ACT Form 68A Math Answer Explanation



2009 December(68A) Math Question 1

Absolute Values

- Absolute Value Expressions

1. |4-1| - |1-4| =?

Choice D

$$=\mid3\mid-\mid-3\mid$$ $$=3-3$$ $$=0$$


Choice G

$$y=2x+1=3$$ $$2x+1=3$$ $$x=1$$


2009 December(68A) Math Question 3

Word Problems

- 2 Variable Addition

3. To attend an annual banquet, members y $16 per ticket while nonmembers pay $22 per ticket.  What is the total amount, in dollars, from the sale of 80 member tickets and n nonmember tickets?

Choice A

$$=80\times16+n\times22$$


Choice J

$$=50\times1.5=75$$


2009 December(68A) Math Question 5

Sequences and Patterns

- Logical Expressions

5. A box contains several marbles. Bill draws a marble at random from the box, notes that it is blue, and places the marble back in the box. Then Aiko draws a marble at random from the box, notes that her marble is blue, and places it back in the box. Which of the following is necessarily true?

Choice E

Draw the blue one twice, …


Choice J

$$3\ pieces=x,\ y,\ z$$ $$x=2y=2(2z)=4z$$ $$y=2z$$ $$\Rightarrow x+y+z=4z+2z+z=21$$ $$\Rightarrow 7z=21$$ $$\Rightarrow z=3$$ $$x=4z=12$$


Choice C

$$(8+3)2=22$$


2009 December(68A) Math Question 8

Angles

- Vertex Angles & Bisectors

8. In the figure below, AE and BD intersect at C. What is the measure of ∠E?


2009 December(68A) Math Question 9

Trigonometry

- Pythagorean Theorem & sin cos tan

9. The diagonal BD divides quadrilateral ABCD into a right triangle and an equilateral triangle, as shown below. The given dimensions are in centimeters. How many centimeters long is AD?

Choice C

$$\triangle BCD is a 6-8-10 triangle.$$ $$\overline{BD}=10$$ $$\underline{AB}=\overline{BD}=\overline{AD}=10$$


2009 December(68A) Math Question 10

Probability

- One Event

10. The circular face of a fair spinner has a red section, a blue, section, and 2 yellow sections, as shown below.
After being spun, the arrow of the spinner is equally likely to be pointing to any I of the 4 sections. What is the probability that, after the arrow has been spun, the section the arrow is pointing to is yellow?

Choice F

The yellow area accounts for 1/2 of the area, so the probability is 1/2.


Choice A

$$2(3x+5)=6x+10$$ $$=6\ squares+10\ circles$$


2009 December(68A) Math Question 12

Equation of a Line

- Equations that Describe a Line

12. What is the slope-intercept form of 5x-y-9=0?

Choice H

$$y=mx+b$$ $$5x-y-9=0$$ $$\Rightarrow -y=9-5x$$ $$\Rightarrow y=5x-9y$$


2009 December(68A) Math Question 13

Equation of a Line

- Distance & Midpoint

13. The points S(4, 10)and T(6, -2)lie in the standard (x, y)
coordinate plane. What is the midpoint of ST?

Choice D

$$\Rightarrow\frac{S+T}{2}$$ $$=\frac{(4,10)+(6,-2)}{2}$$ $$=\frac{(10,8)}{2}$$ $$=(5,4)$$


Choice K

$$\frac{3}{5}\times\frac{1}{2}=\frac{3}{10}$$


Choice C

$$\frac{1}{a}=2$$ $$\frac{1}{b}=3$$ $$\frac{1}{2}(\frac{1}{a}+\frac{1}{b})$$ $$=\frac{1}{2}(2+3)$$ $$=\frac{5}{2}$$


2009 December(68A) Math Question 16

Word Problems

- Cost

16. Guban went shopping at a department store that was having a storewide sale where every item was 10% off its marked price, including already discounted items Guban found a sweater with an original price of $30.00marked with a discounted price of 15% off its original price. There was a 6% sales tax on the final price of all items. How much did Guban pay for the sweater?

Choice J

$$30\times0.85\times0.9\times1.06=24.327\approx24.34$$


2009 December(68A) Math Question 17

Angles

- Parallel Lines

17. In the figure below, parallel lines I1, I2, and l3 , intersect transversal t. What is the value of a+b+c+d?

Choice E

68a

$$a=c$$ $$b=d$$ $$a+b=180°$$ $$c+d=180°$$ $$a+b+c+d=360°$$


Choice G

$$5000\div50=100$$


Choice E

$$(x+2)(x-1)$$ $$=x^{2}-x+2x-2$$ $$=x^{2}+x-2$$


Choice G

$$A=1000(1+0.05)^{2}$$ $$=1000(1.1025)$$ $$=1102.5\approx 1103$$


2009 December(68A) Math Question 21

Equation of a Line

- Distance & Midpoint

21. A motorist traveled at an average speed of 50 miles per hour and arrived at his destination in exactly
20 minutes. Which of the following is closest to the number of miles he traveled?

Choice C

$$20\ mins=\frac{1}{3}\ hr$$ $$50\times\frac{1}{3}=16.666\approx17$$


2009 December(68A) Math Question 22

Word Problems

- 2 Variable Addition

22. Inez has 20 stamps. Some of the stamps are 40g stamps and the others are 25g stamps. The value of the 20 stamps is $6.95. How many 40g stamps does Inez have?

Choice H

$$$1=100¢$$ $$0.40\times x+0.25\times(20-x)=6.95$$ $$0.4x+5-0.25x=6.95$$ $$0.15x=1.95$$ …


2009 December(68A) Math Question 23

Equation of a Line

- Equations that Describe a Line

23. A line in the standard (x, y) coordinate plane has equation 3x-2y =6. What is the slope of this line?

Choice B

$$3x-2y=6$$ $$-2y=6-3x$$ $$2y=3x-6$$ $$y=\frac{3}{2}x-3$$ $$y=mx+b$$ …


2009 December(68A) Math Question 24

Trigonometry

- Pythagorean Theorem & sin cos tan

24. The right triangle shown below has a hypotenuse of 4 inches. The measure of the angle indicated is 6. Which of the following is closest to the length, in inches, of the side opposite the 6 angle?

Choice G

$$\sin 6°=0.1045=\frac{?}{4}$$ $$?=0.1045\times4=0.418$$


2009 December(68A) Math Question 25

Sequences and Patterns

- General Sequences

25. A supermarket soup can display consists of 5 levels of stacked cans. The figure below shows an overhead view of the top level of cans(circles shaded and with solid borders) and the level of cans below it(circles with dashed borders)。 In the display, each level has
4 more cans than the level above it. How many cans are in the display?

Choice E

$$Level\ 1=4\times5$$ $$Level\ 2=4\times6$$ $$Level\ …


Choice J

$$F.\sqrt{-5^{2}}=5$$ $$G.\sqrt{-2^{2}}=2$$ $$H.\sqrt{\frac{1}{2}^{2}}=\frac{1}{2}$$ $$\frac{1}{2}\ is\ …


Choice B

$$16(16-6)(\frac{1}{2})=80$$


2009 December(68A) Math Question 28

Number Types & Properties

- Order of Values

28. For real numbers x and y such that x-y<0, which of the following must be true?

Choice K

$$x-y\lt0$$ $$Let\ x=1,\ y=2$$ $$F.\ …


Choice E

Move down two units, the …


2009 December(68A) Math Question 30

Inequalities

- Inequality Expressions


Choice G

$$-6+7k \geq 3-8k$$ $$\Rightarrow 15k …


Choice C

68a

$$\Rightarrow \overline{AB}=4\overline{GH}$$


Choice F

$$2\frac{1}{3}\times\frac{1}{2}=1\frac{1}{6}$$


2009 December(68A) Math Question 33

Word Problems

- Interesection

33. A greenhouse contained both 6-inch-high geraniums and 3-inch-high geraniums. The 6-inch-high geraniums were taken out of the greenhouse in early spring an d grew at a rate of I inch per week. The 3-inch-high geraniums remained in the greenhouse and grew at a rate 1 inches per week. After the 6-inch-high geraniums were taken outside, in how many weeks were both groups of geraniums the same height?

Choice B

$$6+1\times x=3+1\frac{1}{2}x$$ $$3=\frac{1}{2}x$$ $$x=6$$


Choice K

$$\Rightarrow x^{3}-4x^{2}-x^{2}+5x+3x+6$$ $$\Rightarrow x^{3}-5x^{2}+8x+6$$


2009 December(68A) Math Question 35

Trigonometry

- Pythagorean Theorem & sin cos tan

35. A right triangle is shown in the figure below. Which of the following expressions gives 8?

Choice A

$$\tan \theta=\frac{0.5}{1}$$ $$\Rightarrow \tan \theta=\frac{1}{2}$$ …


Choice G

68a


2009 December(68A) Math Question 37

Perimeter, Area, Volume

- Mixed Shapes

37. The square below has an area of 64 square centimeters.
The circle inscribed in the square is tangent to the square at A, B, C, and D. What is the area, in square centimeters, of the circle?

Choice C

68a

$$\pi r^{2}=16\pi $$


Choice K

68a


2009 December(68A) Math Question 39

Equation of a Line

- Distance & Midpoint

39. The measure of ∠y is 50. what is the measure of the angle between WX and the y-axis?

Choice B

68a


2009 December(68A) Math Question 40

Perimeter, Area, Volume

- Other Shapes

40. Parallelogram WXYZ is rotated clockwise (c ) by 90°
about the origin. At what ordered pair is the image of Z located?

Choice F

68a


2009 December(68A) Math Question 41

Quadratic Equations

- Factoring

41. A liter is 1.000 cubic centimeters. Which of the following is closest to the height, in centimeters, cylindrical container, shown below, with diameter
10 cm and capacity 1 liter?
(Note: The volume of' a cylinder with radius r and height h is  πr^2h.)

Choice D

$$r=\frac{10}{2}=5$$ $$\pi r^{2}h=\pi 5^{2}h=1000$$ $$h=1000\div25\div\pi=\frac{40}{\pi}\approx\frac{40}{3.14}=12.73$$


2009 December(68A) Math Question 42

Perimeter, Area, Volume

- Mixed Shapes

42. For World Literature class, Lenka must read Anna Karenina in 8 days. She reads of the book each of the first 3 days. For the remaining 5 days, what fraction of the book, on average, must Lenka read per day?

Choice J

$$1-\frac{1}{12}\times3=\frac{9}{12}=\frac{3}{4}$$ $$\frac{3}{4}\times\frac{1}{5}=\frac{3}{20}$$


2009 December(68A) Math Question 43

Statistics

- Mean, Median

43. Which of the following equations shows a correct use of the quadratic formula to solve x²-5x+3=0?

Choice A

$$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$$ $$=\frac{-(-5)\pm\sqrt{(-5)^{2}-4(1)(3)}}{2(1)}$$ $$=\frac{5\pm\sqrt{25-4(1)(3)}}{2(1)}$$


2009 December(68A) Math Question 44

Equation of a Line

- Equations that Describe a Line

44. In the standard (x, y) coordinate plane, the point (1, -6)
is the midpoint of the line segment with endpoints
(9, 13)and(a, b). What is (a , b)?

Choice F

$$(9,-13)+(a,b)=2(1,-6)=(2,-12)$$ $$9+a=2\ \Rightarrow\ a=-7$$ $$-13+b=-12\ …


2009 December(68A) Math Question 45

Trigonometry

- Pythagorean Theorem & sin cos tan

45. A straight 10-foot-tall ladder is leaning against a house at an angle of 75,  as shown in the figure below Which of the following expressions gives the distance, in feet, the base of the ladder is from the house along the level ground?

Choice B

Let the base of the …


2009 December(68A) Math Question 46

Probability

- Combinations

46. One of the following equations is graphed in the standard (x, y) coordinate plane below. Which one?

Choice K

68a

$$(0,-1)\Rightarrow G,\ K$$ $$(K). m=3,\ …


Choice D

$$(10+18)\times 3\times\frac{1}{2}=42$$


Choice G

$$C=99\times h$$ $$m=0.99$$


2009 December(68A) Math Question 49

Angles

- Circles

49. The first 3 terms of a geometric sequence are
4, 10, and 25. What is the next term in the sequence?

Choice D

$$4,10,25$$ $$10\div4=2.5$$ $$25\div10=2.5$$ $$25\times2.5=62.5$$


2009 December(68A) Math Question 50

Perimeter, Area, Volume

- Other Shapes

50. The volume of a cube is 64 cubic centimeters. What is the total surface area, in square centimeters, of the cube?

Choice J

$$64=x^{3}$$ $$x=4$$ $$Total\ surface\ area=4\times4\times6=96$$


Choice C

$$2\pi r=2 \pi(4)=8\pi$$ $$\frac{7 \pi}{8\pi}=\frac{315}{360}$$ …


2009 December(68A) Math Question 52

Probability

- One Event

52. In the standard (x, y) coordinate plane, the graph of which of the following equations is a circle with center(4, -2)and radius 3 coordinate units?

Choice K

The equation for a circle …


2009 December(68A) Math Question 53

Trigonometry

- Advanced sin, cos, tan functions

53.In△ABC, the measure of ∠A is47°, the measure of ZB is 76, and the length of BC is 18 centimeters.
Which of the following is an expression for the length, in centimeters, of AC?
(Note: The law of sines states that for any triangle, the ratios of the lengths of the sides to the sines of the angles opposite those sides are equal.)

Choice D

68a

$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$ $$\Rightarrow …


Choice F

68a

$$The\ side\ of\ square1:1+l$$ $$The\ …


2009 December(68A) Math Question 55

Quadratic Equations

- Factoring

55 If x 2, then |x-2|=?

Choice E

$$If\ \ x\leq2\ , then\ …


2009 December(68A) Math Question 56

Word Problems

- Fixed Cost + Variable Cost

56. There are 25 buildings on Elm Street. Of these
25 buildings, 10 have fewer than 6 rooms, 10 have more than 7 rooms, and 4 have more than 8 rooms.
What is the total number of buildings on Elm Street that have 6, 7,or 8 rooms?

Choice H

More than 6 rooms:25-10=15

6 …


Choice B

$$\Rightarrow (\cos x)^{2}=(-\frac{1}{3})^{2}=\frac{1}{9}$$ $$\frac{1}{9}=\frac{1+\cos 2x}{2}$$ …


2009 December(68A) Math Question 58

Graphing Properties

- Create Graphs or Figures

58. Let f (x)= √x and g (x)=10x +b In the standard (x ,y)
coordinate plane, y=f(g(x)passes through(4,6).What is the value of b?

Choice F

$$y=f(g(x))=\sqrt{10x+b}$$ $$(4,6)\Rightarrow 6=\sqrt{10\times4+b}$$ $$\Rightarrow 36=40+b$$ …


2009 December(68A) Math Question 59

Equation of a Line

- Equations that Describe a Line

59. A plane contains 11 horizontal lines and 1l vertical lines. These lines divide the plane into disjoint regions How many of these disjoint regions have a finite, nonzero area?

Choice A

68a


2009 December(68A) Math Question 60

Angles

- Other Shapes

60. When a, b, and c are real numbers and a b^2 c^4 >0, which of the following must be greater than 0

Choice F

$$b^{2}\gt 0;$$ $$(c^{2})^{2}\gt 0;$$ $$\Rightarrow …