ACT Form 72C Math Answer Explanation


2014 June(72C) Math Question 1

Statistics Practice

- Mean, Median

1. The monthly fees for single rooms at 5 colleges are $370, $310, $380, $340, and $310, respectively. What is the mean of these monthly fees?

Choice C

To calculate the mean of n numbers, take the sum of the n numbers and divide it by n. $$\frac{$370+$310+$380+$340+$310}{5}=$342$$

2014 June(72C) Math Question 2

Word Problems Practice

- Cost

2. Disregarding sales tax, how much will you save when you buy a $12.00 compact disc that is on sale for 25% off?

Choice H

A discount is a percentage that is subtracted from a number.

For example, a 10% discount of $30 is $3 (10% converted to a decimal is .10 and .10 x 30 is 3).

$12 x 25% = $3

Choice A

$$450 = c\cdot 10^3 \\$$ $$ \Rightarrow x=0.45$$

2014 June(72C) Math Question 4

Word Problems Practice

- Cost

4. Jorge's current hourly wage for working at Denti Smiles is $12.00. Jorge was told that at the beginning of next, month, his new hourly wage will be an increase of 6% of his current hourly wage. What will be Jorge's new hourly wage?

Choice H

% increase = Increase ÷ Original Number × 100

$12 × 106% = $12.72

Choice E

In a Geometric Sequence each term is found by multiplying the previous term by a constant.

The constant for the sequence in question is -3. Therefore the sequence is:

1, -3, 9, -27, 81, -243, 729, ...

Choice K

A square root of a number a is a number y such that y^2 = a; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is a. $$\sqrt{a} = 36 \\ ⇒a = 36^2 = 1296$$

Choice C

$10 + $0.65 × 15 = $19.75

2014 June(72C) Math Question 8

Statistics Practice

- Mean, Median

8. The table below shows the number of cars Jing sold each month last year. What is the median of the data in the table?

Choice K

The median of a set of numbers is the middle number in the set (after the numbers have been arranged from least to greatest) -- or, if there are an even number of data, the median is the average of the middle two numbers.

In descending order, the set of numbers is:

13, 15, 16, 19, 19, 22, 25, 25, 26, 27, 28, 29

Median = (22 + 25) / 2 = 23.5

Choice C

A relationship of direct proportionality that, when plotted on a graph, traces a straight line. In linear relationships, any given change in an independent variable will always produce a corresponding change in the dependent variable.

Thus, d = at + k

14 = 0⋅a + k

20 = 1⋅a + k

k = 14

a = 6

d = 6t + 14

Choice K

x = 5/24 - 5/8 = 5/24 - 15/24 = - 10/24 = - 5/12

Choice A

The absolute value of a number is its distance from zero on a number line . For instance, and have the same absolute value (). So, the absolute value of a positive number is just the number itself, and the absolute value of a negative number is its opposite.

|-0.4|= 0.4

|-0.042| = 0.042

|-0.0048| = 0.0048

|0.04| = 0.04

|0.047| = 0.047

2014 June(72C) Math Question 12

Angles Practice

- Vertex Angles & Bisectors

12. In the figure below, C is the intersection of AD and BE. If it can be determined, what is the measure of LBAC ?

Choice G

72c

The sum of measures of three angles of any triangle is invariably equal to 180°.

Vertical Angles are angles opposite each other when two lines cross. They are always equal.

∠A = 180° - 35° - 45° = 100°

2014 June(72C) Math Question 13

Word Problems Practice

- 2 Variable Addition

13. This month, Kami sold 70 figurines in 2 sizes. The large figurines sold for $12 each, and the small figurines sold for $8 each. The amount of money he received from the sales of the large figurines was equal to the amount of money he received from the sales of the small figurines. How many large figurines did Kami sell this month?

Choice B

Let the number of large figurines be L;

Let the number of small figurines be S;

L + S = 70

12⋅L = 8⋅S

L = 28, S= 42

Choice K

√(2x) = 1+11 = 12

⇒ 2x = 144

⇒ x = 72

2014 June(72C) Math Question 15

Angles Practice

- Circles

15. Antwan drew the circle graph below describing his time spent at school in 1 day. His teacher said that the numbers of hours listed were correct, but that the central angle measures for the sectors were not correct. What should be the central angle measure for the Core subjects sector?

Choice C

A circle has a total of 360 degrees all the way around the center.

So if that central angle determining a sector has an angle measure of 60 degrees, then the sector takes up 60/360 or 1/6, of the degrees all the way around.

$$\frac{4}{4+1+1+3}\times 360°= 160°$$

Choice K

To find the area of a rectangle, multiply the length by the width. The formula is: A = L * W where A is the area, L is the length, W is the width, and * means multiply.

The perimeter of a rectangle is equal to the sum of all the sides. However, since a rectangle's opposite sides are congruent, we only need to know the length and width.

L × W = 32

L = 2W

⇒ W = 4, L = 8

⇒ Perimeter = (4 + 8) × 2 = 24

2014 June(72C) Math Question 17

Word Problems Practice

- Distance, Speed, Time

17. A car accelerated from 88 feet per second (fps) to 220 fps in exactly 3 seconds. Assuming the acceleration was constant, what was the car's acceleration, in feet per second per second, from 88 fps to 220 fps ?

Choice C

Given the final speed, vf (which is 220 fps), and the initial speed, vi (which is 88 fps), and you know the time needed (3 seconds), you can find the acceleration, a. Because vf – vi = a⋅t

vf – vi = a⋅t

⇒ 220 - 88 = 3a

⇒ a = 44

Choice J

Scientific notation is a way of writing very large or very small numbers. A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10. For example, 650,000,000 can be written in scientific notation as 6.5 ✕ 108

670,000,000 + 700,000,000 = 1,370,000,000 = 1.37 ✕ 109

2014 June(72C) Math Question 19

Angles Practice

- Vertex Angles & Bisectors

19. In a plane, the distinct lines AB and CD intersect at A, where A is between C and D. The measure of ∠BAC is 47°. What is the measure of ∠BAD ?

Choice D

72c

The angle measure of a straight line is 180 degrees.

Choice H

Rule name Rule Example
Product rules a na m = a n+m 23 ⋅ 24 = 23+4 = 128
a nb n = (a b) n 32 ⋅ 42 = (3⋅4)2 = 144
Quotient rules a n / a m = a nm 25 / 23 = 25-3 = 4
a n / b n = (a / b) n 43 / 23 = (4/2)3 = 8
Power rules (bn)m = bn⋅m (23)2 = 23⋅2 = 64
bnm = b(nm) 232 = 2(32)= 512
m√(bn) = b n/m 2√(26) = 26/2 = 8
b1/n = nb 81/3 = 38 = 2
Negative exponents b-n = 1 / bn 2-3 = 1/23 = 0.125
Zero rules b0 = 1 50 = 1
0n = 0 , for n>0 05 = 0
One rules b1 = b 51 = 5
1n = 1 15 = 1
Minus one rule (-1)n = -1 , n odd v
(-1)n = 1 , n even
(-1)5 = -1
Derivative rule (xn) = nx n-1 (x3) = 3⋅x3-1
Integral rule xndx = xn+1/(n+1)+C x2dx = x2+1/(2+1)+C

$$\Rightarrow \frac{(3x)^2}{x^5}=\frac{9x^2}{x^5}=\frac{9}{x^{5-2}}=\frac{9}{x^{3}}$$

Choice A

⇒ 4x - 2x = …

2014 June(72C) Math Question 22

Angles Practice

- Other Shapes

22. For trapezoid ABCD shown below, AB || DC, the measures of the interior angles are distinct, and the measure of ∠D is x°. What is the degree measure of ∠A in terms of x ?

Choice F

When parallel lines get crossed …

Choice A

$$\require{cancel} \frac{1}{2}\cdot y^{2}(6x+2y+12x-2y)$$ $$\Rightarrow \frac{1}{2}\cdot …

2014 June(72C) Math Question 24

Word Problems Practice

- % Interest

24. Sara and Behzad are saving to make a down payment on a house. With an initial deposit of $8,000, they have opened an account that compounds interest at an annual rate of 2.1%. Assuming that Sara and Behzad make no additional deposits or withdrawals, which of the following expressions gives the dollar value of the account 4 years after the initial deposit? (Note: For an account with an initial deposit of P dollars that compounds interest at an annual rate of r%, the value of the account t years after the initial deposit is P(1 + 00) dollars.)

Choice F

Plugging in the numbers in …

Choice D

72c

The Pythagorean theorem states that, …

2014 June(72C) Math Question 26

Quadratic Equations Practice

- Factoring

26. An artist makes a profit of (500p — p^2) dollars from selling p paintings. What is the fewest number of paintings the artist can sell to make a profit of at least $60,000 ?

Choice H

The phrase "At least" means …

Choice B

Among the categories of expenditures, …

2014 June(72C) Math Question 28

Inequalities Practice

- Intersection of Lines

28. When the system of inequalities below is graphed in the standard (x,y) coordinate plane, one of the following graphs is that of the solution set of the system. Which graph?

Choice G

To graph a linear inequality, …

Choice B

To obtain an angle from …

Choice K

An Isosceles Right Triangle is …

Choice C

72c

As we can see from …

Choice G

To obtain an angle from …

2014 June(72C) Math Question 33

Word Problems Practice

- % Interest

33. A formula to estimate the monthly payment, p dollars, on a short-term loan is ... where a dollars is the amount of the loan, r is the annual interest rate expressed as a decimal, and y years is the length of the loan. When a is multiplied by 2, what is the effect on p ?

Choice D

The formula can be rearranged …

2014 June(72C) Math Question 34

Probability Practice

- One Event

34. To make a 750-piece jigsaw puzzle more challenging, a puzzle company includes 5 extra pieces in the box along with the 750 pieces, and those 5 extra pieces do not fit anywhere in the puzzle. If you buy such a puzzle box, break the seal on the box, and immediately select 1 piece at random, what is the probability that it will be 1 of the extra pieces?

Choice J

The probability of selecting an …

Choice B

The perimeter of a rectangle …

Choice H

Plugging the data in the …

Choice B

It is given in the …

Choice H

72c

As we can see from …

Choice D

The cabinet charge out of …

Choice G

Given any two points on …

Choice A

The rule for a reflection …

Choice K

72c

A line perpendicular to the …

2014 June(72C) Math Question 43

Matrices Practice

- Addition & Subtraction

43. Given that for some real number a, what is x + z?

Choice D

72c

To multiply a matrix by …

Choice G

72c

With the supplemented lines, we …

Choice E

The perimeter of a triangle …

2014 June(72C) Math Question 46

Statistics Practice

- Mean, Median

46. The difference (larger minus smaller) between 2 numbers is 15. If n represents the larger number, which expression below represents the average (arithmetic mean) of the 2 numbers?

Choice K

To calculate the mean of …

Choice B

72c

To add fractions there are …

Choice J

72c

The center of rectangle is …

2014 June(72C) Math Question 49

Inequalities Practice

- Intersection of Lines

49. The shaded region in the graph below represents the solution set to which of the following systems of inequalities?

Choice A

To graph inequalities, first, graph …

Choice F

72c

It should be noted that …

Choice D

The area of a parallelogram …

Choice K

72c

$$A_{1}=\frac{1}{2}\cdot\frac{1}{3}Area_{Square} \\$$ $$ A_{2}=\frac{1}{3}\cdot\frac{1}{3}Area_{Square} \\$$ …

2014 June(72C) Math Question 53

Quadratic Equations Practice

- Factoring

53. Which of the following is a quadratic equation that has

Choice A

Factoring 9x2+12x+4=0, we …

2014 June(72C) Math Question 54

Word Problems Practice

- % Concentration

54. Bonkosi mixes 60 milliliters of Solution A with 40 milliliters of Solution X. Solution A has a 40% hydrochloric acid concentration; Solution X has an unknown hydrochloric acid concentration. When Bonkosi tests the resulting 100-milliliter solution, she finds that it has a 36% hydrochloric acid concentration. What is the hydrochloric acid concentration of Solution X ?

Choice H

$$40\% \times 60 + a\times40=36\%\times100 …

Choice E

The square root must be …

Choice F

It can be observed that …

Choice E

The absolute value of a …

Choice J

The smallest angle must be …

Choice D

To reduce a fraction to …

Choice J

72c

As we can see from …