ACT Form 70A Math Answer Explanation



2011 December(70A) Math Question 1

Word Problems

- Cost

1. Karol had a meal that cost $16.00. She tipped the waiter 20% of that amount for service. How much was the tip?

Choice E

$$16\times0.2=3.2$$


Choice J

$$\frac{15}{3.25}=\frac{10.4}{x}$$ $$15x=10.4\times3.25=33.8$$ $$x=\frac{33.8}{15}=2.25$$


Choice C

$$4x-6x=-7+11$$ $$-2x=4$$ $$x=-2$$


2011 December(70A) Math Question 4

Word Problems

- 1 Variable Equations

4. Neka earns his regular pay of $9.00 per hour for up to
40 hours of work in a week. For each hour over
40 hours of work in a week. Neka is paid 1(1/2) times his regular pay. How much does Neka earn for a week in which he works 43 hours?

Choice H

$$9\times40+9(1\frac{1}{2})(3)$$ $$=360+40.5$$ $$=400.5$$


Choice D

$$\Rightarrow 90\times13+n\times15$$


Choice K

$$10\times20\times14=2800$$


Choice A

$$2(3)^{2}(-5)-5(3)(4)^{2}$$ $$=2(9)(-5)-5(3)(16)$$ $$=-90-240$$ $$=-330$$


Choice J

$$(2\times4)(x^{3+5}y^{1+3})$$ $$=8x^{8}y^{4}$$


Choice B

$$4-\frac{1}{4}-1\frac{1}{2}$$ $$=4-\frac{1}{4}-1\frac{2}{4}$$ $$=2\frac{1}{4}$$


2011 December(70A) Math Question 10

Absolute Values

- Absolute Value Expressions

10. What is the value of | -9 | - | 7 -42 |?

Choice G

$$=9-35$$ $$=-26$$


2011 December(70A) Math Question 11

Angles

- Collinear Points & Triangles

11. In right triangle ACD shown below, B lies on side AC. Some angle measures are given. What is the measure of ∠BDC?

Choice B

70a

$$\angle BDC=90°-\angle CBD$$ $$\angle CBD=180°-25°-30°=55°$$ $$\Rightarrow \angle BDC=90°-55°=35°$$


2011 December(70A) Math Question 12

Equation of a Line

- Equations that Describe a Line

12 What is the slope of the line given by the equation
2x + 9y = -18?

Choice H

$$2x+9y=-18$$ $$\Rightarrow y=-\frac{2}{9}x-2$$ $$m=-\frac{2}{9}$$


Choice D

$$0.0000378=\frac{378}{10000000}=378\times10^{-7}$$


Choice F

When two numbers (or variables) are multiplied together within parentheses, the exponent the quantity in the parentheses is raised to can be distributed to both terms.

$$xy^{a}=x^{a}y^{a}$$


2011 December(70A) Math Question 15

Quadratic Equations

- Factoring

15. Which of the following expressions is a factor of the polynomial x-x-12?

Choice A

$$=(x-4)(x+3)$$


Choice G

$$\overline{PQ}:\overline{QR}=3:8$$ $$\overline{PQ}:\overline{PR}=3:3+8=3:11$$


Choice A

$$(A) 5^{2}-13=25-13=12$$


2011 December(70A) Math Question 18

Trigonometry

- Pythagorean Theorem & sin cos tan

18. In the isosceles right triangle below, AB= 10 feet.
what is the length, in feet, of AC

Choice K

$$45-45-90 triangle=1:1:\sqrt{2}$$ $$\overline{AC}=10\sqrt{2}=\sqrt{200}$$


Choice A

$$3x+y+4=14+2x$$ $$y=14-4+2x-3x=10-x$$ $$\Rightarrow y=-x+10$$


Choice K

$$\overline{AC}=\frac{5}{6}\overline{JK}=\frac{5}{6}\times18=15$$ $$\frac{5}{6}=\frac{x}{18}\Rightarrow x=15$$


2011 December(70A) Math Question 21

Perimeter, Area, Volume

- Mixed Shapes

21. The contractor determines that 36 bundles of shingles are needed to cover the surface of the roof of the garage. What is the price the contractor will pay for these shingles?

Choice E

Since the cost of each …


2011 December(70A) Math Question 22

Perimeter, Area, Volume

- Mixed Shapes

22. The contractor will paint the ceiling and inside walls of the garage. The area to be painted, rounded to the nearest square foot, is 2, 650 square feet. What is the minimum number of cans of paint that will be needed to paint this area with 2 coats of paint

Choice K

$$$2,650\times2=$5,330$$ $$$5,330\div400=$13.25$$


Choice B

If the hour hand of …


2011 December(70A) Math Question 24

Perimeter, Area, Volume

- Mixed Shapes

24. In square inches, what is the area of quadrilateral ABCD shown below?

Choice G

$$(4\times4)+(3\times4\times\frac{1}{2})$$ $$=16+6$$ $$=22$$


2011 December(70A) Math Question 25

Graphing Properties

- Interpret Graphs or Figures

25. The sides of a square are 6 cm long. One vertex of the square is at(2, 1)on a square coordinate grid marked in centimeter units. Which of the following points could also be a vertex of the square?

Choice E

Look for a point that …


Choice K

The table contains 11 scores …


2011 December(70A) Math Question 27

Statistics

- Mean, Median

27. What group has the greatest range in exam scores given in the table and what is that range, in points?

Choice E

The range of the scores …


2011 December(70A) Math Question 28

Statistics

- Mean, Median

28. After the 6th student in each group takes the exam, an average score will be calculated for each group.
Suppose the 6th student in Group B earns 100 points on her exam. what is the minimum exam score, in points, the 6th student in Group A would need to earn for Group a to have a higher average exam score than Group B with the 6th score?

Choice G

$$(85+86+87+88+89+100)\div6=89.167$$ $$(88+80+100+99+98+x)\div6\gt89.167$$ $$(465+x)\div6\gt89.167$$ $$(465+x)\gt535$$ $$x\gt70$$


2011 December(70A) Math Question 29

Angles

- Collinear Points & Triangles

29. The points A, B, and D are collinear, B is between A and D, and C is a point not on AD. Which of the following statements about the measures of the angles formed must be true
(Note: The figure below shows I possible arrangement of these points.)

Choice D

By definition, a linear pair …


2011 December(70A) Math Question 30

Quadratic Equations

- FOIL

30.For x2≠121,  ?

Choice F

$$=\frac{(x-11)(x-11)}{(x+11)(x-11)}$$ $$=\frac{(x-11)}{(x+11)}$$


2011 December(70A) Math Question 31

Quadratic Equations

- Factoring

31.The quadratic equation 12x = 28x can be solved by factoring. Which of the following states the solutions?

Choice B

$$12x^{2}=28x$$ $$x(12x-28)=0$$ $$x=0$$ $$12x=28$$ $$x=\frac{28}{12}=\frac{7}{3}$$


2011 December(70A) Math Question 32

Sequences and Patterns

- Logical Expressions

32. Which of the following statements is a logical conclusion from the 3 true statements given below?
All meeps are spools.
 All larps are spools.
 All bins are meeps.

Choice H

If “all blins are meeps,” …


2011 December(70A) Math Question 33

Inequalities

- Inequality Expressions

33. Which of the following inequalities is equivalent to
3+2>127 ?

Choice C

$$\Rightarrow -\frac{5x}{3}\gt10 $$ $$\Rightarrow -5x\gt30 …


Choice J

$$70=7\times10$$ $$110=11\times10$$ $$60=2\times3\times10$$ $$L.C.M=2\times3\times7\times11\times10=4,620$$


2011 December(70A) Math Question 35

Equation of a Line

- Equations that Describe a Line

35. What is the y-intercept of the graph of 3x+ 2y= 12 in the standard (r, y) coordinate plane?

Choice D

$$3x+2y=12$$ $$\Rightarrow 2y=-3x+12$$ $$\Rightarrow y=-\frac{3}{2}x+6$$ …


Choice K

$$Slope =\frac{rise}{run} = \frac{(change\ in\ …


Choice C

$$ 2(x + 5) + …


Choice G

$$200(1.5)^{0.5\cdot 4}=200(1.5)^{2}=200\times2.25=450$$


2011 December(70A) Math Question 39

Graphing Properties

- Interpret Graphs or Figures

39. Twp copy machines copy at the same rate and will both be used to make copies of a report. At 8:00 am,both machines begin copying. One of the machines breaks down at 11:00 a.m. and begins copying again at
2:00 p.m. Both machines finish copying at
5:00pm when the copying of the report is complete. One of the following graphs shows n, the number of copies made as h function of f, the time at any given point during the copying. Which graph is it?

Choice A

Based on the information, the …


2011 December(70A) Math Question 40

Graphing Properties

- Interpret Graphs or Figures

40. Consider the 4th-degree polynomial function graphed in the standard (x,y) coordinate plane below. The function has a local maximum at b, local minimums at a and d, and x intercepts at a, c, and e. On what interval(s)of x is the function increasing?

Choice J

The function is increasing on …


2011 December(70A) Math Question 41

Equation of a Line

- Distance & Midpoint

41. Point A(2, 1) is on a circle centered at (5, -1) in the standard (x,y) coordinate plane. What are the coordinates of the other end of the diameter through A?

Choice E

$$\frac{2+x}{2}=5\Rightarrow x=8$$ $$\frac{1+y}{2}=-1\Rightarrow y=-3$$ $$\Rightarrow …


Choice H

l: the length of the …


2011 December(70A) Math Question 43

Quadratic Equations

- Factoring

43. In the standard(x,y)coordinate plane, which of the following numbers CANNOT be the x-coordinate of a point on the graph of y=2+x-2?

Choice B

You can set the denominator …


2011 December(70A) Math Question 44

Sequences and Patterns

- General Sequences

44. What is the 135th digit to the right of the decimal point in the repeating decimal 0.428571?

Choice K

$$\frac{135}{6}=22...3$$ $$0.\overline{428571}$$

The third number …


Choice D

There are two things that …


2011 December(70A) Math Question 46

Graphing Properties

- Interpret Graphs or Figures

46. The graph of y=16-1x-2lI is shown in the stahdard (x,y)coordinate plane below. The turning points are A, B, and C. Which of the following statements is true regarding AB and AC

Choice F

$$A:\ (2, 6)$$ $$The\ slope\ …


2011 December(70A) Math Question 47

Inequalities

- Inequality Expressions

47. Keisha's weight does not vary more than 5 pounds from p pounds.
Which of the following inequalities gives the range of Keisha's weight, w, in pounds?

Choice A

According to the problem, p …


2011 December(70A) Math Question 48

Trigonometry

- Advanced sin, cos, tan functions

48. A surveyor is finding the dimensions of the triangular lot shown below. The length of I side is 80 feet. which of the following is an expression for the length, in feet, of the side of the lot opposite the 110 angle?
(Note: The law of sines states that in every triangle, the ratio of a sides length to the sine of the angle opposite that side is equal for all 3 sides.)

Choice J

$$\frac{80}{\sin24˚}=\frac{x}{\sin110˚}$$ $$x= 80\times\frac{\sin110˚}{\sin24}$$


2011 December(70A) Math Question 49

Trigonometry

- Advanced sin, cos, tan functions

49. A well in the form of a right circular cylinder is 25 feet deep and 3 feet in diameter. In Figure I below, the Sun is shining directly down on the bottom of the well.
Later in the day, Earth rotates so that the bottom of the well is just in shadow, creating the right triangle shown in Figure 2 below. Which of the following expressions gives 8?

Choice D

$$\tan=\frac{opposite}{adjacent}$$ $$\tan\theta=\frac{3}{25}$$ $$\theta=\tan^{-1}\frac{3}{25}$$


2011 December(70A) Math Question 50

Angles

- Circles

50. Points A and B lie on a circle with center C. The circle has a radius of 16 inches. The measure of ∠ACB is 90°.
 What is the length, in inches, of minor arc AB?

Choice G

$$C=2\pi r$$ $$2\pi r\times\frac{90˚}{360˚}=2\pi(16)(\frac{1}{4})=8\pi$$


2011 December(70A) Math Question 51

Trigonometry

- Pythagorean Theorem & sin cos tan

51. Whenever sin thete = 1, the value of cos thete is:

Choice C

$$sin0°=0;sin90°=1;sin180°=0;sin270°=-1;sin360°=0;$$ $$cos0°=1;cos90°=0;cos180°=-1;cos270°=0;cos360°=1;$$


Choice K

An obtuse triangle is a …


2011 December(70A) Math Question 53

Trigonometry

- Graphing Trig Functions

53. The graph of the function y = 3 sin(2x)is shown below in the standard (x,y) coordinate plane. What are the amplitude and the period of the function?

Choice A

70a

The amplitude is equal to …


2011 December(70A) Math Question 54

Sequences and Patterns

- Geometric Sequences

54. The height that a certain ball rebounds after it hits the ground is directly proportional to the height from which it falls. When the ball falls from an initial height of 270 cm, it hits tha ground for the first time and rebounds to a height of 180 cm. Arranged in order starting with the first bounce, the heights, in centimeters, that the ball rebounds form a sequence Which of the following characterizes this sequence?

Choice H

The equation for y varying …


Choice B

If this system has infinite …


2011 December(70A) Math Question 56

Probability

- One Event

56. Three circles have the same center. Their radii measure
1, 2, and 3 inches, respectively. If a point is chosen at random in the interior of the largest circle, what is the probability that the point is also in the interior of the smallest circle?

Choice H

$$Area\ of\ a\ circle:\ \pi …


Choice A

$$\frac{a}{b}=\frac{7}{11}$$ $$\Rightarrow a:b=7:11$$ $$a=7r,\ b=11r$$ …


2011 December(70A) Math Question 58

Perimeter, Area, Volume

- Mixed Shapes

58. Each of the 2 rectangular prisms shown below has height h units, and the areas of the shaded sides are equal. Which of the following measures for the 2 prisms must be equal?

Choice F

The volume of a rectangular …


2011 December(70A) Math Question 59

Logarithms

- Logarithm Expressions

59. For all x> 2, log(r-2)+log x=?

Choice C

$$\log_{a}{x}+\log_{a}{y}=\log_{a}{xy}$$ $$\log_{}{(x-2)x}=\log_{}{x^{2}-2x}$$


2011 December(70A) Math Question 60

Graphing Properties

- Interpret Graphs or Figures

60. Fifteen students took a test. All their scores were integers in the ranges shown in the bar graph below.
What is the highest possible value for the median score for these 15 students on this test?
(Note: The median of an odd number of scores is the middle score when the scores are ordered from least to greatest)

Choice J

70a

The median score is the …