ACT Form C02 Math Answer Explanation



Choice B

$$60=2\times2\times3\times5$$ $$84=2\times2\times3\times7$$ $$126=2\times3\times3\times7$$ $$(64,84,126)=2\times3=6$$


Choice J

$$=(4)(3)(n^{7})(n^{5})$$ $$=12(n^{7+5})$$ $$=12(n^{12})$$


2020 June(C02) Math Question 3

Word Problems

- Cost

3. Devon bought running shoes at a price that was t off the original price of $88. He paid a sales tax of 7% on the discounted price and gave the clerk four $20 bills How much change should he receive

Choice C

$$$88\times\frac{3}{4}=$66$$ $$($20\times4)-($66\times1.07)$$ $$=$80-$70.62$$ $$=$9.38$$


2020 June(C02) Math Question 4

Word Problems

- 2 Variable Addition

4. Brandon is having a bake sale at school to raise S140.00 to donate to the local animal shelter. He sells brownies for $1. 00 each and cookies for $0.50 each Given that Brandon sells 82 brownies, and all sales go to the donation, how many cookies does he need to sell to reach his goal?

Choice J

$$82\times$1+x\times$0.5=$140$$ $$\Rightarrow 0.5x=140-82=58$$ $$\Rightarrow x=116$$


2020 June(C02) Math Question 5

Angles

- Vertex Angles & Bisectors

5. In the figure below, AB is congruent to BC, and AE intersects BF at C. What is the measure of ZB?

Choice D

c02

$$\because \overline{AB}=\overline{BC}$$ $$\therefore \angle BAC=\angle BCA=38°$$ $$\Rightarrow \angle B=180°-38°-38°=104°$$


2020 June(C02) Math Question 6

Statistics

- Mean, Median

6. Patty, Carla, Shada, and Ling ran a race. The bar graph below gives each girl's running time, in seconds. How many of the girls ran the race in less time than the average of the 4 running times?

Choice G

$$(81+84+62+94)\div4=80.25$$

Only one girl ran the race in less time than the average of the 4 running times.


2020 June(C02) Math Question 7

Statistics

- Mean, Median

7. Between 9:00 a, m. and 10:20 a. m. 18 000 visitors entered the Family Fun Amusement Park, Between
9:00 a. m. and 10:20 a. m, an average of how many visitors per minute entered the park?

Choice D

From 9:00 a.m. to 10:20 a.m., a total of 80 minutes.

$$\frac{18000}{80}=225$$


2020 June(C02) Math Question 8

Graphing Properties

- Vectors

8. Given that u and v are vectors such that u=(-1. 3)and v=(5, 8), what is the component form of the vector u+v?

Choice H

$$\Rightarrow (-1+5,3+8)=(4,11)$$


Choice B

$$5-\frac{1}{2}-1\frac{3}{4}$$ $$=5-\frac{2}{4}-1\frac{3}{4}$$ $$=2\frac{3}{4}$$


Choice G

$$r=-3(4)+2$$ $$r=-12+2$$ $$r=-10$$


Choice A

$$f(x)=mx+b$$ $$6=m(8)+b---(1)$$ $$9=m(12)+b---(2)$$ $$(2)-(1)$$ $$\Rightarrow 3=4m$$ $$\Rightarrow m=\frac{3}{4}$$ $$f(x)=\frac{3}{4}x+b$$ $$6=6+b$$ $$b=0$$ $$f(n)=\frac{3}{4}n$$


Choice G

$$f(-5)=-4(-5)^{2}=-4(25)=-100$$


Choice B

$$=4(-1)^{3}-2(-1)^{2}$$ $$=-4-2$$ $$=-6$$


Choice H

c02

$$9\times6+4\times11$$ $$=54+44$$ $$=98$$


Choice C

c02

The slope is negative and intersects the Y axis at a position where Y is greater than 0, so the Y axis intercept is positive.


2020 June(C02) Math Question 16

Absolute Values

- Absolute Value Expressions

16. Given4x+2=-10,then|5-x2|=?

Choice G

$$4x+2=-10$$ $$\Rightarrow 4x=-12$$ $$\Rightarrow x=-3$$ …


2020 June(C02) Math Question 17

Trigonometry

- Graphing Trig Functions

17. The graph of y= 3 sin(x +2)is shown in the standard
(,y) coordinate plane below. What is the maximum value of this function?

Choice B

c02

According to the picture provided, it can be observed that the maximum value of this function, that is, the maximum value of Y is 3.


2020 June(C02) Math Question 18

Statistics

- Mean, Median

18. Renata took 9 quizzes in German class. Her scores, in order, were 6.7.7.6.8.7.8. 10, and 9. She discovered a scoring error on the 9th quiz, and her score on that quiz was corrected to 10. Which of the following measures of central tendency changed as a result of the correction?
I. Mean IL. Median III. Mode

Choice F

The mean is the average of the numbers.

The Median is the "middle" of a sorted list of numbers.

The mode is the number that is repeated more often than any other.

$$6,7,7,6,8,7,8,10,9$$

Rearrange the 9 scores in order.

$$\rightarrow 6,6,7,7,7,8,8,9,10$$ $$Mean=(6+6+7+7+7+8+8+9+10)\div9=68\div9\approx7.55$$ $$Median=7$$ $$Mode=7$$

After correcting the 9th score to 10.

$$\rightarrow 6,6,7,7,7,8,8,10,10$$ $$Mean=(68+1)\div9\approx7.66$$ $$Median=7$$ $$Mode=7$$

Only the mean is changed.


Choice E

Parallel lines have the same slope and will never intersect.

$$y=\frac{2}{3}x+4$$ $$\Rightarrow Slope=\frac{2}{3}$$

Only the E option has the same slope.


2020 June(C02) Math Question 20

Trigonometry

- Pythagorean Theorem & sin cos tan

20. For AABC shown below, what is the value of tan B?

Choice H

$$\tan \theta=\frac{Opposite}{Adjacent}$$ $$\tan B=\frac{1}{\sqrt{2}}$$


2020 June(C02) Math Question 21

Sequences and Patterns

- Logical Expressions

21. Given the true statement"If I live in Chicago, then I live in Illinois, which of the following statements must be true?

Choice E

c02

It's very clear after drawing …


Choice J

1 foot is equal to …


2020 June(C02) Math Question 23

Perimeter, Area, Volume

- Mixed Shapes

23. What is the minimum number of square floor tiles, each 9 inches on a side, that could be used to cover the floor of a rectangular hallway 15 feet long and 6 feet wide?

Choice E

1 foot is equal to …


2020 June(C02) Math Question 24

Trigonometry

- Pythagorean Theorem & sin cos tan

24. Graphed in the standard (x,y)coordinate plane below is line I and the circle with equation(x-2)+y=l Line I passes through 0(0,0)and is tangent to the circle at A. and B is the center of the circle. what is the asure of∠AOB?

Choice H

c02

$$2^{2}=1^{2}+x^{2}$$ $$\Rightarrow x=\sqrt{3}$$ $$1:\sqrt{3}:2$$

It …


2020 June(C02) Math Question 25

Perimeter, Area, Volume

- Mixed Shapes

25. One square has a side whose length is x centimeters.
and a second square has a side whose length is
(r-2)centimeters. What expression below represents the sum of the areas of the 2 sq wares, In square centimeters?

Choice E

$$x\times x+(x-2)(x-2)$$ $$=x^{2}+x^{2}-4x+4$$ $$=2x^{2}-4x+4$$


Choice H

$$\frac{5}{2}x+\frac{3}{4}y=-\frac{1}{2}$$ $$\Rightarrow 4\times\frac{5}{2}x+4\times\frac{3}{4}y=4\times(-\frac{1}{2})$$ $$\Rightarrow 10x+3y=-2$$ …


2020 June(C02) Math Question 27

Word Problems

- Fixed Cost + Variable Cost

Use the following information to answer questions 27-29
Kojo has an Internet site where his classmates can sell items in online auctions. For each item, a student pays Kojo a listing fee, based on the item's starting price, and a selling fee calculated as a percent of the selling price, as shown in the tables below Starting price Listing fee s0.01-s4.99$0.25
s5.00-s199950.5020.00-$49.9951.00
$50.00 and up$2.00
Selling price Selling fee S 0.$49.99 5% of selling price
$50.00 and up 3% of selling price
27. Lucie sold a jacket on Kojo's site. The starting price of the jacket was $6.25, and its selling price was $34.20
What is the sum of the listing fee and selling fee Lucie paid to sell the jacket?

Choice D

$$0.5+34.2(0.05)$$ $$=0.5+1.71$$ $$=2.21$$


2020 June(C02) Math Question 28

Word Problems

- Fixed Cost + Variable Cost

28. For the items his classmates listed on his site last Friday, Kojo was paid listing fees that totaled $5.75.
What is the maximum number of the items listed last Friday whose starting prices could have been in the range of s5.00-S19,99?

Choice F

$$5.75\div0.5=11.5$$


Choice D

$$x+y=116---(1)$$ $$0.03x+0.05y=4.34---(2)$$ $$(1)\times5-(2)\times100$$ $$(5x+5y)-(3x+5y)=580-434$$ $$2x=146$$ …


2020 June(C02) Math Question 30

Perimeter, Area, Volume

- Other Shapes

30. A formula for the volume, v, of a right circular cylinder is V=trh, where r is the radius and h is the height. The cylindrical tank shown below has radius 6 meters and height 5 meters and is filled with water.
Given that the weight of 1 cubic meter of water is approximately 2,205 pounds, the weight, in pounds, of the water in the tank:

Choice J

$$V=\pi r^{2}h$$ $$V=3.14(6)^{2}(5)=565.2$$ $$565.2\times2,205=1,246,266$$

It's …


2020 June(C02) Math Question 31

Word Problems

- 2 Variable Addition

31. Admission to a carnival is s4 for children and $6 for adults. A group of 21 people pays $90 for admission to the carnival. What is the ratio of the number of children to the number of adults in this group

Choice C

$$Children: x$$ $$Adults: y$$ $$x+y=21---(1)$$ …


Choice K

$$(\frac{2x^{3}y^{-5}z^{8}}{8x^{-2}y^{6}z^{3}})^{-2}$$ $$=(\frac{2}{8})(\frac{x^{3}y^{-5}z^{8}}{x^{-2}y^{6}z^{3}})^{-2}$$ $$=(\frac{1}{4})(x^{3-(-2)}y^{-5-6}z^{8-3})^{-2}$$ $$=(2^{-2})(x^{5}y^{-11}z^{5})^{-2}$$ $$=(2^{(-2)(-2)})(x^{5(-2)}y^{(-11)(-2)}z^{5(-2)})$$ …


Choice D

$$g(-1)=(-1)^{2}-1=1-1=0$$ $$f(g(-1))=f(0)=2(0)+3=3$$


2020 June(C02) Math Question 34

Probability

- Combinations

34. In a window display at a flower shop, there are 3 spots for I plant each. To fill these 3 spots, Adam has
7 plants to select from m, each of a different type Selecting from the 7 plants, Adam can make how many possible display arrangements with I plant in each spot?
(Note:The positions of the unselected plants do not matter.

Choice J

Choose a pot from the …


2020 June(C02) Math Question 35

Perimeter, Area, Volume

- Other Shapes

35. In quadrilateral ABCD shown below, BC AD BC= 20 inches, AD= 28 inches, and the distance between BC and AD is 7 inches. What is the area. in square inches, of quadrilateral ABCD?

Choice B

Area of a trapezoid is …


Choice H

First calculate the area of …


2020 June(C02) Math Question 37

Perimeter, Area, Volume

- Mixed Shapes

37. A rectangle that is c inches by d inches is in the interior of a rectangle that is a inches by b inches. as shown below. The area of the shaded region is what fraction of the area of the large rectangle, in terms of a, b.c. and d

Choice A

$$The\ large\ rectangle=ab$$ $$The\ shaded\ …


2020 June(C02) Math Question 38

Probability

- Combinations

38. A company prints contest codes on its fun-size bags of candy. Each 6-character code consists of the letter A followed by the letter H followed by 4 of the digits o through 9. The digits may repeat. Which of the following expressions gives the number of different
6-character codes that are possible?

Choice F

$$Code\rightarrow A-H-Number-Number-Number-Number$$ $$Number:(1,2,3,4,5,6,7,8,9,0)$$ $$\Rightarrow 1(1)(10)(10)(10)(10)$$


2020 June(C02) Math Question 39

Statistics

- Mean, Median

39. The mean of the daily high temperatures for a 5-day period in a certain city was recorded as being 4.0F was later determined that the high temperature for I of these 5 days was recorded incorrectly. If that day's high temperature was 2 F higher than originall recorded. what is the difference between. the incorrectly recorded mean and the correct mean?

Choice A

$$2\div5=0.4$$


2020 June(C02) Math Question 40

Probability

- One Event

40. A box contains 6 objects. Of those, 3 are disks (2 blue and I red) and 3 are triangles(I blue, I red, and I yellow)。 If the probability of drawing each object is the same, what is the probability that an object drawn from the box is a blue object or a triangle?

Choice K

The box cotains 6 objects. …


2020 June(C02) Math Question 41

Logarithms

- Logarithm Expressions

41. When log4 x=-3, what is x

Choice A

$$\log_{4}{x}=-3$$ $$\Rightarrow \log_{4}{4^{-3}}=-3$$ $$\Rightarrow x=4^{-3}=\frac{1}{64}$$


Choice J

c02

$$41-50\rightarrow 2$$ $$51-60\rightarrow 3$$ $$61-70\rightarrow …


Choice D

c02

The Median is the "middle" …


Choice F

$$4\times2=8$$ $$8-3=5$$


2020 June(C02) Math Question 45

Absolute Values

- Absolute Value Expressions

45. The solution set of the equation Ix-l|=x-I is the set of all values of x such that

Choice B

$$x-1=x-1$$ $$-(x-1)=x-1$$ $$-x+1=x-1$$ $$\Rightarrow x=1$$ …


2020 June(C02) Math Question 46

Trigonometry

- Advanced sin, cos, tan functions

46. A surveyor needs to find the length from point A to point C across a lake as shown in the figure below. The measurements of which of the following angles and side lengths are sufficient for the surveyor to determine the length of AC using only the law of sines?

Choice F

We require that the value …


2020 June(C02) Math Question 47

Conics

- Ellipse

47. Which of the following equations determines the ellipse shown in the standard (x, y) coordinate plane below?

Choice C

c02

$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ $$\Rightarrow \frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=1$$ $$\Rightarrow 9x^{2}+52y^{2}=1\times9\times25$$ …


2020 June(C02) Math Question 48

Quadratic Equations

- Factoring

48. Given that x=-2 is a solution to x'+bx-6=0,which of the following polynomials is a factor of x+bx-6?

Choice F

$$(-2)^{2}+(-2)b-6=0$$ $$\Rightarrow 4-2b-6=0$$ $$\Rightarrow -2=2b$$ …


2020 June(C02) Math Question 49

Perimeter, Area, Volume

- Mixed Shapes

49. Square ABCE, shown below, has a side length of
10 inches. Point D is the midpoint of CE, and F is the midpoint of AE. What is the ratio of the area of ADEF shown shaded, to the area of pentagon ABCDF?

Choice C

c02

$$\Rightarrow 1:7$$


2020 June(C02) Math Question 50

Absolute Values

- Absolute Value Expressions

50. For some positive integer k, the sum of the absolute values of all the integers from -k through k is 12. What is the value of k?

Choice G

$$12\div2=6$$ $$1+2+3=6$$ $$\mid-3\mid+\mid-2\mid+\mid-1\mid+\mid0\mid+\mid1\mid+\mid2\mid+ \mid3\mid$$ $$=3+2+1+0+1+2+3$$ …


Choice C

c02

$$f(g(x))$$ $$=\sqrt{(x-3)^{3}}+1$$

If x < 3, you would be taking the square root of a negative number, so x ≥ 3.


2020 June(C02) Math Question 52

Equation of a Line

- Distance & Midpoint

52. A highway engineer is using a road map to lay out a detour for the westbound lane of a section of highway that, on the map, is a straight line going east and west On the map, the detour goes 4 miles straight north, I mile straight west, 2 miles straight north, 6 miles straight west, 3 miles straight south, I mile straight east, and finally 3 miles straight south, back to the highway. According to the map, how many more miles ll a westbound driver travel by taking the detour than he would if he could stay on the highway?

Choice G

$$(0,0)$$ $$\rightarrow(0+4,0)=(4,0)$$ $$\rightarrow(4,0-1)=(4,-1)$$ $$\rightarrow(4+2,-1)=(6,-1)$$ $$\rightarrow(6,-1-6)=(6,-7)$$ …


Choice B

A Rational Number can be …


2020 June(C02) Math Question 54

Number Types & Properties

- Order of Values

54. For all real values of x such that-I<x<0, which of the following expressions has the greatest value?

Choice K

Let X be -0.5 and …


2020 June(C02) Math Question 55

Trigonometry

- Pythagorean Theorem & sin cos tan

55. A 6-foot awning that extended 2 feet horizontally from a vertical building at 8:00 a m, was adjusted to extend
3 feet horizontally from the building at 12:00 p. m,as shown below. Which of the following expressions equals the positive difference in the measures of the angle between the awning and the building at 8:00 a.m and at 12:00 p.m.?

Choice E

c02

$$8:00\ a.m.\ \rightarrow \sin \theta=\frac{2}{6}$$ …


2020 June(C02) Math Question 56

Number Types & Properties

- Complex Number

56. In the complex numbers, where i=-1, 33+1=?

Choice K

$$\frac{2-i}{-3+i}$$ $$=\frac{(2-i)(-3-i)}{(-3+i)(-3-i)}$$ $$=\frac{-6-2i+3i+i^{2}}{(-3)^{2}-(i)^{2}}$$ $$=\frac{-6+i-1}{9-(-1)}$$ $$=\frac{(-7+i)}{10}$$ …


2020 June(C02) Math Question 57

Angles

- Other Shapes

57. The degree measures of the interior angles of a certain pentagon are in the ratio 2:3:4:4:5. What is the measure of the largest interior angle of this pentagon?

Choice E

The sum of the internal …


2020 June(C02) Math Question 58

Absolute Values

- Graphing Absolute Values

58. The function y=f(r) is graphed in the standard (x,y)
coordinate plane below. The domain of f is the set of all positive real numbers, One of the following graphs is the graph of y=-lfx)I. Which one is it?

Choice K

c02

The absolute value sign is …


2020 June(C02) Math Question 59

Word Problems

- Cost

59. The sale price of a jacket is 10%o off the original price The clearance price of the jacket is 30% off the sale price. The clearance price is what percent off the original price?

Choice D

$$The\ clearance\ price$$ $$=The\ sale\ …


2020 June(C02) Math Question 60

Perimeter, Area, Volume

- Mixed Shapes

60. How long, in centimeters, is I side of a square whose perimeter is equal to the circumference of a circle with a radius of 2 centimeters?

Choice F

$$2\pi r=2 \pi\times2=4\pi$$ $$\frac{4\pi}{4}=\pi$$