ACT Form 68C Math Answer Explanation



2010 June(68C) Math Question 1

Probability

- One Event

1. The face of a fair spinner is divided into 24 sectors, each of which is I of 5 colors:Blue, Purple, Green Orange, or Red. The table below gives each color and the probability that the spinner, when spun, will stop on that color On which color shown in the table is the spinner LEAST likely to stop?

Choice C

$$Blue=\frac{1}{4}=\frac{6}{24}$$ $$Purple=\frac{1}{8}=\frac{3}{24}$$ $$Green=\frac{1}{12}=\frac{2}{24}$$ $$Orange=\frac{3}{8}=\frac{9}{24}$$ $$Red=\frac{1}{6}=\frac{4}{24}$$

The chance of stopping in green is the lowest.


2010 June(68C) Math Question 2

Word Problems

- Cost

2. Saul goes shopping and decides to buy 4 shirts at a price of $20.08 cach. Which of the following expressions gives the total price, in dollars, of the
4 shirts?

Choice K

$$4\times$20.08=4($20)+4($0.08)$$


2010 June(68C) Math Question 3

Quadratic Equations

- Factoring

3. Which of the following expressions is equivalent to x2-9?

Choice C

$$(x-3)(x+3)=x^{2}+3x-3x+3^{2}=x^{2}+9$$


2010 June(68C) Math Question 4

Angles

- Collinear Points & Triangles

4. In the figure below, E is on AD, EB is perpendicular to EC, and the measure of ZCED equals 38. What is the measure of∠AEB?

Choice H

$$\angle BAE=180°-90°-38°=52°$$


Choice D

$$70\ beats \times 60\ minutes\times24\ …


Choice K

$$\frac{(-7)^{2}-1}{(-7)+1}$$ $$=\frac{49-1}{-6}$$ $$=\frac{48}{-6}$$ $$=-8$$


Choice C

$$(-6,11)+(+11,-6)=(5,5)$$


Choice G

$$7+4x=31$$ $$4x=24$$ $$x=6$$ $$3x=18$$


2010 June(68C) Math Question 9

Graphing Properties

- Create Graphs or Figures

9. When Jamar purchased a houseplant for his room, it was 4 inches in height. For the next 6 weeks, the houseplant grew taller at a rate of 1 inches per week.
Which of the following graphs best represents the houseplant's height, h inches, versus the number of weeks, w, after Jamar purchased the houseplant?

Choice E

$$Weeks=0,\ height=4$$ $$Slope=1\frac{1}{2}$$ $$h=4+1\frac{1}{2}w$$


Choice H

$$h(a,b)=a^{b}-b^{a}$$ $$h(3,2)=3^{2}-2^{3}=9-8=1$$


2010 June(68C) Math Question 11

Angles

- Collinear Points & Triangles

11. In the figure below, ∠ BAC measures35°, ∠ABC measures 115, and points B, C, and D are collinear What is the measure of ∠ACD?

Choice E

$$\angle ACD=180°-\angle ACB$$ $$\angle ACD=35°+115°=150°$$


2010 June(68C) Math Question 12

Inequalities

- Inequality Expressions

12. Koji had $50.00 to spend on school supplies. He bought a schoolbag for $18.50, a box of pencils for
$2.50, and a package of pens for $4.00. Koji still must buy b binders at $3.50 each and n spiral notebooks at s0.75 each. Which of the following inequalities represents how many binders and spiral notebooks he can buy with the remaining money?
(Note: No tax is charged on school supplies.)

Choice K

$$$50-$18.5-$2.5-$4=$25$$ $$3.5b+0.75n\leq25$$


2010 June(68C) Math Question 13

Angles

- Collinear Points & Triangles

13. In the figure below, ZAEB is congruent to ZDEC and BE is congruent to CE. One of the following angle congruences completes what is needed to prove as directly as possible that AABE is congruent to ADCE by applying the Angle-Side-Angle(ASA) property.
Which one?

Choice B

68c

Angel-Side-Angel(ASA)


2010 June(68C) Math Question 14

Probability

- One Event

14. Of the 40 marbles in Dakarai's bag, all are solid in color and 16 are blue. The probability that he will randomly choose a yellow marble from the bag How many yellow marbles are in Dakarai's bag?

Choice H

$$40\times\frac{1}{8}=5$$


2010 June(68C) Math Question 15

Perimeter, Area, Volume

- Mixed Shapes

15. A square and a rectangle have the same area. The length of the rectangle is 36 centimeters, and the width of the rectangle is 4 centimeters. What is the length, in centimeters, of a side of the square?

Choice B

$$36\times4=144$$ $$144=x^{2}$$ $$x=12$$


2010 June(68C) Math Question 16

Statistics

- Mean, Median

16. Arita played in 7 softball games. In the first 4 games she had 2, 3, 1, and 3 hits, respectively. In the last
3 games, she had a total of 6 hits. What was her average number of hits per game in the 7 games?

Choice G

$$\frac{2+3+1+3+6}{7}=\frac{15}{7}$$


2010 June(68C) Math Question 17

Word Problems

- 2 Variable Addition

17. A test has 50 questions and is worth 130 points. It consists of x 5-point questions and y 2-point questions Which of the following systems of equations, when solved, gives the number of each type of question that is on the test?

Choice C

$$x+y=50---50\ questions$$ $$x\cdot5+y\times2=130---130\ points$$ $$\Rightarrow 5x+2y=130$$


Choice F

$$4.5=\frac{1}{t}$$ $$t=\frac{1}{4.5}=\frac{2}{9}$$


Choice A

$$=(4a-6a)+(7b-6b)+(6c+3c)$$ $$=-2a+b+9c$$


Choice J

$$50=2\times5\times5$$ $$30=2\times3\times5$$ $$70=2\times5\times7$$ $$[50,30,70]=2\times3\times5\times5\times7=1050$$


Choice C

$$B=7LH$$ $$420=7\times30\times H$$ $$H=2$$


Choice G

$$y=mx+b$$ $$5x+7y=7$$ $$\Rightarrow 7y=-5x+7$$ $$\Rightarrow …


Choice E

$$\triangle A=\frac{1}{2}bh$$ $$h=\overline{AB}$$ $$20=\frac{1}{2}\times5\times \overline{AB}$$ …


2010 June(68C) Math Question 24

Equation of a Line

- Distance & Midpoint

24. Reuben is standing 300 yards due east from the flagpole at his school and 400 yards due south from the oak tree in his front yard. What is the straight-line distance, in yards, from the flagpole to the oak tree?

Choice J

68c

$$3-4-5\ triangle$$ $$\Rightarrow 300:400:500$$ $$x=500$$


2010 June(68C) Math Question 25

Absolute Values

- Absolute Value Expressions

25.|9(-6)+5(4)|=?

Choice D

$$=\mid (-54)+20\mid$$ $$=\mid -34 \mid$$ …


Choice G

$$6x-y=x+5y$$ $$\Rightarrow 5x=6y$$ $$\Rightarrow y=\frac{5x}{6}$$


Choice E

$$2:7=10:x$$ $$x=\frac{7\times10}{2}=35$$


Choice G

$$=x^{5+-(3^{2})}$$ $$=x^{5+(-9)}$$ $$=x^{-4}$$


2010 June(68C) Math Question 29

Word Problems

- Cost

Use the following information to answer questions 29-31
Kyla purchased 25 pieces of candy from Laszko's Candy Shop. Her purchase consists of 7 lollipops, 4 candy bars,
10 licorice sticks, and 4 gumballs. The unit price of each candy item is shown in the table below. Kyla's purchase without sales tax, totaled $10.30. Laszko's charges an
8% sales tax on each purchase, which is calculated by multiplying the purchase total by 0.08 and rounding to the nearest $0. 01

29. Kyla gave the shop clerk $15.00. How much change should Kyla have received?

Choice A

$$10.3(1.08)=11.124$$ $$15-11.124=-3.876\approx-3.88$$


2010 June(68C) Math Question 30

Statistics

- Mean, Median

30. Without sales tax, what was the average price Kyla paid per piece of candy, to the nearest $0.01?

Choice J

$$10.3\div25$$ $$=0.412\approx0.41$$


2010 June(68C) Math Question 31

Probability

- Combinations

31. Kyla offers to sell 2 of the 25 pieces of candy to her brother Virgil. She lets Virgil have a choice of 2 pieces of the same candy item or l piece each of 2 different candy items. Kyla will have Virgil pay the same total cost for the 2 pieces that he would pay for the 2 pieces at Laszko's. How many different total costs(in dollars)
are possible for Virgil's choice of 2 pieces?

Choice B

Lollipop: L

Candy bar: C …


Choice G

$$2r=36$$ $$\Rightarrow 2\pi r=36\pi$$


2010 June(68C) Math Question 33

Quadratic Equations

- FOIL

33. For all real x, ?

Choice D

$$\frac{[3(x-2)]^{2}}{3}$$ $$=\frac{3^{2}(x-2)^{2}}{3}$$ $$=3(x-2)^{2}$$ $$=3(x^{2}-4x+4)$$ $$=3x^{2}-12x+12$$


2010 June(68C) Math Question 34

Inequalities

- Inequality Expressions

34. Which of the following is a solution statement for the inequality 6x-6 < 7+10x?

Choice F

$$6x-6 \lt 7+10x$$ $$\Rightarrow -6-7 …


2010 June(68C) Math Question 35

Perimeter, Area, Volume

- Other Shapes

35. The perimeter of a parallelogram is 68 inches, and I side measures 14 inches. If it can be determined what are the lengths, in inches, of the other 3 sides?

Choice C

The two sets of opposite …


Choice J

$$Slope=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$ $$\Rightarrow m=\frac{4-2}{9-5}=\frac{2}{4}=\frac{1}{2}$$


2010 June(68C) Math Question 37

Angles

- Circles

37. The hour and minute hands of the clock shown below move at constant speeds. What is the measure of the angle formed by the hour and minute hands at exactly 1:30?

Choice E

68c

At 1:30, the hour hand …


2010 June(68C) Math Question 38

Trigonometry

- Pythagorean Theorem & sin cos tan

Use the following information to answer questions 38-40.
Shown below is an inflatable display (on level ground)of a Lions Crest Cola can. A blower pumps air into the display at an average rate of 600 cubic feet per minute as it inflates the display. When fully inflated, the display is a right circular cylinder with a diameter of 12 feet and a height of
20 feet. Also shown below is 1 of the support ropes that anchors the display. The rope is attached to the top of the display and has an angle of elevation of 55. The volume of air in the fully inflated display is equal to the displays height times the area of l of the circular bases.

38. Which of the following is closest to the length, in feet.
of the support rope?
Note:sin55°-0.82;cs55°=0.57;tan55°=1.43)

Choice F

$$\sin 55°\approx 0.82=\frac{20}{Support\ rope}$$ $$\Rightarrow …


2010 June(68C) Math Question 39

Perimeter, Area, Volume

- Other Shapes

39. The blower starts to pump air into the completely deflated display and continues until the display is fully inflated To the nearest 0. 1 minute, for how many minutes does the blower pump air?

Choice C

$$\pi r^{2}\times20$$ $$=3.14\times 6^{2}\times20$$ $$=2260.8$$ …


Choice G

$$20:35=12:x$$ $$x=21$$


Choice C

$$x(2x)=162$$ $$2x^{2}=162$$ $$x^{2}=81$$ $$x=9$$ $$2(x+2x)=6x=54$$


2010 June(68C) Math Question 42

Word Problems

- Cost

42. While shopping, Ashton decided to purchase a shirt on sale that was marked down 40% from its original price of $10.00. At the checkout, he used an in-store coupon for 10% off the sale price. What was the final price of the shirt before any tax was added?

Choice K

$$$10\times0.6\times0.9=5.4$$


Choice C

$$(C)\ y=\frac{4x+3}{x-2}$$

Since I can't …


2010 June(68C) Math Question 44

Matrices

- Addition & Subtraction

44. For what(x,y) pair is the matrix equation below true?

Choice J

$$3+x=y$$ $$\frac{x}{2}+5=y$$ $$\Rightarrow \frac{x}{2}+5=3+x$$ $$\Rightarrow …


2010 June(68C) Math Question 45

Statistics

- General Statistics

45. The marketing students at Fort Link Business Academy took a placement test for an accounting class. The mean of the test scores is 100 and the standard deviation is 15. In order to be accepted into the accounting class, a student must have attained a test score that is at least 1 standard deviation above the mean. What is the lowest test score a student could attain and still be accepted into the accounting class?

Choice D

$$100+15=115$$


2010 June(68C) Math Question 46

Angles

- Circles

46. Two angles are coterminal if they have the same initial and terminal sides. For instance, the angles shown below with measures 45 and 405 are coterminal.
An angle with measure 30 is coterminal with a second angle. Which of the following measures could be that of the second angle?

Choice K

$$405°=360°+45°$$ $$360°+30°=390°$$


2010 June(68C) Math Question 47

Equation of a Line

- Distance & Midpoint

47. In the standard (x,y) coordinate plane, the midpoint of CD is at(5,-7)and point C is at(3, 11). For point D at a, b), what is the value of a+b?

Choice D

$$(a,b)+(3,-11)=(5,-7)$$ $$\frac{a+3}{2}=5 \Rightarrow a=7$$ $$\frac{b+(-11)}{2}=-7 …


2010 June(68C) Math Question 48

Trigonometry

- Pythagorean Theorem & sin cos tan

48. An airplane flies the 200 miles from Fairhaven to Locksboro along a straight line in a direction(shown below) that is 25 clockwise(C) from due north. To the nearest mile, Locksboro is how many miles due east and how many miles due north from Fairhaven?
Note: sin25°=0423,cos25°=0.906

Choice F

68c

$$\sin25°=0.423,\ \cos25°=sin(90-25)°=\sin65°=0.906$$ $$\frac{200}{\sin 90°}=\frac{x}{\sin25°}=\frac{y}{\sin65°}$$ $$\Rightarrow …


Choice B

68c

A graph is said to …


2010 June(68C) Math Question 50

Perimeter, Area, Volume

- Mixed Shapes

50. In the figure below, the square has side length
8 centimeters. Each vertex of the square is the center of a circle. The circles are congruent, and each circle is tangent to 2 of the other circles. The region that is interior to the square and exterior to all 4 circles is shaded. What is the area, to the nearest square centimeter, of the shaded region?

Choice K

68c

$$\Rightarrow 8\times8=64$$ $$64-\pi (4)^{2}=64-3.14(16)=13.76\approx14$$


Choice B

$$(width+length)2=perimeter$$ $$width=w$$ $$length=w+x$$ $$\Rightarrow (w+w+x)2=P$$ …


2010 June(68C) Math Question 52

Trigonometry

- Pythagorean Theorem & sin cos tan

52. An angle with measure 0 such that sin 8= i3 is standard position with its terminal side extending into Quadrant II as shown in the standard (x,y) coordinate plane below What is the value of cos θ ?

Choice F

68c

$$13^{2}=5^{2}+x^{2}$$ $$\Rightarrow x=12$$ $$\cos \theta=\frac{-12}{13}$$


2010 June(68C) Math Question 53

Quadratic Equations

- FOIL

53. Let a and b be real numbers. If (a +b)2=a2+b2.it must be true that:

Choice A

$$(a+b)^{2}=a^{2}+b^{2}$$ $$\Rightarrow a^{2}+2ab+b^{2}=a^{2}+b^{2}$$ $$\Rightarrow 2ab=0$$ …


Choice J

$$ $1\times3+$3\times0=$3 $$ $$ $1\times4+$3\times1=$7 …


2010 June(68C) Math Question 55

Angles

- Collinear Points & Triangles

55. In the figure below, lines a and b are parallel, the angle measure y is in degrees, and ∠1 and ∠2 have the same measure. In terms of y, which of the following is equivalent to the degree measure of ∠3?

Choice A

$$\angle 1=\angle 2$$ $$\angle 1+\angle …


2010 June(68C) Math Question 56

Equation of a Line

- Distance & Midpoint

56. In the figure below, the vertices of rectangle BCDE lie on the circle and the vertices of rhombus FGHI are the midpoints of the sides of rectangle BCDE Point A is at the center of the circle, rectangle, and rhombus.
The radius of the circle is 4 feet. How long is a side of the rhombus, in feet?

Choice H

68c


Choice C

$$f(g(x))$$ $$=\frac{3}{(x+1)^{2}-1}$$ $$=\frac{3}{x^{2}+2x+1-1}$$ $$=\frac{3}{x^{2}+2x}$$ $$=\frac{3}{x(x+2)}$$ …


2010 June(68C) Math Question 58

Absolute Values

- Absolute Value Expressions

58. Which of the following is equivalent to | x-1 | < B?

Choice F

$$\mid x-1 \mid \lt B$$ …


2010 June(68C) Math Question 59

Graphing Properties

- Create Graphs or Figures

59. The area, A, of a circle varies directly with the square of the radius, r. One of the following graphs shows the area, A, of a circle as the radius varies in the (x,A)
coordinate plane Which graph is it?
(Note: The (r, A) plane is the standard (x,y )coordinate lane with r along the horizontal axis and A along the vertical axis

Choice A

68c

$$Area\ of\ a\ circle:\pi r^{2}$$ …


2010 June(68C) Math Question 60

Quadratic Equations

- Factoring

60. Susan discovered by trial and error that one root of the equation 2x³-9x²+ 16x-12=0 is 2. What are the other roots?

Choice J

68c