ACT Form 69A Math Answer Explanation



2010 December(69A) Math Question 1

Probability

- One Event

1. Mr. Chiang announced the grade distribution for this week's book reports. Of the 24 students in the class
8 received A's for their book reports, 1l received B's and 5 received Cs. When a student is chosen at random to be the first one to read his or her book report to the class, what is the probability that the student chosen had received an A for the book report?

Choice B

The answer is 8/24(=1/3).


Choice H

The area of a rectangle:

$$3\times4=12$$ $$12\times$3=$36$$


Choice C

$$(-5+8,8-5)=(3,3)$$


Choice H

$$18000\times2=36000$$ $$83000-36000=47000$$ $$\frac{47000}{7000}=6.7$$

You will need 7 trucks.


Choice D

$$15=3\times5$$ $$25=5\times5$$ $$30=2\times3\times5$$ $$\Rightarrow 2\times3\times5\times5=150$$


Choice F

$$180°-96°=84°$$ $$84°\div2=42°$$


Choice C

$$-2(x-7)=20$$ $$\Rightarrow x-7=\frac{20}{-2}=-10$$ $$\Rightarrow x=-3$$


Choice F

$$=x^{3+3+3+3}$$ $$=x^{12}$$


Choice B

$$=1\times0.1\times0.1=0.01$$


2010 December(69A) Math Question 10

Statistics

- Mean, Median

Use the following information to answer questions 10-12.
The table below gives the price per gallon of unleaded gasoline at Gus's Gas Station on January 1 for 5 consecutive years in the 1990s. At Gus's, a customer can purchase a car wash for $4.00.

10. What is the mean price per gallon, to the nearest $0.01, on January 1 for the 5 years listed in the table?

Choice J

$$\frac{1.34+1.41+1.41+1.25+1.36}{5}=\frac{6.77}{5}=1.354$$


2010 December(69A) Math Question 11

Word Problems

- Cost

11. The price for gas on January I of Year 6 was 3%
higher than the price on January 1 of Year 5. To the nearest $0.01, how much was the price per gallon on January I of Year 6?

Choice B

$$0.03\times1.36=0.04$$ $$1.36+0.04=1.4$$


2010 December(69A) Math Question 12

Word Problems

- Fixed Cost + Variable Cost

12. On January 1 of Year 5, Anamosa bought gas and a car wash at Gus's. She put 11.38 gallons of gas in her car and 1.85 gallons of gas in a container for her snowblower. To the nearest so. 01. how much did Anamosa pay for the gas for her car and snowblower, and a car wash?

Choice K

$$11.38 + 1.85 = 13.23 gallons$$ $$13.23\times$1.36=$17.99$$ $$$17.99+4=$21.99$$


2010 December(69A) Math Question 13

Absolute Values

- Absolute Value Expressions

13. What is the value of | x + y | +(x+y)2?when r=2 and y=-3?

Choice E

$$\mid 2+(-3)\mid+(2+(-3))^{2}$$ $$=\mid-1\mid+(-1)^{2}$$ $$=1+1$$ $$=2$$


Choice F

$$3.9\times 10^{-1}=0.39$$ $$3.9\times 10^{-2}=0.039$$ $$3.9\times 10^{-3}=0.0039$$

There will always be one less zero in front of the three and the nine.


Choice C

$$525=7\times15\times H$$ $$\Rightarrow H=5$$


Choice K

$$g(2)=(-3)^{2}+3=9+3=12$$


2010 December(69A) Math Question 17

Quadratic Equations

- FOIL

17.(5c-3d)(2d-5c)is equivalent to:

Choice D

$$=5c(2d-5c)-3d(2d-5c)$$ $$=10cd-36c^{2}-6d^{2}+15cd$$ $$=25cd-36c^{2}-6d^{2}$$


2010 December(69A) Math Question 18

Angles

- Collinear Points & Triangles

8. In the figure below, AC and BE intersect at D. What is the measure of ∠EAD?

Choice G

$$\angle ADB=180°-30°-20°=130°$$ $$\angle EDA=180°-130°=50°$$ $$\angle EAD=180°-83°-50°=47°$$


Choice D

$$x + y = 21 $$ $$x – y = 15$$ $$\Rightarrow 2x=21+15=36$$ $$x=18$$ $$y=3$$ $$xy=18\times3=54$$


2010 December(69A) Math Question 20

Word Problems

- Fixed Cost + Variable Cost

20. After visiting Mountain State University during spring, Meredith rents a car for 1 day  to travel around the area. She has $100 to spend on car rental. Green Tree Car Rental charges per day and s0. 25 per mile. Big Rock Car Rental charges $60 per day and s0. 20 per mile. Which company, if either, allows her to travel more miles, and how many miles more?
(Note: Taxes are already included in the rental charges)

Choice K

For Green Tree, the equation is 50 + 0.25m = 100, so m = 200.

For Big Rock, the equation is 60 + 0.2m= 100, so m = 200.

Therefore, she can drive the same amount of miles with either company.


2010 December(69A) Math Question 21

Quadratic Equations

- FOIL

21.For x2≠25,

Choice E

$$\frac{(x-5)^{2}}{x^{2}-25}$$ $$=\frac{(x-5)(x-5)}{(x-5)(x+5)}$$ $$=\frac{(x-5)}{(x+5)}$$


2010 December(69A) Math Question 22

Word Problems

- 2 Variable Addition

22. The directions for punch call for 3 cups of juice concentrate to be mixed with 5 cups of water. if the directions are followed, how many cups of Juice concentrate will be needed to make 80 cups of punch?

Choice F

$$\frac{3}{8}=\frac{x}{80}$$ $$x=30$$


2010 December(69A) Math Question 23

Number Types & Properties

- Order of Values

23. Which of the following inequalities is true for the fractions 3/7, 5/12 and 4/9?

Choice B

Just get the decimal equivalents …


Choice H

$$t=0,y=12, $$

The y-intercept is …


Choice C

T is the speed of …


2010 December(69A) Math Question 26

Graphing Properties

- Interpret Graphs or Figures

26. The line shown below in the standard (x,y) coordinate plane is represented by one of the following equations Which one is it?

Choice F

$$x=0,y=-3 \Rightarrow (F)、(H)$$ $$x=2,y=0 \Rightarrow …


Choice D

$$25\times 3^{5}=6075$$


Choice K

$$\frac{9}{36}=\frac{16}{x}$$ $$\Rightarrow x=64$$


2010 December(69A) Math Question 29

Word Problems

- 2 Variable Addition

29. At a produce market, apples sell for $1.00 per bag and oranges for $1.75 per bag. The fruit is only sold in whole bags. You plan to buy at least I bag of each fruit and spend exactly $16.00. What is the maximum number of bags of oranges you can buy?

Choice D

You can spend exactly $14 …


2010 December(69A) Math Question 30

Conics

- Parabola

30. In the standard (r,y)coordinate plane, what is the vertex of the parabola with the equation y=2(x-3)2-5?

Choice H

$$y=ax^{2}+bx+c$$ $$y=2x^{2}-12x+4$$ $$x=\frac{-b}{2a}=\frac{12}{4}=3$$ $$\Rightarrow x=3,y=-5$$


2010 December(69A) Math Question 31

Equation of a Line

- Distance & Midpoint

31. A diameter of a circle in the standard (r, y) coordinate lane has endpoints at (2, 8)and (2, 6)。 which of the following points is the center of the circle?

Choice B

$$\frac{(2,8)+(-2,6)}{2}=(0,7)$$


Choice J

$$c=2\pi r=12\pi$$


2010 December(69A) Math Question 33

Trigonometry

- Pythagorean Theorem & sin cos tan

33. How many inches long is BC?

Choice C

This is a 3-4-5 right …


2010 December(69A) Math Question 34

Trigonometry

- Pythagorean Theorem & sin cos tan

34. Which of the following is an expression for sin ∠CDA?

Choice F

$$\sin=\frac{opposite}{hypotenuse}$$ $$\angle CDA = \frac{\overline{AC}}{\overline{AD}}$$


2010 December(69A) Math Question 35

Equation of a Line

- Distance & Midpoint

35. Point B is reflected across the line(not shown) that connects the midpoints of AE and CD. This reflection of B is labeled F. How many inches from AC is the intersection of CF and BD?

Choice A

DCBF is a rectangle, with …


2010 December(69A) Math Question 36

Perimeter, Area, Volume

- Mixed Shapes

36. In the diagram below, B, F, and H are on AC, AE, and BF, respectively, and GH 1 BF. The area of square ABGF is the area of square ACDE. The area of AFGH is what fraction of the area of ACDE?

Choice J

$$\triangle FGH=\frac{1}{4}\Box ABGH$$ $$\Box ABGH=\frac{1}{4}\Box …


2010 December(69A) Math Question 37

Sequences and Patterns

- Arithmetic Sequences

37. The first 3 terms of an arithmetic sequence are 2(1/6), 3(1/3),
and 4(1/2),  in that order. What is the fourth term of the sequence?

Choice D

$$2\frac{1}{6},3\frac{1}{3},4\frac{1}{2}$$

The common difference is …


Choice G

$$B=x+2$$ $$y=A-4 \Rightarrow A=y+4$$ $$B+A=(x+2)+(y+4)=x+y+6$$


2010 December(69A) Math Question 39

Perimeter, Area, Volume

- Other Shapes

39. The perimeter of a parallelogram is 64 inches, and 1 side measures 10 inches. If it can be determined, what are the lengths, in inches, of the other 3 sides?

Choice C

$$10,10,x,x$$ $$20+2x=64$$ $$x=22$$ $$\Rightarrow 10,22,22$$


2010 December(69A) Math Question 40

Number Types & Properties

- Order of Values

40. Given that m, n, and a are positive integers, which of the following statements is true whenever a<(-a)?

Choice H

69a


2010 December(69A) Math Question 41

Equation of a Line

- Distance & Midpoint

41. In quadrilateral ABCD shown in the standard (x,y) coordinate plane below, what is the distance, in coordinate units, from the midpoint of AD to the midpoint of CD?

Choice B

69a

The midpoint of AD to …


2010 December(69A) Math Question 42

Perimeter, Area, Volume

- Other Shapes

42. In rhombus ACBD below, AB is 8 inches long and CD is 6 inches long. What is the area, in square inches, of ACBD?

Choice G

The area of a rhombus: …


2010 December(69A) Math Question 43

Probability

- Combinations

43. In how many distinct orders can 5 students stand in line to buy yearbooks?

Choice D

$$5\times4\times3\times2\times1=120$$


2010 December(69A) Math Question 44

Trigonometry

- Pythagorean Theorem & sin cos tan

44. A group of hikers put up the tent shown below. They draped a rectangular tarp that is 10 feet by 12 feet over a rope stretched tightly between 2 trees so that the tent's rectangular sides and base are each 6 feet wide and 10 feet long. What is the height of the tent, to the nearest foot?

Choice G

69a

$$3^{2}+h^{2}=6^{2}$$ $$h=3\sqrt{3}\approx 3\times1.73=5.19$$


2010 December(69A) Math Question 45

Number Types & Properties

- Order of Values

45. Which of the following statements gives the rea.
number values of x for which x<x is true?

Choice A

$$x^{2}\lt x$$

Only for x …


2010 December(69A) Math Question 46

Probability

- One Event

46. A teacher asked all the students in the junior class about the number of cats and/or dogs their family had The results are given in the table below. How many students answered that their family had l or more cats?

Choice G

$$48+66=114$$


2010 December(69A) Math Question 47

Perimeter, Area, Volume

- Mixed Shapes

47. The Worthwhile Company's logo consists of
2 concentric circles. The radius of the outer circle of the logo on Worthwhile's building is 4 feet, and the distance between the outer circle and the inner circle is
1.75 feet. Which of the following is an expression for the area, in square feet, of the inner circle of the logo on Worthwhile's building

Choice A

If the radius of the …


Choice F

$$C^{2}=D \Rightarrow C=\sqrt{D}$$ $$C=\sqrt{B} \Rightarrow …


2010 December(69A) Math Question 49

Perimeter, Area, Volume

- Other Shapes

49. The volume, V, of a right circular cylinder with radius r and height h is given by the formula V= th. The right circular cylinder shown below has radius R and height H. A second right circular cylinder has radius
2R and height 3H. The volume of the second right circular cylinder is how many times the volume of the first right circular cylinder?

Choice D

If you double the radius …


2010 December(69A) Math Question 50

Trigonometry

- Pythagorean Theorem & sin cos tan

50. For right triangle  △ABC, sin∠A =  2/3, What is cos ∠A?

Choice J

69a

$$2^{2}+x^{2}=3^{2}$$ $$x=\sqrt{5}$$ $$\cos \angle A=\frac{\sqrt{5}}{3}$$


2010 December(69A) Math Question 51

Equation of a Line

- Equations that Describe a Line

51. The line with equation x-4y=12 in the standard (xy)
coordinate plane crosses the x-axis at which of the following points?

Choice E

The point where a line …


2010 December(69A) Math Question 52

Quadratic Equations

- Factoring

52. Alexia knows that the height of an object propelled vertically from a height of 48 feet can be modeled by h=-16r+32r+48, where h is the height, in feet, and r is the time, in seconds. Using this model, how many seconds will it take the object to reach a height of 64 feet?

Choice F

$$-16t^{2}+32t-16=0$$ $$\Rightarrow t^{2}-2t+1=0$$ $$t=1,-2$$

The …


2010 December(69A) Math Question 53

Angles

- Parallel Lines

53. In the figure below.
E is parallel to GD, B lies on AD and EC, F lies on GD, the measure of  ∠ABC is 132°, and ∠EBG ≅ ∠GBF ≅ ∠FBD. What is the measure of ∠BFD?

Choice C

69a

$$\angle CBD=180°-132°=48°$$ $$132°=3x$$ $$x=44°$$ $$\angle …


2010 December(69A) Math Question 54

Trigonometry

- Pythagorean Theorem & sin cos tan

54. For a certain angle with measure θ, sin θ = 0.4. What is csc θ?

Choice F

$$\csc =\frac{hypotenuse}{opposite}=\frac{1}{\sin}$$ $$=\frac{1}{0.4}=\frac{5}{2}$$


2010 December(69A) Math Question 55

Graphing Properties

- Create Graphs or Figures

55. Simpson Manufacturing tested the hydraulic pressure capacity of 2 brands of valves, valve a and valve b on their hydraulic pressure pump. valve B allowed a slower decrease in pressure than Valve A followed by a slower increase in pressure than Valve A. The minimum pressure for Valve A was less than the minimum pressure for Valve B. One of the following graphs best illustrates the test results. Which graph is

Choice E

If B has a slower …


2010 December(69A) Math Question 56

Probability

- Combinations

56. Happy Soup Company stamps a 6-character product code on each can of soup it produces. Each product code consists of 5 letters (from the 26-letter alphabet)
followed by a single digit(from the digits 0 to 9). The letters may repeat. How many such product codes are possible?

Choice K

Each of the first five …


2010 December(69A) Math Question 57

Absolute Values

- Absolute Value Expressions

57. For all real numbers x, the value of x-|x| is

Choice E

If x is positive or …


2010 December(69A) Math Question 58

Trigonometry

- Pythagorean Theorem & sin cos tan

58. For all values of x such that sin x >0 and cos x>0
which of the following expressions is equivalent to sin x > 1/2 cos x ?

Choice J

It's in the first quadrant, …


Choice A

Picture a vector of the …


Choice K

$$=\frac{\sqrt{r}}{\sqrt{s}}+\frac{\sqrt{s}}{\sqrt{r}}$$ $$=\frac{\sqrt{r}\times\sqrt{r}}{\sqrt{s}\times\sqrt{r}}+\frac{\sqrt{s}\times\sqrt{s}}{\sqrt{r}\times\sqrt{s}}$$ $$=\frac{r+s}{\sqrt{rs}}$$