ACT Form 72C Math Answer Explanations
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Explain to me:
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?
To calculate the mean of n numbers, take the sum of the n numbers and divide it by n.
$$\frac{$370+$310+$380+$340+$310}{5}=$342$$
Choice C.
2. Disregarding sales tax, how much will you save when you buy a $12.00 compact disc that is on sale for 25% off?
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 H.
3. Given f = cd3, f= 450, and d = 10, what is c
$$450 = c\cdot 10^3 \\ \Rightarrow x=0.45$$
Choice A.
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?
% increase = Increase ÷ Original Number × 100
$12 × 106% = $12.72
Choice H.
5. The first term is 1 in the geometric. sequence 1, —3, 9, —27, …. What is the SEVENTH, term of the geometric sequence?
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 E.
6. If /a = b and b= 36, a = ?
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 K.
7. The shipping rate for customers of Ship Quick consists of a fee per box and a price per pound for each box. The table below gives the fee and the price per pound for customers shipping boxes of various weights.
$10 + $0.65 × 15 = $19.75
Choice C.
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?
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 K.
9. Students studying motion observed a cart rolling at a constant rate along a straight line. The table below gives the distance, d feet, the cart was from a reference point at 1-second intervals from t = 0 seconds to t= 5 seconds.
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 C.
10. If x + 5/8 = 5/24, then x = ?
x = 5/24 – 5/8 = 5/24 – 15/24 = – 10/24 = – 5/12
Choice K.
11. The absolute value of which of the following numbers is the greatest?
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
Choice A.
12. In the figure below, C is the intersection of AD and BE. If it can be determined, what is the measure of LBAC ?
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°
Choice G.
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?
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 B.
14. Given that √2x — 11 = 1, x = ?
√(2x) = 1+11 = 12
⇒ 2x = 144
⇒ x = 72
Choice K.
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?
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 C.
16. The area of a rectangular sheet of paper is 32 square inches. The length of the sheet of paper is twice its width. What is the perimeter, in inches, of the sheet of paper?
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
Choice K.
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 ?
Given the final speed, v_{f} (which is 220 fps), and the initial speed, v_{i} (which is 88 fps), and you know the time needed (3 seconds), you can find the acceleration, a. Because v_{f} – v_{i} = a⋅t
v_{f} – v_{i} = a⋅t
⇒ 220 – 88 = 3a
⇒ a = 44
Choice C.
18. In scientific notation, 670,000,000 + 700,000,000 = ?
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 ✕ 10^{8}
670,000,000 + 700,000,000 = 1,370,000,000 = 1.37 ✕ 10^{9}
Choice J.
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 ?
The angle measure of a straight line is 180 degrees.
Choice D.
20. Which of the following expressions is equivalent to (3x)^2/x^5 ?
Rule name | Rule | Example |
---|---|---|
Product rules | a^{ n} ⋅ a^{ m} = a^{ n+m} | 2^{3} ⋅ 2^{4} = 2^{3+4} = 128 |
a^{ n} ⋅ b^{ n} = (a ⋅ b)^{ n} | 3^{2} ⋅ 4^{2} = (3⋅4)^{2} = 144 | |
Quotient rules | a^{ n} / a^{ m} = a^{ n}^{–m} | 2^{5} / 2^{3} = 2^{5-3} = 4 |
a^{ n} / b^{ n} = (a / b)^{ n} | 4^{3} / 2^{3} = (4/2)^{3} = 8 | |
Power rules | (b^{n})^{m} = b^{n⋅m} | (2^{3})^{2} = 2^{3⋅2} = 64 |
_{b}n^{m} _{= b}(n^{m}) | _{2}3^{2} _{= 2}(3^{2})_{= 512} | |
^{m}√(b^{n}) = b ^{n/m} | ^{2}√(2^{6}) = 2^{6/2} = 8 | |
b^{1/n} = ^{n}√b | 8^{1/3} = ^{3}√8 = 2 | |
Negative exponents | b^{-n} = 1 / b^{n} | 2^{-3} = 1/2^{3} = 0.125 |
Zero rules | b^{0} = 1 | 5^{0} = 1 |
0^{n} = 0 , for n>0 | 0^{5} = 0 | |
One rules | b^{1} = b | 5^{1} = 5 |
1^{n} = 1 | 1^{5} = 1 | |
Minus one rule | (-1)^{n} = -1 , n odd v (-1)^{n} = 1 , n even |
(-1)^{5} = -1 |
Derivative rule | (x^{n})‘ = n⋅x^{ n}^{-1} | (x^{3})‘ = 3⋅x^{3-1} |
Integral rule | ∫ x^{n}dx = x^{n}^{+1}/(n+1)+C | ∫ x^{2}dx = x^{2+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 H.
21. Whenever 4x + 7 = 2x – g, which of the following expressions must be equal to x ?
⇒ 4x – 2x = -g – 7
⇒ 2x = -g – 7
…
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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 ?
When parallel lines get crossed by another line (which is called a Transversal), the sum of the measures of the consecutive interior angles …
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23. Which of the following expressions is equivalent to 1/2 y^2(6x + 2y + 12x -2y) ?
1/2 ⋅ y^{2 }(6x + 2y + 12x – 2y) =1/2 ⋅ y^{2 }(6x + 2y + 12x – 2y) = …
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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.)
Plugging in the numbers in the given formula, we have:
…
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25. Right triangle △RST has its right angle at vertex S. The length of ST is 6.0 feet and the length of RS is 2.5 feet. Which of the following values is closest to the length, in feet, of RT ?
The Pythagorean theorem states that, in a right triangle, the square of the length of the hypotenuse (the side across from the right angle) is equal to the sum of the squares of the other two sides. So if the length of the hypotenuse is c and the lengths of the other two sides are a and b, then c^{2 } = a^{2} + b^{2}.
…
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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 ?
The phrase “At least” means “greater than or equal to” (≥).
…
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27. Last month, Lucie had total expenditures of $900. The pie chart below breaks down these expenditures by category. The category in which Lucie’s expenditures were greatest is what percent of her total expenditures, to the nearest 1 % ?
Among the categories of expenditures, the expenditure on clothes is the greatest. Therefore, …
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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?
To graph a linear inequality, first, graph the “equals” line, then shade in the correct area.
There are three steps:
- Rearrange the equation so “y” is on the left and everything else on the right.
- Plot the “y=” line (make it a solid line for y≤ or y≥, and a dashed line for y< or y>)
- Shade above the line for a “greater than” (y> or y≥)
or below the line for a “less than” (y< or y≤).
…
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29. Janelle is loading a truck by using a ramp, as shown below. The ramp is 8 feet long, and the end of the ramp that is resting on the truck is 2.5 feet above the level ground. Which of the following expressions gives the angle of inclination of the ramp?
To obtain an angle from any of the angle’s trigonometric ratios, we can use the inverse trigonometric functions.
…
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30. What is the perimeter, in inches, of the isosceles right triangle shown below, whose hypotenuse is 8√2 inches
An Isosceles Right Triangle is a right triangle with the two legs (and their corresponding angles) equal. An isosceles right triangle therefore has angles of 45 degrees, 45 degrees, and 90 degrees. For an isosceles right triangle with side lengths a, the hypotenuse has length √2a.
…
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31. The equation y = ax^2 + bx + c is graphed in the standard (x,y) coordinate plane below for real values of a, b, and c. When y = 0, which of the following best describes the solutions for x ?
As we can see from the graph, the two solutions …
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32. The radius of the base of the right circular cone shown below is 5 inches, and the height of the cone is 7 inches. Solving which of the following equations gives the measure, 0, of the angle formed by a slant height of the cone and a radius?
To obtain an angle from any of the angle’s trigonometric ratios, we can use the inverse trigonometric functions.
…
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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 ?
The formula can be rearranged as:
$$p=\frac{(\frac{1}{2}ry+1)a}{12y}$$
…
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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?
The probability of selecting an extra piece (of which there are 5 options) out of ..
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35. The length of a rectangle is 3 inches more than twice the width of the rectangle. The perimeter of the rectangle is 36 inches. What is the width of the rectangle, in inches?
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36. The solid shown below is composed of a right circular cylinder and a right circular cone with base diameters and heights given in centimeters. The cylinder and the cone have equal base diameters. What is the volume, in cubic centimeters, of the solid?
Plugging the data in the formulae, the volume of the solid is …
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37. Gianna is converting a 12-foot-by-15-foot room in her house to a craft room. Gianna will install tile herself but will have CC Installations build and install the cabinets. The scale drawing shown below displays the location of the cabinets in the craft room (0.25 inch represents 2 feet). Cabinets will be installed along one of the 12-foot walls from floor to ceiling, and 4 cabinets that are each 3 feet tall will be installed in the middle of the room. These are the only cabinets that will be installed, and each of them will be 2 feet wide and 2 feet deep. CC Installations has given Gianna an estimate of $2,150.00 for building and installing the cabinets. A 15-foot wall is how many inches long in the scale drawing?
It is given in the question that “0.25 inch represents 2 feet”. Therefore, …
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38. Gianna will install tile on the portion of the floor that will NOT be covered by cabinets. What is the area, in square feet, of the portion of the floor that will NOT be covered by cabinets?
As we can see from the graph, area = …
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39. CC Installations’ estimate consists of a $650.00 charge for labor, plus a fixed charge per cabinet. The labor charge and the charge per cabinet remain the same for any number of cabinets built and installed. CC Installations would give Gianna what estimate if the craft room were to have twice as many cabinets as Gianna is planning to have?
The cabinet charge out of the total estimate cost is $2,150 – $650 = $1,500;
Were the craft room to have twice as many cabinets, the cabinet charge would have been …
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40. Trapezoid ABCD is graphed in the standard (x,y) coordinate plane below. What is the slope of CD ?
Given any two points on the line, its slope is given by the formula
$$slope = \frac{A_{y}-B_{y}}{A_{x}-B_{x}}$$
where:
A_{x} the x coordinate of point A
A_{y} the y coordinate of point A
B_{x} the x coordinate of point B
B_{y} the y coordinate of point B
Thus, …
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41. When ABCD is reflected over the y-axis to A’B’C’D’, what are the coordinates of D’?
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42. Which of the following vertical lines cuts ABCD into 2 trapezoids with equal areas?
A line perpendicular to the x-axis would have cut the trapezoid ABCD into two right trapezoids. The area of a right trapezoid is A = (1/2)(h)(a + b). Since the two new right trapezoids share the same height, their areas will be the same if the sums of the top length and the bottom length (a + b ) are the same. …
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43. Given that for some real number a, what is x + z?
To multiply a matrix by a single number:
…
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44. The points E(6,4) and F(14,12) lie in the standard (x,y) coordinate plane shown below. Point D lies on EF between E and F such that the length of EF is 4 times the length of DE. What are the coordinates of D ?
With the supplemented lines, we have two similar triangles with sides in the proportion of 1:4.
…
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45. A certain triangle has a perimeter of x meters. One side of the triangle is 60 meters long, another side is 1/3 the length of the perimeter, and the third side is 1/4 the length of the perimeter. What is the perimeter, in meters, of the triangle?
The perimeter of a triangle is the total distance around the outside, which can be found by adding together the length of each side. Or as a formula:
Perimeter = a + b + c
where:
a, b and c are the lengths of each side of the triangle.
…
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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?
To calculate the mean of n numbers, take the sum of the n numbers and divide it by n.
…
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47. 4/√2+2/√3=?
To add fractions there are Three Simple Steps:
- Step 1: Make sure the bottom numbers (the denominators) are the same
- Step 2: Add the top numbers (the numerators), put that answer over the denominator
- Step 3: Simplify the fraction (if needed)
…
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48. A square in the standard (x,y) coordinate plane has vertices at (1,3), (2,1), (4,2), and (3,4). Where do the diagonals of the square intersect?
The center of rectangle is the mid point of the diagonal end points of rectangle.
…
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49. The shaded region in the graph below represents the solution set to which of the following systems of inequalities?
To graph inequalities, first, graph the “equals” lines, then shade in the correct area.
…
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50. In the standard (x,y) coordinate plane, line a contains the points (-4,2) and (-1,-3), and line b contains the points (3,0) and (7,0). At what point does line a intersect line b ?
It should be noted that the line b is the x-axis. Thus the question is to find the point at which line a intersects with the x-axis.
To find the linear function of line a:
…
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51. In the figure below, the side lengths and the length of an altitude of parallelogram BCDE are given in feet. What is the area, in square feet, of BCDE ?
The area of a parallelogram is given by the formula
where
b is the length of any base
a is the corresponding altitude
…
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52. The square below is divided into 3 rows of equal area. In the top row, the region labeled A has the same area as the region labeled B. In the middle row, the 3 regions have equal areas. In the bottom row, the 4 regions have equal areas. What fraction of the square’s area is in a region labeled A ?
…
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53. Which of the following is a quadratic equation that has
Factoring 9x^{2}+12x+4=0, we have:
…
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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 ?
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55. What are the real number values of x that make the equation 4√x^12 = x^3 true?
The square root must be great than or equal to 0. …
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56. The functions y = sin x and y = sin(x + a) + b, for constants a and b, are graphed in the standard (x,y) coordinate plane below. The functions have the same maximum value. One of the following statements about the values of a and b is true. Which statement is it?
It can be observed that the function is not moved vertically, but only laterally on the left. …
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57. Which of the following number line graphs shows the solution set to the inequality |x — 5| < -1?
The absolute value of a number is its distance from zero on a number line.
…
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58. The sides of an acute triangle measure 14 cm, 18 cm, and 20 cm, respectively. Which of the following equations, when solved for 0, gives the measure of the smallest angle of the triangle?
The smallest angle must be opposite to the smallest side.
…
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59. For all values of x where the expression is defined,
To reduce a fraction to lowest terms, we first need to make the denominators of the two fractions above and below the fraction bar the same:
…
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60. Mr. Martin wants to plant 7 trees evenly spaced along a straight fence 300 feet long, with 1 of the trees at each end of the fence. About how many feet apart should he plant the trees?
…
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