ACT Form B05 Math Answer Explanation



2018 December(B05) Math Question 1

Graphing Properties

- Interpret Graphs or Figures

1. Marine biologists collected a sample of 40 adult sea turtles from Hammerhead Bay and estimated the age in whole years, of each turtle. The histogram below shows the frequency of sea turtles for each of the estimated age groups. What is the maximum possible number of sea turtles that could have had an estimated age of 45 years?


Choice G

$$t=10 \Rightarrow 180(t-2)=180(10-2)=180\times8=1440$$


Choice D

$$(-8,-3)\Rightarrow (-8+8,-3+3)=(0,0)$$


Choice C

$$\frac{1}{3}\pi r^{2}h,\ r=3,\ h=6$$ $$\Rightarrow \frac{1}{3}\pi\times3^{2}\times6=18\pi$$


Choice G

$$(x^{4})^{6}=x^{4\times6}=x^{24}$$


2018 December(B05) Math Question 7

Word Problems

- 1 Variable Equations

7. Milo earns his regular pay of $12.00 per hour for up to 40 hours of work per week. For each hour over
40 hours of work per week, Milo earns 11 times his regular pay. How much does Milo earn in a week in which he works 45 hours?

Choice C

$$$12\times40+$12\times1\frac{1}{2}\times5$$ $$=$480+$90$$ $$=$570$$


2018 December(B05) Math Question 8

Matrices

- Addition & Subtraction

8. Which of the following matrices is equal to ?

Choice F

$$=\begin{bmatrix}5+(-6) & 7+3 \\(-4)+6 & 4+8 \end{bmatrix}$$ $$=\begin{bmatrix}-1 & 10 \\2 & 12 \end{bmatrix}$$


2018 December(B05) Math Question 9

Probability

- Combinations

9. Tomi has 6 pairs of shoes, 4 pairs of pants, and 6 shirts, which can be worn in any combination. He needs to choose a clothes combination to wear to the school dance. How many different combinations consisting of I of his 6 pairs of shoes, I of his 4 pairs of pants, and I of his 6 shirts are possible for Tomi to wear to the dance?

Choice E

$$6\times4\times6=144$$


2018 December(B05) Math Question 10

Conics

- Parabola

10. In the standard (x,y)coordinate plane below, the grap of the equation y=-2(x+1)2+8 intersects the x-axis at points (3, 0)and (a, 0)and has its vertex at point(-1. 8)。 What is the value of a?

Choice G

b05

$$(a,0)=(-1+2,0)=(1,0)$$ $$\Rightarrow a=1$$


Choice B

b05

$$3\times8\times7\times1\times2=336$$


2018 December(B05) Math Question 12

Equation of a Line

- Equations that Describe a Line

12. In the standard (x,y) coordinate plane, what is the slope of the line through(-7,3) and (2, 4)?

Choice J

$$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{4-3}{2-(-7)}=\frac{1}{9}$$


Choice A

$$60\times\frac{1}{2}=30$$ $$60\times\frac{1}{4}=15$$ $$60\times\frac{1}{6}=10$$ $$60-30-15-10=5$$


2018 December(B05) Math Question 14

Inequalities

- Inequality Expressions

14. What is the greatest integer solution to 6x-2 ≤ 11.2?

Choice J

$$6x-2\leq11.2$$ $$\Rightarrow 6x\leq13.2$$ $$\Rightarrow x\leq2.2$$

2 is the greatest solution.


2018 December(B05) Math Question 15

Word Problems

- Fixed Cost + Variable Cost

15. Classics Online charges a onetime registration fee of $17.50 and sells classical music downloads for $0. 70 per song. Ian has $50.00 that he will use to pay the registration fee and buy classical music from Classics Online. What is the maximum number of songs lan can buy?

Choice D

$$17.5+0.7x\leq50$$ $$\Rightarrow 0.7x\leq 32.5$$ $$\Rightarrow x\leq46.42...$$ $$Ans:\ x=46$$


Choice K

$$2:9=8:x$$ $$\Rightarrow x=\frac{8}{2}\times9=36$$


2018 December(B05) Math Question 17

Statistics

- Mean, Median

7. The mean age of the 5 people in a room is 30 years One of the 5 people, whose age is 50 years, leaves the room. What is the mean age of the 4 people remaining in the room?

Choice C

$$Total=30\times5=150$$ $$150-50=100$$ $$100\div4=25$$


2018 December(B05) Math Question 18

Angles

- Parallel Lines

18. In the figure below, B lies on AC, E lies on DF, AC‖DF,△ EBF is isosceles with BE ~ BF, and
∠ CBF measures 32°. What is the measure of ∠BED?

Choice K

b05

$$\because\overline{BE}\equiv\overline{BF}$$ $$\therefore ABE=\angle CBF=32°$$ $$\because\overline{AC}\parallel\overline{DF}$$ $$\therefore\angle ABE=\angle BEF=32°$$ $$\angle BED=180°-32°=148°$$


Choice C

$$x=-1, y=2, $$ $$x^{3}-2x^{2}y-4xy^{2}+8$$ $$=(-1)^{3}-2(-1)^{2}\times2-4(-1)\times2^{2}+8$$ $$=-1-2\times2+4\times4+8$$ $$=-1-4+16+8$$ $$=19$$


Choice H

$$A: 10\times5+12\times10+5\times6=200$$ $$B: 10\times5.5+12\times9.5+5\times5.75$$ $$=55+114+28.75$$ $$=197.75$$ $$200-197.75=2.25$$


2018 December(B05) Math Question 21

Equation of a Line

- Equations that Describe a Line

21. Line I is shown in the standard (x,y) coordinate plane below. Alicia drew line p with a slope that is 1/2 of the slope of line I and with a y-intercept that is 3 times the y-intercept of line l. One of the following equations represents line p. Which one?

Choice E

b05

$$Slope\ l=1$$ $$\Rightarrow Slope\ p=1\times\frac{1}{2}=\frac{1}{2}$$ …


2018 December(B05) Math Question 22

Trigonometry

- Pythagorean Theorem & sin cos tan

22. One angle measure and 2 side lengths, in inches; are given in the right scalene triangle below. In terms of a and b, what is sin theta?

Choice H

b05

$$\sin\theta=\frac{a}{\sqrt{a^{2}+b^{2}}}$$


Choice E

b05

$$50\times50=2500$$ $$10\times x=2500$$ $$\Rightarrow x=250$$


2018 December(B05) Math Question 24

Algebraic & Function Expressions

- Nested Functions

24. If f(x)=5x2-6x+1 and (x)=x-2, which of the following expressions represents (fg)(x)?

Choice J

$$(fg)(x)=(5x^{2}-6x+1)(x^{2}-2)$$ $$=(5x^{2}-6x+1)(x^{2})-(5x^{2}-6x+1)2$$ $$=(5x^{4}-6x^{3}+x^{2})-10x^{2}+12x-2$$ $$=5x^{4}-6x^{3}-9x^{2}+12x-2$$


2018 December(B05) Math Question 25

Probability

- One Event

25. A bag contains 12 red marbles, 14 yellow marbles, and
8 green marbles. How many additional red marbles must be added to the 34 marbles already in the bag so that the probability of randomly drawing a red marble is 3/5?

Choice B

$$Total\ marbles:\ 12+14+8=34$$ $$p(red)=\frac{12+x}{34+x}=\frac{3}{5}$$ $$5(12+x)=3(34+x)$$ …


Choice H

b05

$$Megan:\frac{120}{40}=3$$ $$Louisa:\frac{120}{70}=1\frac{5}{7}\approx1.71$$


Choice E

$$\require{cancel} $$ $$3x-(x+6)+8=2x+14$$ $$\Rightarrow3x-x-6+8=2x+14$$ $$\Rightarrow\cancel{2x}+2=\cancel{2x}+14$$ …


2018 December(B05) Math Question 28

Statistics

- Mean, Median, Mode Combinations

28. The first 7 terms in an arithmetic sequence are listed below. What is the difference between the mean and the median of the 7 terms?
1/2, 1, 3/2, 2, 5/2, 3, 7/2

Choice F

$$Mean:(\frac{1+2+3+4+5+6+7}{2})\div7$$ $$=\frac{[(1+7)7]\div2}{2}\div7$$ $$=\frac{56\div2}{2}\div7$$ $$=\frac{28}{2}\div7$$ $$=14\div7$$ …


2018 December(B05) Math Question 29

Number Types & Properties

- Complex Number

29. What is the product of the complex numbers (-2i+5) and (2i+5) ?

Choice C

$$(-2i+5)(2i+5)=(-2i)2i+(-2i)5+5(2i)+5(5)$$ $$=-4i^{2}-10i+10i+25=25-4i^{2}$$ $$Since\ i^{2}=-1,$$ $$\Rightarrow …


Choice H

$$3.8\times10^{5}+6.4\times10^{4}$$ $$=3.8\times10^{1}\times10^{4}+6.4\times10^{4}$$ $$=38\times10^{4}+6.4\times10^{4}$$ $$=(38+6.4)\times10^{4}$$ $$=44.4\times10^{4}$$ …


Choice B

b05


Choice F

b05

$$350\ miles\ of\ highway\ driving.$$ …


Choice B

$$$2400\div$4/gallon=600$$ $$600\times25=15000$$


Choice H

$$2400\times5=12000$$ $$12000+3500=15500$$ $$15500\div5=3100$$


2018 December(B05) Math Question 35

Word Problems

- Distance, Speed, Time

35. A certain race car has a maximum speed of 240 miles per hour. Which of the following is an expression for his maximum speed in feet per second?
(Note: 1mile= 5,280 feet)

Choice C

$$240\ miles=240\times5280\ feet$$ $$1\ hour=3600\ …


2018 December(B05) Math Question 36

Word Problems

- % Concentration

36. A chemist needs 1 ounce of element X. The only way which the chemist can get element X is to buy compound y, which contains 10% X Compound costs $2. 40 per pound(16 ounces)。 How much must the chemist pay in order to ensure that she receives 1 ounce of element X?

Choice H

$$\frac{$2.4}{16}=$0.15\ /ounces$$ $$$0.15\times10=$1.5$$


2018 December(B05) Math Question 37

Graphing Properties

- Interpret Graphs or Figures

37. Emi traveled to 3 locations during a workday Emi remained at each location a whole number of hours The graph below shows the relationship between time, in hours, into her workday and total distance, in kilometers, traveled. Which of the following values is closest to Emis average speed, in kilometers per hour, for the parts of the workday when she was traveling?

Choice D


Choice G

The denominator of any fraction …


2018 December(B05) Math Question 39

Statistics

- Mean, Median

39. A new band asked its audience to rate the bands performance on a scale from 1(poor) through
5(excellent). The table below gives the percentage of the audience that gave each of the ratings. To the nearest 0.1, what was the mean rating given by this audience?

Choice E

$$0.1\times3+0.7\times4+0.2\times5$$ $$=0.3+2.8+1=4.1$$


Choice K

Cannot be determined from the …


2018 December(B05) Math Question 41

Absolute Values

- Absolute Value Expressions

41. Let a and b represent real numbers with the property l a-b-1 l >0. Which of the following statements about a and b CANNOT be true?

Choice B

$$\mid a-b-1 \mid \gt0$$ $$\Rightarrow …


Choice G

$$2\pi r=2\times3.14\times5=31.4$$ $$150-31.4=118.6$$ $$118.6\div4=29.65$$


Choice D

b05

$$3\times4=12$$ $$3\times2=6$$ $$2\times4=8$$ $$(12+6+8)\times2=52$$ $$52\times$0.01=$0.52$$


2018 December(B05) Math Question 44

Word Problems

- 1 Variable Equations

44. The cost of processing cans and bottles at the recycling center is $0.03 per can and $0.02 per bottle. After paying the processing cost and the payment to customers, what is the recycling center's profit on the resale of 200 cans and 300 bottles to XYZ Inc.?

Choice F

$$Cans:0.05+0.03=$0.08$$ $$Bottles:0.1+0.02=$0.12$$ $$200($0.15-$0.08)+300($0.18-$0.12)$$ $$=200($0.07)+300($0.06)$$ $$=$14+$18=$32 …


2018 December(B05) Math Question 45

Word Problems

- Cost

45. To the nearest 1%,  the recycling center's payment to a customer for a bottle is what percent of the resale price of a bottle sold to XYZ Inc.

Choice B

$$\frac{0.1}{0.18}\approx. 0.56$$


2018 December(B05) Math Question 46

Word Problems

- 2 Variable Addition

46. In I shipment, the recycling center sold a total of
2,700 cans and bottles to XYZ Inc for $441. 00. How many bottles were in the shipment?

Choice F

The recycling center sold x …


2018 December(B05) Math Question 47

Quadratic Equations

- FOIL

47. Given constants c, d, m, and n such that x'+mx +chas factors of (x+2)and (x+ 4) and x+nx+d has factors of (x +3)and (x+7),  what is mn?

Choice D

$$x^{2}+mx+c=(x+2)(x+4)$$ $$=x^{2}+2x+4x+8$$ $$=x^{2}+6x+8$$ $$\Rightarrow m=6,\ …


2018 December(B05) Math Question 48

Trigonometry

- Graphing Trig Functions

48. For every angle theta, measured in radians, which of the following is equal to sin(2pi + theta)?

Choice G

b05


2018 December(B05) Math Question 49

Conics

- Circles

49.  A small circle and a large circle are tangent at T, as shown in the figure below. The center, O, of the large circle lies on the small circle. The diameter of the large circle is 12 cm. What is the ratio of the area of the small circle to the area of the large circle?

Choice A

b05

$$Small\ circle\ :\ Large\ circle$$ …


Choice J

$$\frac{2a}{b}+\frac{b}{2a}$$ $$=\frac{2a\times2a}{b\times2a}+\frac{b\times b}{2a\times b}$$ $$=\frac{4a^{2}+b^{2}}{2ab}$$


2018 December(B05) Math Question 51

Graphing Properties

- Vectors

51. The vector i represents 1 mile per hour east, and the vector j represents 1 mile per hour north. According to her GPS, at a particular instant, Tia is biking 30 west of north at 16 miles per hour. One of the following vectors represents Tia's velocity, in miles per hour, at that instant. Which one?

Choice B

b05


2018 December(B05) Math Question 52

Algebraic & Function Expressions

- Nested Functions

52. Let (f g)(x)=Vx+ 1-2 and g(x)=x+1.Which of the following expressions defines(g f)(x)?

Choice G

$$f(g(x))=\sqrt[3]{x+1}-2$$ $$g(x)=x+1$$ $$f(x)=\sqrt[3]{x}-2$$ $$g(f(x))=(\sqrt[3]{x}-2)+1=\sqrt[3]{x}-1$$


2018 December(B05) Math Question 53

Sequences and Patterns

- Logical Expressions

53. Consider sets A, B, C, and D such that B is a subset of A, C is a subset of B, and D is a subset of C. Whenever x is an element of B, x must be an element of:

Choice A

b05


Choice J

$$x^{\frac{4}{6}}\cdot x^{\frac{4}{3}}$$ $$=x^{\frac{4}{6}}\cdot x^{\frac{8}{6}}$$ $$=x^{\frac{4}{6}+\frac{8}{6}}$$ …


Choice A

$$\frac{3}{2}:\frac{2.5}{1.5}$$ $$=\frac{3}{2}:\frac{25}{15}$$ $$=\frac{3}{2}:\frac{5}{3}$$ $$=\frac{3}{2}\times\frac{3}{5}$$ $$=\frac{9}{10}$$


2018 December(B05) Math Question 56

Probability

- One Event

56. Each of 100 distinct playing cards is 1 of 5 solid colors and is numbered with 1 integer. There are 20 each of blue, red, yellow, green, and orange cards numbered
1-20. One of the 100 cards will be selected at random What is the probability that the selected card will be blue OR numbered 17

Choice J

$$P(Blue\ Cards)+P(17)-P(Blue\ 17)$$ $$=\frac{20}{100}+\frac{1}{20}-\frac{1}{100}$$ $$=\frac{20}{100}+\frac{5}{100}-\frac{1}{100}$$ …


Choice C

$$City's population:\ x$$ $$x\times1.2\times1.3\times0.8=1.248x$$ $$1.248x-x=0.248x$$


2018 December(B05) Math Question 58

Probability

- More than One Event

58. Four golfers will be randomly split into 2 groups of 2 for a tournament. If Jill and Ramona are among the 4. what is the probability that they will be paired together?

Choice K

b05

$$\Rightarrow\frac{1}{3}$$


2018 December(B05) Math Question 59

Trigonometry

- Pythagorean Theorem & sin cos tan

59. Jed is standing 10 meters from a maple tree that is
14 meters from a willow tree, as shown in the figure below, in which the measure of an angle is given Which of the following equations, when solved for x gives the distance, x meters, between Jed and the willow tree?
(Note:For a triangle with sides of length a, b, and c that are opposite angles∠A, ∠B and ∠C, respectively, c2=a2 +b2 -2ab cos∠C)

Choice A

b05

$$c^{2}=a^{2}+b^{2}-2ab\cos\angle C$$ $$\Rightarrow 10^{2}=x^{2}+14^{2}-2ab\cos\angle 34°$$


2018 December(B05) Math Question 60

Conics

- Ellipse

60. Suppose the equations (x-4)+(y-3)2=4 and
4+16-1 are graphed in the same standard (x,y) coordinate plane. How many points of intersection do these graphs share?

Choice F

b05