ACT Form Z15 Math Answer Explanation



Choice A

(r+b-g)(b+g)

=(4+2-(-5))(2+(-5))

=(4+2+5)(2-5)

=11(-3)

=-33


2019 April(Z15) Math Question 2

Word Problems

- Fixed Cost + Variable Cost

2. Jeralyn purchases 1 box of granola bars and 6 boxes of chocolate bars for a total price of $22.00. The price of each box of granola bars is $2.50, and the price of each box of chocolate bars is n dollars. which of the following equations models Jeralyn's purchase?

Choice J

$$2.5\times1+6\times n=22$$


Choice B

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$$\angle CAB=\angle CBA=48°$$ $$\angle ACB=180°-48°-48°=84°$$


Choice H

$$20\div4=5$$ $$5\times5=25$$


2019 April(Z15) Math Question 5

Probability

- One Event

5. A bag contains exactly 21 solid-colored buttons ns:3 red 6 blue, and 12 white. What is the probability of randomly selecting I button that is NoT white?

Choice C

$$P(Not\ white)$$ $$=1-P(White)$$ $$=1-\frac{12}{21}$$ $$=\frac{9}{21}$$ $$=\frac{3}{7}$$


2019 April(Z15) Math Question 6

Absolute Values

- Absolute Value Expressions

6. What is the value of 1-7-17-29?

Choice G

$$=7-22$$ $$=-15$$


2019 April(Z15) Math Question 7

Word Problems

- Fixed Cost + Variable Cost

7. A store's revenue is the amount of money received for goods sold. A store's cost is the amount of money the store pays for the goods plus all the store's operating costs like rent, utilities, wages, etc. A store' s net profit s the difference between revenue and cost. During I month, a grocery store paid $30,000 for goods that were sold for $39, 500. With operating costs as shown below, what was the stores net profit for that month?
Operating costs Amount Rent, utilities, and telephone$2,500
Taxes and insurance s370
Interest on business loan
400
Grocers own wages
1,500
Wages for part-time help s630
Miscellaneous
400

Choice A

$$ $39500-$30000=$9500 $$ $$ $2500+$370+$400+$4500+$630+$400=$8800 $$ $$ $9500-$8800=$700 $$


2019 April(Z15) Math Question 8

Word Problems

- Distance, Speed, Time

8. When Jorge began a driving trip, his car's odometer read 42 miles. After Jorge drove for 3 hours, the odometer read 165 miles. Which of the following values is closest to Jorge's average driving speed, in miles per hour, during those 3 hours?

Choice G

$$(165-42)\div3=123\div3=41$$


Choice A

$$5-\frac{3}{4}-2\frac{1}{2}$$ $$=\frac{20}{4}-\frac{3}{4}-\frac{10}{4}$$ $$=\frac{7}{4}$$ $$=1\frac{3}{4}$$


Choice F

$$\frac{25^{x}}{5^{2}}=5^{4} $$ $$\Rightarrow \frac{5^{2x}}{5^{2}}=5^{4} $$ $$\Rightarrow 2x-2=4 $$ $$\Rightarrow 2x=6 $$ $$\Rightarrow x=3 $$


Choice B

$$f(x)=5\cdot 2^{x}$$ $$g(x)=10x$$ $$f(3)=5\cdot 2^{3}=5\cdot8=40$$ $$g(3)=10\cdot3=30$$ $$\Rightarrow f(3)-g(3)=40-30=10$$


2019 April(Z15) Math Question 12

Word Problems

- 2 Variable Addition

12. Tanisha, a manager at a state park, counted the money in the cash register at the end of her shift, and then she deposited the money in the bank. When she went back to her office, she accidentally shredded the deposis ip. She remembered that there were only $5 and $10 bills. She also recalled that there were 27 bills totaling S205. How many $5 bills were in Tanisha's cash register at the end of her shift?

Choice F

$$$5\ bill:x$$ $$$10\ bill:y$$ $$5x+10y=205---(1)$$ $$x+y=27---(2)$$ $$(1)-(2)\times5$$ $$\Rightarrow (5x+10y)-(5x+5y)=205-27\times5$$ $$\Rightarrow 5y=70$$ $$\Rightarrow y=14$$ $$\Rightarrow x=13$$


Choice A

$$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{3-2}{8-0}=\frac{1}{8}$$


2019 April(Z15) Math Question 14

Trigonometry

- Pythagorean Theorem & sin cos tan

14. For an angle with measure a in a right triangle, sin a-l and tan a-B. What is the value of cos a

Choice H

$$\sin^{2}(\theta)+\cos^{2}(\theta)=1$$ $$\Rightarrow \sin^{2}(\alpha)+\cos^{2}(\alpha)=1$$ $$\Rightarrow (\frac{15}{17})^{2}+\cos^{2}(\alpha)=1$$ $$\Rightarrow \frac{225}{289}+\cos^{2}(\alpha)=1$$ $$\Rightarrow \cos^{2}(\alpha)=\frac{64}{289}$$ $$\Rightarrow \cos(\alpha)=\sqrt{\frac{64}{289}}=\frac{8}{17}$$


2019 April(Z15) Math Question 16

Angles

- Vertex Angles & Bisectors

16. In the figure below,  xw intersects vz at y,  the measure of ∠XYV  is(3x+5)°,  and the measure of
∠ZYW is(4x-6)°.  What is the measure of ∠XYZ?

Choice J

$$(3x+5)=(4x-6)$$ $$\Rightarrow x=5+6=11$$ $$\Rightarrow (3x+5)=(4x-6)=38$$ $$\angle XYZ=180°-38°=142°$$


2019 April(Z15) Math Question 17

Probability

- One Event

17. The table below shows the letter grades 60 students earned on the final exam in American Literature. The highest possible grade is A:the lowest possible grade is F.
A student from this group will be chosen at random What is the probablity that the student's final exam grade is C or higher?

Choice E

$$Grade\ C\ or\ higher:10+26+18=54$$ $$\Rightarrow\frac{Grade\ C\ or\ higher}{Total}=\frac{54}{60}=\frac{9}{10}=0.9$$


2019 April(Z15) Math Question 18

Quadratic Equations

- Factoring

18. For what value of n does the quadratic equation x-2r+n=0 bave solutions of x=4 and x=-2?

Choice F

$$x^{2}-2x+n=0$$ $$x=4,$$ $$\Rightarrow 4^{2}-2\cdot4+n=0$$ $$\Rightarrow 16-8+n=0$$ $$\Rightarrow n=-8$$ $$x=-2,$$ $$\Rightarrow (-2)^{2}-2(-2)+n=0$$ $$\Rightarrow 4+4+n=0$$ $$\Rightarrow n=-8$$


Choice E

$$Circumference = π × diameter = 2 × π × radius.$$ $$2\pi r=12 \pi $$ $$\Rightarrow r=6 $$ $$Area=\pi r^{2}=\pi 6^{2}=36\pi $$


2019 April(Z15) Math Question 20

Probability

- Combinations

20. The app plication for a license plate states that the license late number has 3 letters followed by a 3-digit number, for exampie, AEE123 The letters O and I and the digit 0 cannot be part of the license plate number.
Any of the other letters and digits may be used up to 3 times. Which of the following expressions represents how many different license plate numbers are possible?

Choice H

$$26-2=24$$ $$10-1=9$$ $$\Rightarrow24\times24\times24\times9\times9\times9$$


Choice E

$$3(a+b)-2(a-5b)$$ $$=3a+3b-2a+10b$$ $$=a+13b$$


2019 April(Z15) Math Question 22

Perimeter, Area, Volume

- Squares in a Shape

22. In the figure below, all of the small squares are equal in area, and the area of rectangle ABCD is I square unit. Which of the following expressions represents the area, In square units, of the shaded region?

Choice K

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$$\Rightarrow\frac{8}{9}\times\frac{6}{7}$$


2019 April(Z15) Math Question 23

Angles

- Collinear Points & Triangles

23. Marie is building a kite. In a drawing of her kite.
shown below BC, AD=DC, the measure of ∠ABC is80°, and the measure of ∠ADC is50°.What is the measure of ∠BAD?

Choice D

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$$\overline{AB}=\overline{BC}$$ $$\angle CBD=\angle ABD=\frac{1}{2}\angle ABC=40°$$ …


Choice K

$$\because\triangle ABC\sim\triangle DEF$$ $$\therefore 2:4=3:x$$ …


Choice C

$$\because1\ foot =12\ inches$$ $$\therefore …


Choice F

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$$\Rightarrow 10:5=3:x$$ $$x=3\times\frac{1}{2}=1.5$$


Choice B

$$t=10,\ v=5$$ $$t=20,\ v=3$$ $$\frac{3-5}{20-10}=\frac{-2}{10}=-\frac{1}{5}$$


2019 April(Z15) Math Question 28

Word Problems

- Distance, Speed, Time

28. Calen started his bike ride earlier than Shasta. During the first 15 seconds of Shasta s ride, Calen was traveling at a constant speed equal to 5 of Shasta's maximum speed during that same time period. How far, in meters, did Calen travel during the first 15 seconds of Shasta's ride?

Choice H

$$Shasta's\ maximum\ speed:\ 5$$ $$\frac{1}{2}\times5\times15=37.5$$


2019 April(Z15) Math Question 29

Graphing Properties

- Create Graphs or Figures

29. Which of the following graphs best represents th portion of Shasta's ride beginning at t=20 seconds?

Choice D

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Choice J

$$\frac{2}{3}-\frac{5}{6}(\frac{2}{5}+\frac{1}{10})$$ $$=\frac{2}{3}-\frac{5}{6}(\frac{4}{10}+\frac{1}{10})$$ $$=\frac{2}{3}-\frac{5}{6}\times\frac{5}{10}$$ $$=\frac{2}{3}-\frac{5}{12}$$ $$=\frac{8}{12}-\frac{5}{12}$$ …


Choice E

$$(x+2)(2x+7)$$ $$=2x^{2}+7x+4x+14$$ $$=2x^{2}+11x+14$$


2019 April(Z15) Math Question 32

Number Types & Properties

- Order of Values

32. Which one of the following inequalities is true?

Choice H

$$F.\ 2\lt\sqrt{3}\lt4 $$ $$\Rightarrow 2=\sqrt{4} …


2019 April(Z15) Math Question 33

Probability

- More than One Event

33. Two fair coins are repeatedly tossed simultaneously.
What is the probability that both coins land heads up on the 36th toss?

Choice E

$$P(Heads\ up)=\frac{1}{2}$$ $$\Rightarrow \frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$$


2019 April(Z15) Math Question 34

Statistics

- Mean, Median

34. Suppose a student's course grade is determined solely by that students scores on 8 tests, which are worth 100 points each. If Bane has an average of exactly 88 points on the first 6 tests, how many points must he average on the last 2 tests to earn exactly a 90-point course grade?

Choice G

$$\frac{88\times6+2x}{8}=90$$ $$\Rightarrow 528+2x =720$$ $$\Rightarrow …


Choice C

$$A.\ (62+(-\frac{1}{65}))\div2=31-\frac{1}{130}$$ $$B.\ 62\div(-\frac{1}{65})=-62\times65=4030$$ $$\checkmark …


Choice F

$$A:\left\{0,1,2,3\right\}$$ $$B:\left\{1,3,5,7\right\}$$ $$F.\ f(x)=2x+1$$ $$f(0)=2\times0+1=1$$ …


Choice D

$$x=\frac{3}{4}+\frac{4}{3}=\frac{9+16}{12}=\frac{25}{12}$$ $$y=\frac{2}{3}+\frac{3}{2}=\frac{4+9}{6}=\frac{16}{6}=\frac{32}{12}$$ $$z=1+1=\frac{24}{12}$$ $$\Rightarrow \frac{32}{12}\gt\frac{25}{12}\gt\frac{24}{12}$$ …


2019 April(Z15) Math Question 38

Trigonometry

- Pythagorean Theorem & sin cos tan

38. Isosceles triangle AABC has an altitude of h inches. a base of 36 inches, and 2 base angles measuring 76 each, as shown in the figure below. What is the value of h?

Choice G

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$$tan\ 76°=\frac{h}{18}$$ $$h=18\times tan\ 76°$$


Choice E

$$216=2^{3}\times 3^{3}$$ $$108=2^{2}\times 3^{3}$$ $$(A).[2,108]=108$$ …


2019 April(Z15) Math Question 40

Conics

- Parabola

40. In the standard (x,) coordinate plane, given Parabola A with equation y= 3x. Parabola B is the image of Parabola A after a shift of 7 coordinate units to the left and 4 coordinate units down. Parabola B has which of the following equations?

Choice J

z15

$$\Rightarrow y=3(x+7)^{2}-4$$


Choice E

$$Less\ than\ $100$$ $$\Rightarrow(Less\ than\ …


Choice G

$$At\ least\ $150\ each\ week$$ …


2019 April(Z15) Math Question 43

Angles

- Circles

43. A pollster will create a circle graph using the information in the table. One sector of the circle grap will represent the percent of adults in the sample who said they spend an average of $300 or more on food each week. What will be the measure of the central angle for that sector?

Choice C

$$$300\ or\ more:\ 0.1$$ $$\Rightarrow …


2019 April(Z15) Math Question 44

Trigonometry

- Pythagorean Theorem & sin cos tan

44. A spherical droplet of ink strikes a vertical wall, as modeled in the figure below. The angle of impact is indicated by e in the figure ink droplet一wall The stain the droplet leaves on the wall is oval-shaped Scientists can measure the maximum length and maximum width of the stain to determine the angle of Impact according to the formula sin e maximum width maximum length The figure below models such a stain. What was the impact angle of the droplet that left this stain?

Choice F

$$\sin\theta=\frac{maximum\ width}{maximum\ length}$$ $$=\frac{0.6}{1.2}=\frac{1}{2}$$

Take …


2019 April(Z15) Math Question 45

Statistics

- Mean, Median, Mode Combinations

45. The graph below shows the distribution of a data set consisting of 16 positive integers. Which of the following statements about the mean, median. and mode of the data set is true?

Choice A

$$1,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8$$ $$Mode=3$$ $$Mean=\frac{1+2\times3+3\times4+4\times3+5\times2+6+7+8}{16}$$ $$=\frac{1+6+12+12+10+6+7+8}{16}$$ $$=\frac{62}{16}=3.875$$ …


Choice K

$$Yolanda:\ y$$ $$Xavier:\ x$$ $$Zach:\ …


2019 April(Z15) Math Question 47

Equation of a Line

- Distance & Midpoint

47. What is the distance, in coordinate units, between the points (-2, 1)and(1, 10)in the standard (x, y)coordinate plane?

Choice D

$$1-(-2)=3$$ $$10-1=9$$ $$\sqrt{3^{2}+9^{2}}=\sqrt{9+81}=\sqrt{90}$$


Choice J

$$100=10\times10\times1$$ $$\Rightarrow 20\times20\times2=800$$


2019 April(Z15) Math Question 49

Absolute Values

- Absolute Value Expressions

49. The set of all values of x that satisfies l x-2 l <7
the same as the set of all values of x that satisfies:

Choice D

$$\mid x-2 \mid\lt7$$ $$\Rightarrow -7\lt …


Choice G

$$380,204,032$$ $$F.\ 1\ and\ 10$$ …


2019 April(Z15) Math Question 51

Matrices

- Multiplication

51. Given A=  , B=  , and C=  , if it is possible to calculate C+AB, which of the following matrices is the result?

Choice B

$$AB$$ $$=\begin{bmatrix}2(-3)+0(4)+3(1) & 2(1)+0(1)+3(2) \\(-1)(-3)+5(4)+(-2)1 …


Choice H

$$\frac{x+y}{y}$$ $$=\frac{x}{y}+\frac{y}{y}$$ $$=\frac{x}{y}+1$$

Clearly that …


2019 April(Z15) Math Question 53

Graphing Properties

- Interpret Graphs or Figures

53. The graph below gives the speed, in knots(nautical miles per hour), of a cruise ship during a 5-hour period. which of the following values is closest to the rate of change, in knots per hour. of the speed of the ship between hours 2 and 4?

Choice C

$$t=2\ hours,\ v_{2}=15$$ $$t=4\ hours,\ …


2019 April(Z15) Math Question 54

Probability

- More than One Event

54. A fair spinner with 4 equally sized regions and an arrow has regions numbered 1,2,3 and 4 respectively,
and a second fair spinner with 5 equally sized regions and an arrow has regions numbered 1, 2, 3, 4 and 5.
respectively. The arrows are both spun at the same time, and the numbers the 2 arrows land on are multiplied together.  What is the probability that this product is an odd number?

Choice K

$$(1).\ 1,2,3,4 \rightarrow Odd:1,3$$ $$(2).\ …


Choice A

$$Area=630\ square\ feet$$ $$Depth=5\ ft$$ …


2019 April(Z15) Math Question 56

Sequences and Patterns

- General Sequences

56. For all positive integers a and b. the expression(a!)b is equivalent to one of the he following expressions. Which one?

Choice K

$$3!=3\times2\times1$$ $$N!= N(N-1)(N-2)...1$$ $$\Rightarrow (a!)^{b}=[a(a-1)(a-2)...1]^{b}$$


2019 April(Z15) Math Question 57

Statistics

- General Statistics

57. The standard normal probability distribution function
( =0 and o=l) is graphed in the standard(xy)
coordinate plane below. Which of the following percentages is closest to the percent of the data points that are within 2 standard deviations of the mean in any normal distribution?

Choice D

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2019 April(Z15) Math Question 58

Matrices

- Determinants

58. For what value of b will the determinant of the matrix
[    ] have a value of 18?

Choice G

$$\begin{bmatrix}a & b \\c & …


2019 April(Z15) Math Question 59

Quadratic Equations

- Factoring

59. What are the solutions to x-2x +17=0?

Choice C

$$x^{2}-2x+17=0$$ $$\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} $$ $$\Rightarrow \frac{-(-2)\pm\sqrt{(-2)^{2}-4\times1\times17}}{2\times1} …


2019 April(Z15) Math Question 60

Perimeter, Area, Volume

- Squares in a Shape

60. Shown below are the top, front, and right side views of a stack of 1-centimeter cubes. The labels T. fand R spe ecify where the top front, and right sides are located with respect to the view. What is the volume. in cubic centimeters, of the stack of cubes?

Choice G

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