ACT Form C03 Math Answer Explanation



Choice C

$$f(x,y)=3x^{2}-4y$$ $$f(3,2)=3\times3^{2}-4\times2=27-8=19$$


2019 December(C03) Math Question 2

Angles

- Collinear Points & Triangles

2. In the figure below.∠ BAC measures35°,∠ABC measures 95, and points B, C, and D are collinear.
What is the measure of /ACD?

Choice H

c03

$$\angle ACB=180°-35°-95°=50°$$ $$\angle ACD=180°-50°=130°$$


2019 December(C03) Math Question 4

Word Problems

- Fixed Cost + Variable Cost

4. At a certain airline company. the cost to transfer mileage points from one persons account to another person's account is $0. 75 for every 100 mileage points transferred plus a onetime $20 processing fee. What is the cost to transfer 7,000 mileage points from one account to another at that airline company?

Choice H

$$\frac{7000}{100}\times$0.75+$20$$ $$=70\times$0.75+$20$$ $$=$52.5+$20$$ $$=$72.5$$


Choice E

$$f(x)=4x^{2}-11x$$ $$f(-5)=4\times(-5)^{2}-11\times(-5)$$ $$=4\times25-(-55)$$ $$=100+55=155$$


2019 December(C03) Math Question 6

Word Problems

- 1 Variable Equations

6. Taho earns his regular pay of $11 per hour for up to 40 hours of work per week. For each hour over
40 hours of work per week, Taho earns I5 times his regular pay. How much does Taho earn in a week in which he works 50 hours?

Choice G

Taho works 40 hours+10 hours.

$$ 40\ hours\Rightarrow $11\times 40=$440 $$ $$ 10\ hours\Rightarrow $11(1\frac{1}{2})\times 10=$16.5\times 10=$165 $$ $$ $440+$165=$605 $$


2019 December(C03) Math Question 7

Probability

- One Event

7. A science class has nios and 4 seniors. The teacher will randomly ct 2 students, one at a time to represent the class in a committee at the school Given that the first student selected is a junior, what is the probability that the second student selected will be a senior

Choice E

$$8\ juniors+4\ seniors\Rightarrow 7\ juniors+4\ seniors$$ $$P(Seniors)=\frac{4}{7+4}=\frac{4}{11}$$


2019 December(C03) Math Question 8

Word Problems

- Temperature

8. When Tyrone fell asleep one night, the ter perature was 24 F. When Tyrone awoke the next morning. the temperature was F. Letting denote a rise in temperature and- denote a drop in temperature. what was the change in temperature from the time Tyrone fell asleep until the time he awoke?

Choice F

$$-12°F-24°F=-36°F$$


Choice C

$$$35\times6+$0.425\times350 $$ $$=$210+$148.75 $$ $$=$358.75 $$


2019 December(C03) Math Question 10

Equation of a Line

- Equations that Describe a Line

10. In the standard (x, y)coordinate plane, what is the slope of the line through (-6, 4) and(1.3)?

Choice H

$$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{3-4}{1-(-6)}=\frac{-1}{7}$$


2019 December(C03) Math Question 11

Probability

- One Event

11. One morning at a coffee shop, each customer ordered either decaf or regular coffee, and each ordered it either with milk or without milk. The number of customers who ordered each type of coffee with or without milk is listed in the table below Order Decaf Regular Total With milk
820
Without milk
6106
Total
1818
A customer will be randomly selected from all
36 customers for a prize. What is the probability that the selected customer will have ordered a regular coffee without milk?

Choice B

$$P=\frac{Regular\ coffee\ without\ milk}{Total}$$ $$=\frac{10}{36}=\frac{5}{18}$$


2019 December(C03) Math Question 12

Inequalities

- Inequality Expressions

12. Which of the following inequalities describes the solution set for 3x-5< 2x+1?

Choice J

$$3x-5\lt2x+1$$ $$\Rightarrow 3x-5-2x+5\lt2x+1-2x+5$$ $$\Rightarrow x\lt6$$


Choice B

$$4(x+2)+3(2x-1)$$ $$=4x+8+6x-3$$ $$=10x+5$$ $$=5(2x+1)$$


Choice G

$$1.36\times 10^{4}\times\frac{4}{100}$$ $$=1.36\times4\times10^{2}$$ $$=136\times4$$ $$=544$$


Choice B

c03

$$7\times11\times5\times1\times2=770$$


2019 December(C03) Math Question 17

Equation of a Line

- Distance & Midpoint

17. In the standard (x, y) coordinate plane, what is the midpoint of the line segment that has endpoints (-6,9 ) and(2,5)?

Choice B

$$\frac{(-6,9)+(2,5)}{2}$$ $$=\frac{(-6+2,9+5)}{2}$$ $$=\frac{(-4,14)}{2}$$ $$=(-2,7)$$


2019 December(C03) Math Question 18

Quadratic Equations

- Factoring

18. What value of x satisfies the equation =2?

Choice K

$$\require{cancel} $$ $$\frac{x^{2}+2x}{x+2}=2$$ $$\Rightarrow x^{2}+2x=2(x+2)$$ $$\Rightarrow x\cancel{(x+2)}=2\cancel{(x+2)}$$ $$\Rightarrow x=2$$


2019 December(C03) Math Question 19

Statistics

- Mean, Median

Use the following information to answer questions 19-21
A large theater complex surveyed 5,000 adults. The results of the survey are shown in the tables below Age groups Number
21-302.7501,2241-5062551 or older400
Moviegoer category Number Very often
3083
Often
1.650
Sometimes 2.320
Rarely
00
Tickets are $9. 50 for all regular showings and $7.00 for matinee
19. Based on the survey results. what was the average number of moviegoers for each of the 4 categories?

Choice D

$$830+1650+2320+200=5000$$ $$5000\div4=1250$$


2019 December(C03) Math Question 20

Word Problems

- 2 Variable Addition

20. Suppose all the adults surveyed happened to attend I movie each in one particular week. The total amount spent on tickets by those surveyed in that week was S44, 000.00. How many adults attended matinees that week?

Choice G

$$x:\ attend\ regular\ showing$$ $$y:\ attend\ matinees$$ $$x+y=5000$$ $$\Rightarrow y=5000-x$$ $$9.5x+(5000-x)7=44000$$ $$\Rightarrow 9.5x+35000-7x=44000$$ $$\Rightarrow 2.5x=9000$$ $$\Rightarrow x=3600$$ $$\Rightarrow y=5000-3600=1400$$


Choice A

c03

$$\frac{2750}{5000}=0.55$$ $$\frac{1225}{5000}=0.245$$ $$\frac{625}{5000}=0.125$$ $$\frac{400}{5000}=0.08$$


Choice G

c03

$$9\times6=54$$ $$3\times7=21$$ $$54+21=75$$


2019 December(C03) Math Question 23

Angles

- Collinear Points & Triangles

23. In the figure below, E is on CA, and the measures of
∠ BED and∠ AEB are90°andl45°, respectively.Ifit can he determined. what is the measure of ∠CED?

Choice C

c03

$$\angle BEC=180°-\angle AEB=180°-145°=35°$$ $$\angle CED=90°-\angle …


2019 December(C03) Math Question 24

Trigonometry

- Pythagorean Theorem & sin cos tan

24. In the standard (r, y)coordinate plane, the graph of the function y=5 sin(x)-7 undergoes a single translation such that the equation of its image is y= 5 sin(x)-14
Which of the following describes this translation?

Choice G

$$(5\sin(x)-14)-(5\sin(x)-7)=-7$$


Choice C

$$(9^{\frac{1}{2}}+16^{\frac{1}{2}})^{2}$$ $$=(\sqrt{9}+\sqrt{16})^{2}$$ $$=(3+4)^{2}$$ $$=7^{2}$$ $$=49$$


2019 December(C03) Math Question 26

Trigonometry

- Pythagorean Theorem & sin cos tan

26. A right triangle is shown in the figure below. What is the value of sin ?

Choice F

c03

$$12^{2}+x^{2}=13^{2}$$ $$\Rightarrow 144+x^{2}=169$$ $$\Rightarrow x^{2}=169-144=25$$ …


2019 December(C03) Math Question 27

Perimeter, Area, Volume

- Mixed Shapes

27. A 6-inch-by-6-inch square grid shown below is divided into 36 squares, each with a side length of I inch Each vertex of the 2 shaded triangles lies at an intersection of 2 grid lines. What fractional part of the
6-inch-by-6-inch square is shaded?

Choice C

c03

$$\frac{2\times4}{2}=4$$ $$\frac{4\times6}{2}=12$$ $$4+12=16$$ $$6\times6=36$$ $$\Rightarrow\frac{16}{36}=\frac{4}{9}$$


Choice H

$$(4.25\times 10^{2x+4})(6\times 10^{7})=255$$ $$\Rightarrow 4.25\times6\times …


2019 December(C03) Math Question 29

Inequalities

- Inequality Expressions

29. Which of the following inequalities is true for all positive integers m?

Choice D

$$If\ m=2,$$ $$(A)\ 2\leq\frac{1}{2}$$ $$(B)\ …


2019 December(C03) Math Question 30

Perimeter, Area, Volume

- Other Shapes

30. A formula for the volume, v, of a right circular cylinder is v=rh, where r is the radius and h is the height. The cylindrical tank shown below has radius 5 meters and height 3 meters and is filled with water.

Given that the weight of I cubic meter of water is approximately 2,205 pounds. the weight, in pounds, of the water in the tank is:

Choice J

$$V=\pi r^{2}h$$ $$\Rightarrow V=3.14\times 5^{2}\times3=235.5$$ …


2019 December(C03) Math Question 31

Equation of a Line

- Distance & Midpoint

31. Graphed in the standard (x, y)coordinate plane below is a right triangle with vertices(0.0)。(-40. 0) and (0.30)。
What is the length. in otenuse of the triangle coordinate units. of the

Choice D

c03

$$30^{2}+40^{2}=50^{2}$$


Choice F

c03


2019 December(C03) Math Question 33

Conics

- Parabola

33. In the standard (x, y) coordinate plane, the graph of y-30(x+17)2-42 is a parabola. What are the coordinates of the vertex of the parabola?

Choice B

c03

$$y=30(x+17)^{2}-42$$ $$\Rightarrow (-17,-42)$$


Choice H

$$\Box ABCD=15\times15=225$$ $$225\div10=22.5$$


2019 December(C03) Math Question 35

Statistics

- Mean, Median

35. The average weight of 10 boys is 77.0 pounds, If the oungest boy is excluded, the average weight of the remaining boys is 78.0 pounds. What is the weight in pounds, of the youngest boy?

Choice B

$$77\times10=770$$ $$\frac{770-x}{9}=78$$ $$770-x=78\times9=702$$ $$770-702=x=68$$


2019 December(C03) Math Question 36

Logarithms

- Logarithm Expressions

36. The total amount of a certain substance present in a laboratory experiment is given by the formula A-A(2)。 where A is the total amount of the substance h hours after an initial amount (Ao) of the substance began accumulating. Which of the following expressions gives the number of hours it will take an initial amount of 10 grams of this substance to accumulate to 100 grams?

Choice J

$$A=A_{0}(2^{w5})$$ $$100=10(2^{w5})$$ $$10=(2^{w5})$$ $$\log 2^{10}=\log …


2019 December(C03) Math Question 37

Quadratic Equations

- FOIL

37. For all values of x greater than 3. which of the following expressions is equivalent to?

Choice E

$$\require{cancel} $$ $$\frac{x^{2}-x-6}{x^{2}-9}$$ $$=\frac{\cancel{(x-3)}(x+2)}{\cancel{(x-3)}(x+3)}$$ $$=\frac{x+2}{x+3}$$


Choice H

$$\frac{9\times12+4}{2}=\frac{112}{2}=56$$ $$12\times4=48$$ $$56=48+8$$


Choice A

$$\frac{1}{2}+\frac{1}{3}\cdot\frac{1}{4}\div\frac{1}{5}=\frac{x}{y}$$ $$\Rightarrow\frac{1}{2}+\frac{1}{3}\cdot\frac{1}{4}\cdot5=\frac{x}{y}$$ $$\Rightarrow\frac{1}{2}+\frac{5}{12}=\frac{x}{y}$$ $$\Rightarrow\frac{6}{12}+\frac{5}{12}=\frac{11}{12}$$ $$\Rightarrow11+12=23$$


2019 December(C03) Math Question 40

Sequences and Patterns

- General Sequences

40. What is the 358th digit after the decimal point in the repeating decimal 0.3178?

Choice H

c03

$$0.\overline{3178}=0.3178317831783178...$$ $$358\div4=89...2$$


2019 December(C03) Math Question 41

Statistics

- Mean, Median

41. To promote a new brand of shoes, a shoe store will run a promotion using a jar containing 3 red balls marked
“10%off,2 white balls marked“30%of;"and I green ball marked60% off. "Each customer will randomly select I ball from the jar to determine the discount that the customer will receive on any single pair of the new brand of shoes. Given that the new brand of shoes regularly costs $60 per pair. what is the average discount amount, in dollars, that the store can expect to give each customer due to this promotion?

Choice C

$$60\times0.1=6$$ $$60\times0.3=18$$ $$60\times0.6=36$$ $$\frac{6\times3+18\times2+36\times1}{6}$$ $$=\frac{18+36+36}{6}$$ …


Choice G

200/2400=8.33%


2019 December(C03) Math Question 43

Matrices

- Multiplication

43. In this park. the average number of gallons of water consumed per day by each elephant, lion, and giraffe is
50. 5 and 10 respectively. Which of the following matrix products yields the average total number of allons of water consumed per day by all the elephants, lions, and giraffes in the park?

Choice A

c03

$$600\times50+200\times5+400\times10$$


Choice J

$$10000(1+0.02)^{t}$$


2019 December(C03) Math Question 45

Word Problems

- Distance, Speed, Time

45. Anela and Jacob plan to attend a concert in Brady Anela will drive 375 km to Brady at a constant speed of 75 km/hr, stopping one time for a 30-minute break Jacob will start 600 km from Brady and will drive at a constant speed of 90 km/hr for 2 hours. He will take a I-hour break and then drive to brady at a constant speed of 70 km/hr. To the nearest 0 I hour, Jacob must leave how much earlier than Anela in order for them to arrive in Brady at the same time?

Choice D

$$Anela:$$ $$375\div75=5$$ $$\Rightarrow 5hrs\ 30\ …


Choice J

$$\frac{3x+5}{2x}-\frac{7x-3}{2x}$$ $$=\frac{-4x+8}{2x}$$ $$=\frac{-2x+4}{x}$$


Choice B

$$90\ ft\times30\ ft\ wide$$ $$1\ …


2019 December(C03) Math Question 48

Probability

- One Event

48. A rectangle, with its vertex coordinates labeled, is graphed in the standard (r,y) coordinate plane below. A lattice point is a point with coordinates that are both integers. A lattice point inside but NOT on the rectangle will be chosen at random. What is the probability that the sum of the t coordinate and the y-coordinate of the chosen lattice point will be odd?

Choice H

$$x+y$$ $$x:1,2,3,4,5$$ $$y:1,2,3$$ $$Total=5\times3=15$$ $$Even+Odd=Even\ …


2019 December(C03) Math Question 49

Sequences and Patterns

- Arithmetic Sequences

49.The nth term of an arithmetic progression is given by the formula on-a,+(n-1)d, where d is the common difference and a is the first term. If the third term of an arithmetic progression is 5 and the sixth term is what is the seventh term?

Choice A

$$a_{n}=a_{1}+(n-1)d$$ $$a_{3}=\frac{5}{2}=a_{1}+2d---(1)$$ $$a_{6}=\frac{1}{4}=a_{1}+5d---(2)$$ $$(2)-(1)$$ $$\Rightarrow …


2019 December(C03) Math Question 50

Probability

- One Event

50. The probability of Jamie being chosen to bat first in the lineup for his baseball team is - What are the odds in favor of Jamie being chosen to bat first?
(Note:The odds in favor of an event are defined as the ratio of the probability that the event will happen to the probability that the event will NOT happen.)

Choice F

$$\require{cancel} $$ $$1-\frac{1}{9}=\frac{8}{9}$$ $$\frac{\frac{1}{9}}{\frac{8}{9}}=\frac{1}{\cancel{9}}\times\frac{\cancel{9}}{8}=\frac{1}{8}$$


2019 December(C03) Math Question 51

Word Problems

- % Concentration

51. A 120-liter solution that is 5% salt is mixed with an
80-liter solution that is I5% salt. The combined solution is what percent salt?

Choice B

$$\frac{120\times0.05+80\times0.15}{200}$$ $$=\frac{6+12}{200}$$ $$=\frac{18}{200}$$ $$=\frac{9}{100}$$


2019 December(C03) Math Question 52

Trigonometry

- Pythagorean Theorem & sin cos tan

52.A 50-foot-long rectangular swimming pool with vertical sides is 3 feet deep at the shallow end and
10 feet deep at the deep end. The bottom of the pool slopes downward at a constant angle from horzontal alor th e length of the pool, Which of the following expressions gives this constant angle?
(Nore:For

Choice F

c03

$$\tan\theta=\frac{7}{50}$$ $$\theta=\tan^{-1}\frac{7}{50}$$


2019 December(C03) Math Question 53

Conics

- Hyperbola

53. A hyperbola that has vertices (1, 2)and (3, 2) and that passes through the origin is shown below in the which of the following equations? The hyperbola has tandard (x, y) coordinate plane?

Choice A

c03

$$\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1$$ $$\frac{(x-2)^{2}}{a^{2}}-\frac{(y-2)^{2}}{b^{2}}=1$$ $$(A)\frac{(x-2)^{2}}{1}-\frac{3(y-2)^{2}}{4}=1$$ $$(0,0)\Rightarrow \frac{(0-2)^{2}}{1}-\frac{3(0-2)^{2}}{4}=1$$ …


2019 December(C03) Math Question 54

Trigonometry

- Pythagorean Theorem & sin cos tan

54. As shown below, Alli walked her dog 250 feet due east from the entrance of a dog park to a trash can and then walked 700 feet in a straight line 25 north of east to a bench. Which of the following expressions is equal to the distance, in feet, between the entrance and the bench?

Choice K

c03

$$c^{2}=a^{2}+b^{2}-2ab\cos C$$ $$c^{2}=\sqrt{250^{2}+700^{2}-2\times250\times700\times\cos 155°}$$


2019 December(C03) Math Question 55

Absolute Values

- Absolute Value Expressions

55. For real numbers a, h, and c such that a>b>c and b>0, which of the statements below is(are)always true?

Choice A

$$a\gt b\gt c, b\gt0$$ $$(1)a=3,b=2,c=1$$ …


2019 December(C03) Math Question 56

Probability

- More than One Event

56. Kenji and Mary are members of a school committee that will be meeting this afternoon. The 6 members of the committee will be seated randomly around a circular table. What is the probability that Kenji and Mary will NOT sit next to each other at the meeting?

Choice J

c03

$$c_6^2=\frac{6!}{2!(6-2)!}=\frac{6\times5}{1\times2}=15$$ $$P(All)-P(Sit\ next\ to\ each\ …


Choice C

$$2^{88}=................6$$ $$2^{90}=2^{88}\times4 =................6\times4=................4$$


Choice F

c03

$$\frac{1}{2}\times\overline{RT}\times h$$ $$=\frac{1}{2}(c-a)(e-d)$$


2019 December(C03) Math Question 59

Number Types & Properties

- Complex Number

59. In the complex numbers, where i2=-I, what complex number x is a solution to the equation x( 2+30)=1?

Choice A

$$x(2+3i)=1$$ $$\star(a+bi)(a-bi)=a^{2}+b^{2}$$ $$x=\frac{1}{2+3i}\times\frac{2-3i}{2-3i}$$ $$=\frac{2-3i}{4+9}$$ $$=\frac{2-3i}{13}$$ …


Choice J

$$39=13\times h\Rightarrow h=3$$ $$(15\times12)\times3-\frac{1}{2}(5\times12)\times3$$ $$=540-90$$ …