ACT Form A10 Math Answer Explanation


2017 December(A10) Math Question 1

Equation of a Line Practice

- One Event

1. A marble will be randomly selected from a bag of solid-colored marbles. The probability of selecting a red marble is 5/19. The probability of selecting a blue marble is 4/19. What is the probability of selecting a red marble or a blue marble?

Choice B

$$p=\frac{5}{19}+\frac{4}{19}=\frac{9}{19}$$

2017 December(A10) Math Question 2

Number Types & Properties Practice

- One Event

2. The graph below shows the number of students who were present on Thursday from each of the 5 groups in Ms. Meagan's class. What is the probability that a student selected at random from the class on Thursday is in Group 4 ?

Choice G

$$p=\frac{2}{8+12+6+2}=\frac{2}{28}=\frac{1}{14}$$

Choice A

$$k=\frac{7}{5}j+54=40\Rightarrow j=-10$$

2017 December(A10) Math Question 4

Angles Practice

- Absolute Value Expressions

4. What is |3 - x| when x = 8 ?

Choice H

$$|3-x|=|3-8|=|-5|=5$$

2017 December(A10) Math Question 5

Perimeter, Area, Volume Practice

- Temperature

5. When Tyrese fell asleep one night, the temperature was 24°F. When Tyrese awoke the next morning, the temperature was —16°F. Letting + denote a rise in temperature and — denote a drop in temperature, what was the change in temperature from the time Tyrese fell asleep until the time he awoke?

Choice A

Let d be the change in temperature: $$24+d=-16\Rightarrow d=-40$$

Choice G

$$1000+200\times 28-5400=1200$$

Choice E

a10

It can be proved that each side equals to 4. $$perimeter=4\times 6=24$$

Choice J

a10

$$S_1=22\times40=880\\$$ $$S_2=22\times28=616\\$$ $$\Rightarrow S_1+S_2=1496$$

Choice H

$$g(4)=\frac{4^2}{2}=\frac{16}{2}=8\\$$$$\Rightarrow f(g(4))=f(8)=3\times 8+7=31$$

Choice E

$$profit=sales \ revenue-cost \ of \ sales=2h-200>0$$

Choice G

$$3x+8y=5$$ $$\Rightarrow 8y=-3x+5 $$ $$\Rightarrow y=-\frac{3}{8}x+\frac{5}{8}$$

2017 December(A10) Math Question 13

Equation of a Line Practice

- Systems of Equations

13. Which of the following (x,y) pairs is the solution for the system of equations x + 2y = 2 and —2x + y = 16 ?

Choice A

$$x+2y=2\ (given)\Rightarrow x=2-2y \ (a)\\$$ $$-2x+y=16 \ (given) \ and \ (a) \Rightarrow -2(2-2y)+y=16\Rightarrow y=4 \\$$ $$\Rightarrow x= 2-2\times4=-6$$

2017 December(A10) Math Question 14

Word Problems Practice

- Proportions & Ratios

14. On a map, 1/4 inch represents 16 actual miles. Two towns that are 2 3/4 inches apart on this map are how many actual miles apart?

Choice K

$$\frac{x \ miles}{2\frac{3}{4}inches}=\frac{16\ miles}{\frac{1}{4}\ inches}\\$$ $$\Rightarrow \frac{x}{\frac{11}{4}}=\frac{16}{\frac{1}{4}}\\$$ $$\Rightarrow \frac{x}{11}=\frac{16}{1}\\$$ $$\Rightarrow x=16\times11=176 \ miles$$

2017 December(A10) Math Question 15

Word Problems Practice

- Addition & Subtraction

15. Which of the following matrices is equal to 4

Choice E

$$4\begin{bmatrix}-1&2\\0&-4\\ \end{bmatrix}=\begin{bmatrix}-1\times4&2\times4\\0\times4&-4\times4\\ \end{bmatrix}=\begin{bmatrix}-4&8\\0&-16\\ \end{bmatrix}$$

Choice J

$$tan \ A=\frac{15}{8}$$

Choice D

$$x=\frac{8 \ miles}{60 \ minutes}\times12 \ minutes=1\frac{3}{5}$$

2017 December(A10) Math Question 18

Statistics Practice

- Nested Functions

18. A function f(x) is defined as f(x) = —6x^2. What is f(-3) ?

Choice G

$$f(-3)=-6\cdot(-3)^2=-6\cdot9=-54$$

2017 December(A10) Math Question 19

Trigonometry Practice

- Collinear Points & Triangles

19. In the figure below, A is on BE and C is on BD. What is the measure of ZABC ?

2017 December(A10) Math Question 20

Perimeter, Area, Volume Practice

- General Sequences

20. Marcos programs his calculator to evaluate a linear function, but he doesn't say what the function is. When 5 is entered, the calculator displays the value 2. When 15 is entered, the calculator displays the value 6. Which of the following expressions explains what the calculator will display when any number, n, is entered?

Choice F

$$Let \ the \ linear \ function \ be \ y=kx+b \\$$ $$\Rightarrow 2=5k+b\\$$ $$6=15k+b\\$$ $$\Rightarrow k=\frac{2}{5},\ b=0\\$$ $$y=\frac{2}{5}x$$

2017 December(A10) Math Question 21

Perimeter, Area, Volume Practice

- General Sequences

21. On Friday, the temperature at 8:00 a.m. was 49°F and rose at a constant rate of °F per hour until noon. A cold front passed through at noon, and the temperature then fell at a constant rate of 1°F per hour. The temperature first fell below 49°F between:

Choice C

When at noon (12pm), the …

2017 December(A10) Math Question 22

Word Problems Practice

- Percents & Fractions

22. Letter grades in Hugo’s math class are based on the percent of the total possible points on 4 unit exams (each worth 100 points) and the final exam (worth 200 Points) and are assigned according to the chart below.

Choice K

Let the points on the …

Choice A

The area of 10 pins …

Choice G

$$\frac{65\times3\frac{1}{2}}{5\times3\frac{1}{2}}=\frac{13}{1} $$

2017 December(A10) Math Question 25

Graphing Properties Practice

- Mean, Median

25. What score will Halle need to earn in her 3rd game to have an average score of 172 for the 3 games?

Choice D

$$\frac{148+176+x}{3}=172 \\ $$ $$\Rightarrow x=192 …

Choice F

$$l=3w\\$$ $$ S=l\cdot w=3w^2=300 \\$$ …

2017 December(A10) Math Question 27

Word Problems Practice

- Other Shapes

27. A parallelogram has a perimeter of 96 inches, and 1 of its sides measures 16 inches. If it can be determined, what are the lengths, in inches, of the other 3 sides?

Choice C

$$16+16+2x=96 \\$$ $$ \Rightarrow x=32 …

Choice H

a10

8+6+12+8+6+12=52

2017 December(A10) Math Question 29

Trigonometry Practice

- More than One Event

29. At Central High School, 4 out of every 10 students ride the bus to and from school, and 3 out of every 8 who ride the bus are freshmen. If there are 2,500 students at Central, how many of the students are freshmen who ride the bus?

Choice A

$$2500\times \frac{4}{10}\times \frac{3}{8}=375$$

Choice H

$$\sin^2 \theta+\cos^2\theta=1 \\$$ $$\Rightarrow \cos\theta=\pm …

Choice C

$$f(t)=\frac {2}{t+1}=t \\ $$ $$\Rightarrow …

Choice J

a10

$$ BD=\sqrt{BC^2-CD^2}=\sqrt{13000^2-12000^2}=5000\\$$ $$\Rightarrow AB=AD+DB=9000+5000=14000$$

2017 December(A10) Math Question 33

Logarithms Practice

- Equations that Describe a Line

33. Let A, C, and D lie in the standard (x,y) coordinate plane such that A is at (0,0) and D is at (9,000, 0). Which of the following equations represents the line along which the airplane is flying?

Choice C

a10

2017 December(A10) Math Question 34

Probability Practice

- Distance, Speed, Time

34. Which of the following values is closest to the number of seconds it will take for the airplane to fly from C to the point directly above the radar station at A ?

Choice G

$$ t=\frac{9000}{300}=30$$

Choice A

B is apparently decreasing. C …

Choice G

$$\frac{3}{4}\times 4 \times w=50\% \times …

Choice E

$$ x+6=2(x+3)-x \\$$ $$ \Rightarrow …

Choice J

Let L be the length …

2017 December(A10) Math Question 39

Statistics Practice

- General Statistics

39. There are exactly 5 people in a bookstore at 12:00 p.m. Each person earns an annual income that is between $30,000 and $35,000. No one enters or leaves the bookstore until 12:15 p.m., when a professional athlete with an annual income of more than $1,000,000 enters the bookstore and joins the other 5 people. The mean, median, range, and standard deviation of the annual incomes of the 5 people in the bookstore at 12:00 p.m. are calculated and compared to, the same 4 statistics of the annual incomes of the 6 people in the bookstore at 12:15 p.m. If it can be determined, which of the 4 statistics changed the least?

Choice C

The range and the mean …

Choice K

Ruben completed $$1-\frac{2}{5}-\frac{1}{3}=\frac{4}{15}$$ The total …

Choice C

$$\frac{(2a^{-1}\sqrt{b})^4}{ab^{-3}}=\frac{2^4a^{-4}(\sqrt{b})^4}{ab^{-3}}=\frac{16b^2}{a^5b^{-3}}=\frac{16b^5}{a^5}$$

Choice G

There is only one correct …

Choice D

$$\frac{\frac{x}{3}+\frac{1}{2}}{\frac{2}{3}-\frac{1}{4}}=\frac{\frac{2x+3}{6}}{\frac{8-3}{12}}=\frac{\frac{2x+3}{6}}{\frac{5}{12}}=\frac{2x+3}{6}\cdot \frac{12}{5}=\frac{(2x+3)\cdot2}{5}=\frac{4x+6}{5}$$

2017 December(A10) Math Question 44

Equation of a Line Practice

- Combinations

$$\frac{\frac{x}{3}+\frac{1}{2}}{\frac{2}{3}-\frac{1}{4}}=\frac{\frac{2x+3}{6}}{\frac{8-3}{12}}=\frac{\frac{2x+3}{6}}{\frac{5}{12}}=\frac{2x+3}{6}\cdot \frac{12}{5}=\frac{(2x+3)\cdot2}{5}=\frac{4x+6}{5}$$

Choice K

10×10×10×26×26×26=17576000

2017 December(A10) Math Question 45

Conics Practice

- Graphing Trig Functions

45. The function y = f(x) is graphed in the standard (x,y) coordinate plane below.

Choice A

The function is moved to …

Choice K

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

2017 December(A10) Math Question 47

Probability Practice

- LCM, GCF, Prime Numbers

47. Which of the following lists those integer values of D for which the fraction lies between 5 and ?

Choice D

$$\frac{1}{5}=\frac{2}{10} \\$$ $$ \frac{1}{3}=\frac{2}{6}$$ The …

Choice F

The effect of a number …

Choice B

a10

$$ 39^2=38^2+37^2-2\times38\times37cos\ C \\$$ $$ …

Choice K

$$ x+2=\frac{4x+(x+s)}{5} \\$$ $$ \Rightarrow …

2017 December(A10) Math Question 51

Conics Practice

- Vertex Angles & Bisectors

51. The intersection of lines 1 and m forms the 4 angles LA, LB, LC, and LD. The measure of LB is 3f times the measure of LA. Which of the following values is closest to the measure of LA ?

Choice D

a10

As seen from the diagram, …

2017 December(A10) Math Question 52

Probability Practice

- Arithmetic Sequences

52. A sequence is defined for all positive integers by sr, = 2s(„__ ,) + n + 1 and si = 3. What is s4 ?

Choice J

$$ s_2=2s_1+3=2\times3+3=9 \\$$ $$ s_3=2s_2+4=2\times9+4=22 …

2017 December(A10) Math Question 53

Trigonometry Practice

- Absolute Value Expressions

53. If a is an integer less than -1, which of the following orders the expressions I a -a2, and - —a from least value to greatest value?

Choice E

Let a be -2: $$ …

Choice J

$$ Expected \ value \ …

2017 December(A10) Math Question 55

Perimeter, Area, Volume Practice

- General Sequences

55. Which of the following is equivalent to the sum of any 3 consecutive odd integers, x, y, and z, such that x < y < z ?

Choice B

Since x, y and z …

2017 December(A10) Math Question 56

Statistics Practice

- Mean, Median

56. The mean of the set of 5 numbers {42, 3, 11, 27, x} is 24, and the median of the set of 4 numbers {53, 8, 29, y) is 38. If it can be determined, which of the following values is equal to x-y?

Choice G

$$ 24=\frac{42+3+11+27+x}{5} \Rightarrow x=37 \\$$ …

Choice A

For A: $$ \\2\times(W+L)=48 \Rightarrow …

2017 December(A10) Math Question 58

Number Types & Properties Practice

- Circles

58. The equation (x — 7)2 + (y — 8)2 = 10 is that of a circle that lies in the standard (x,y) coordinate plane. One endpoint of a diameter of the circle has y-coordinate 11. What is the y-coordinate of the other endpoint of that diameter?

Choice J

a10

$$(x-7)^2+(y-8)^2=10 \\ $$ $$y=11 \\$$ …

2017 December(A10) Math Question 59

Perimeter, Area, Volume Practice

- Other Shapes

59. The plans for a diving pool call for a rectangular prism that has a length of 30 meters, a width of 25 meters, and a depth of 5 meters. If the plans are changed to increase both the length and the width of the pool by 10%, what will be the increase, to the nearest 1%, in the volume of the pool?

Choice D

$$V_0=30\times25\times5\\$$ $$ V_1=30\times 110\%\times 25\times110\%\times5=(110\%)^2\times30\times25\times5=(110\%)^2\times …

Choice K

a10