,and∠C respectively sin∠A n∠C d c2=a2+b2-2 ab cos∠C.)

Choice K

c01

$$c^{2}=60^{2}+25^{2}-2(60)(25)\cos 110°$$ $$c=\sqrt{60^{2}+25^{2}-2(60)(25)\cos 110°}$$