ACT Form B05 Math Answer Explanation


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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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

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

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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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

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

Choice E

b05

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

Choice H

b05

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

Choice E

b05

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

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 Practice

- 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

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

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 Practice

- 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 Practice

- 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 Practice

- % 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

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

Choice D

Choice G

The denominator of any fraction …

2018 December(B05) Math Question 39

Statistics Practice

- 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 …

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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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

Choice G

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

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 Practice

- 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 Practice

- 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

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

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 Practice

- 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