ACT Form 70A Math Answer Explanation


2011 December(70A) Math Question 1

Word Problems Practice

- 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 Practice

- 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

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

Choice G

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

2011 December(70A) Math Question 11

Angles Practice

- 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

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

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 Practice

- 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

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

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 Practice

- 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 Practice

- 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 Practice

- Mixed Shapes

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

Choice G

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

Choice E

Look for a point that …

Choice K

The table contains 11 scores …

2011 December(70A) Math Question 27

Statistics Practice

- 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 Practice

- 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 Practice

- 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 Practice

- 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 Practice

- Factoring

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

Choice B

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

Choice H

If “all blins are meeps,” …

2011 December(70A) Math Question 33

Inequalities Practice

- 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

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

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

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

Choice A

Based on the information, the …

Choice J

The function is increasing on …

2011 December(70A) Math Question 41

Equation of a Line Practice

- 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 Practice

- 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 Practice

- 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 …

Choice F

$$A:\ (2, 6)$$ $$The\ slope\ …

2011 December(70A) Math Question 47

Inequalities Practice

- 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 …

Choice J

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

Choice D

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

2011 December(70A) Math Question 50

Angles Practice

- 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

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

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 Practice

- 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 …

Choice H

The equation for y varying …

Choice B

If this system has infinite …

2011 December(70A) Math Question 56

Probability Practice

- 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 Practice

- 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 Practice

- Logarithm Expressions

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

Choice C

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

Choice J

70a

The median score is the …