1,296 ACT Practice Questions, 3rd Edition (2013)
Math Practice Section 3
60 Minutes—60 Questions
DIRECTIONS: Solve each problem, choose the correct answer, and then darken the corresponding oval on your answer document.
Do not linger over problems that take too much time. Solve as many as you can; then return to the others in the time you have left for this test.
You are permitted to use a calculator on this test. You may use your calculator for any problems you choose, but some of the problems may best be done without using a calculator.
Note: Unless otherwise stated, all of the following should be assumed:
1. Illustrative figures are NOT necessarily drawn to scale.
2. Geometric figures lie in a plane.
3. The word line indicates a straight line.
4. The word average indicates arithmetic mean.
1. For each of 3 years, the table below gives the number of different routes a runner ran, the number of runs she ran, and the total number of miles she ran.
To the nearest tenth of a mile, what is the average number of miles the runner ran per run in 2005?
2. The lengths of 2 sides are not given in the polygon below. If each angle between adjacent sides measures 90°, then, in meters, what is the perimeter of the polygon?
3. Which of the following inequalities represents the graph shown below on the real number line?
A. 0 < x < 5
B. 0 < x ≤ 4
C. −2 < x ≤ 4
D. 1 < x ≤ 4
E. 0 < x ≤ 4
4. What is the value of 4 + 3x – y when x = 3 and y = −1?
5. For integers x and y such that xy = 14, which of the following is NOT a possible value of x?
6. In cubic meters, what is the volume of a large cube whose edges each measure 6 meters in length?
7. Pat’s Pastries baked 80 apple pies and 50 loaves of apple bread to be sold at a 2-day Fall Festival. The pies were sold for $25 each and the loaves of bread were sold for $10 each. Which of the following expressions gives the total amount of money, in dollars, collected from selling all of the apple pies and B of the loaves of bread?
C. B + 80
D. 10B + 1,250
E. 10B + 2,000
8. In the figure below, W, X, and Z are collinear, the measure of WXY is 4a°, and the measure of YXZ is 11a°. What is the measure of WXY?
9. Each of the following values could represent a probability EXCEPT:
10. For the first several weeks after hiring a private tutor, Teddy’s score on a standardized test increased slowly. As Teddy began to understand the concepts more clearly, though, his standardized test scores improved more rapidly. After several more weeks, Teddy stopped working with his tutor and his scores did not improve any more. Which of the following graphs could represent all of Teddy’s standardized test scores as a function of time, in weeks, after he hired a private tutor?
11. The Northampton Volunteer Association has built a rectangular sandbox for a local elementary school and is ready to fill it with sand. The sandbox is 60 inches wide, 72 inches long, and will be filled 18 inches deep. Under the assumption that 1 bag of sand can fill 3,600 cubic inches of the sandbox, what is the minimum number of bags of sand they will need in order to fill the sandbox?
12. Salvador is trying to scale his rectangular self-portrait down to postcard size. The painting is 9 feet wide by 16 feet long. He is using a scale of inch = 1 foot for the postcard-sized self-portrait. What will be the dimensions, in inches, of Salvador’s postcard-sized self-portrait?
F. 1 by 4
G. 3 by 5
H. 3 by 4
J. 27 by 48
K. 36 by 64
13. The Crestview High School student body is made up only of freshmen, sophomores, juniors, and seniors. 25% of the students are freshmen, 35% are sophomores, and 20% are juniors. If no student can be considered to be in two classes, and there are 150 seniors, how many students make up the Crestview High School student body?
14. The circumference of a car tire is 75 inches. About how many revolutions does this car tire make traveling 225 feet (2,700 inches) without slipping?
15. (2 − 4t + 5t2) − (3t2 + 2t − 7) is equivalent to:
A. 2t2 − 6t + 9
B. 2t2 − 2t + 9
C. 2t4 − 2t2 − 5
D. 8t2 − 6t − 5
E. 8t4 − 6t2 − 5
16. At Blackstone Café, a regular entrée costs $18.00 while an entrée off the children’s menu costs less. Cliff treats his niece to dinner at the café and spends of a gift certificate on her children’s entrée and a drink. Afterwards, she orders a $6.00 dessert and he pays for that as well. When Cliff has paid for all of his niece’s food, he has exactly enough money left on the gift certificate to pay for his regular entrée. How much money was the gift certificate worth?
17. In June, Ms. Kunkel gave her English students 15 books to read over the summer. When classes resumed in September, she asked them what percentage of the books they had finished. Only one of the following values represents a possible percentage of books a student could have completed. Which one is it?
18. A geometric sequence has as its first 4 terms, −0.125, 1, −8, and 64. What is the 5th term of this sequence?
19. Which of the following is equivalent to (a − 5b)2?
A. 2a − 10b
B. a2 − 25b2
C. a2 − 10ab + 25b2
D. a2 − 12ab + 25b2
E. a2 − 25ab + 25b2
20. As shown in the figure below, Tony has determined that he must ride his skateboard down a long ramp to be able to jump a shorter ramp with enough time to complete a new trick. First, he needs to determine the dimensions of both the shorter and longer ramps. Tony is on his skateboard at point K, 20 feet above the ground. He then notes that the vertical height of the shorter ramp is 6 feet above the ground, and the length of the shorter ramp is 9 feet. Approximately how many feet long is the longer ramp?
(Note: In FKG and HJG, FGK is congruent to HGJ.
21. What is the solution to the equation 9x − (3x − 1) = 3?
22. The area of ABC below is 54 square meters. If altitude is 9 meters long, how long is AC, in meters?
23. Given g(x) = 4x2 − 8x + 2, what is the value of g(−5)?
24. A company will reimburse its employees’ personal expenses on weekend business trips. It will reimburse $0.80 for every $1.00 an employee spends, up to $100.00. For the next $200 an employee spends, the company will reimburse $0.70 for every $1.00 spent. For each additional dollar spent, the company will reimburse $0.60. If an employee was reimbursed $400.00, approximately how many dollars must she have spent on a weekend business trip?
25. The following table shows the ages of all the attendees of Camp Wannaboggin.
What percent of the Wannaboggin campers are at least 11 years old?
26. What percent of is ?
27. The newspaper headline below tells about a power outage. If there are 63,000 residences in Springfield, how many residences were affected by the outage?
28. The ratio of a side of square X to the length of rectangle Z is 3:4. The ratio of a side of square X to the width of rectangle Z is 3:2. What is the ratio of the area of square X to the area of rectangle Z?
29. In her Algebra II class, Mrs. Pemdas writes the following statement on the board: “a varies inversely as the product of b2 and c, and directly as d3.” She then asks her students to translate the statement into an equation. Which of the following equations, with k as the constant of proportionality, is a correct translation of Mrs. Pemdas’s statement?
E. a = kb2 cd2
30. In a certain isosceles triangle, the measure of the vertex angle is four times the measure of each of the base angles. What is the measure, in degrees, of the vertex angle?
31. A restaurant decides on the following production model, where N is the number of ounces of flour the restaurant purchases each month, based on the number of ounces, x, the restaurant uses during the preceding month. N = x2 − 600x − 160,000. According to this model, what is the greatest quantity of flour, in number of ounces, that the restaurant can use during a month, without having to purchase any new flour the next month?
Use the following information to answer questions 32−34.
A poor, frustrated artist named Fresco created a plan to make money. He collected trash, repurposed it into sculptures, then asked various celebrities to write and paint on these trash objects, which he then sold on his own as modern high art. The chart below separately shows the cost and revenue of his plan. The linear cost function, C(x), represents the total money spent to make and market the art, while the linear revenue function, R(x), shows the amount of money he has made in sculpture sales.
32. Fresco initially spent money promoting the project in the media. He also had to pay the celebrities to participate. After 6 months, Fresco had created and sold x number of trash sculptures and finally broke even: he hadn’t made or lost any money. How many sculptures did Fresco sell in his first 6 months of the project?
33. The cost function in the chart is determined by a constant production cost per sculpture—in this case the amount Fresco pays each celebrity to participate—as well as a fixed cost, or the initial cost of promoting the project. What is the fixed cost of Fresco’s trash sculpture project?
34. The selling price of each trash sculpture is an integer number of dollars. According to the revenue function, what is the selling price of one trash sculpture?
K. Cannot be determined from the chart
35. Which of the following is a complete factorization of the expression 12b2c + 6bc + 3b?
A. 4bc + 2c + 1
B. 3b (9bc + 2c + 1)
C. 3b (4bc + 2c + 1)
D. 3b (4bc + 2c)
E. 6bc (2b + 6) + 3b
36. Which of the following could be the equation of a line that passes through the points (−2,−7) and (2,17) in the standard (x,y) coordinate plane?
F. 3x − 2y = 8
G. 6x − y = −5
H. 5x − 2y = 7
J. 9x − 2y = −16
K. x + y = 6
37. A circle has a radius that is the same length as the sides of a square. If the square has a perimeter of 64 square inches, what is the area, in square inches, of the circle?
38. What is the y-coordinate of the solution of the following system, presuming the system has a solution?
8x + y = 30
8x + 4y = 96
K. The system has no solution.
Use the following information to answer questions 39−41.
In the figure below, M is on and Q is on The measurements are given in feet. Both NPQM and MQRL are trapezoids. The area, A, of a trapezoid is given by A = h(b1 + b2), where h is the height and b1 and b2 are the lengths of the 2 parallel sides.
39. What is the area of MQRL, in square feet?
40. What is the length of in feet?
41. What is the diameter, in feet, of the largest circle that can be drawn inside MNPQ?
42. The figure below shows a ramp for skateboarders. The base of the ramp is 25 feet long, and it rises at a 10° angle.
Given the trigonometric calculations in the table below, how high off the ground will a skateboarder be at the top of the ramp, rounded to the nearest 0.1 foot?
43. The 12 numbers on a circular clock are equally spaced around the edges of the clock. Belinda chooses an integer, n, that is greater than 1. Beginning at a randomly chosen number, Belinda goes around the circle counterclockwise and paints in every nth number. She continues going around and around the clock, painting in every nth number, until all twelve numbers on the clock are painted. Which of the following could have been Belinda’s integer n?
44. Consider the exponential equation y = , where K and p are positive real constants and x is a positive real number. The value of y decreases as the value of x increases if and only if which of the following statements about p is true?
F. 0 < p < 1
G. 1 < p < 2
H. p > −1
J. p > 0
K. p > 1
45. What is the distance, in coordinate units, between the points M (1,−3) and N (−5,5) in the standard (x,y) coordinate plane?
46. During their daily training race, Carl has to stop to tie his shoes. Melissa, whose shoes are velcro, continues to run and gets 20 feet ahead of Carl. Melissa is running at a constant rate of 8 feet per second, and Carl starts running at a constant rate of 9.2 feet per second to catch up to Melissa. Which of the following equations, when solved for s, gives the number of seconds Carl will take to catch up to Melissa?
F. 8s + 20 = 9.2s
G. 8s − 20 = 9.2s
J. 8s = 20
K. 9.2s = 20
47. Which of the following defines the solution set for the system of inequalities given below?
0 > 3x − 6
−4 < x
A. x > − 4
B. x < 2
C. −4 < x < 18
D. −4 < x < −2
E. −4 < x < 2
48. At the company YouGroove, 35 employees work in the sales department and 50 employees work in the operations department. Of these employees, 15 work in both the sales and the operations departments. How many of the 110 employees at YouGroove do NOT work in either the sales or the operations departments?
49. The slope of a line in the standard (x,y) coordinate plane is 4. What is the slope of a line perpendicular to that line?
50. The point (24,3) on a standard (x,y) coordinate plane is halfway between points (z,2z + 1) and (15z,z − 4). What is the value of z?
51. How many 4-letter orderings, where no letters are repeated, can be made using the letters of the word BADGERS?
52. As shown in the (x,y,z) coordinate space below, the cube with vertices L through S has edges that are 2 coordinate units long. The coordinates of Q are (0,0,0), and S is on the positive x-axis. What are the coordinates of O?
53. Whenever a, b, and c are positive real numbers, which of the following expressions is equivalent to
D. log4(a − c) − log8 2b
E. log4(a − c) − log8b2
54. If −6≤a≤−4 and 3≤b≤7, what is the maximum value of |a − 2b|?
55. The measure of the sum of the interior angles of a regular n-sided polygon is (n − 2)180°. A regular octagon is shown below. What is the measure of the designated angle?
56. Which of the following trigonometric functions has an amplitude of 3?
(Note: the amplitude of a trigonometric function is the nonnegative difference between the maximum and minimum values of the function.)
F. f(x) = sin x
G. f(x) = cos 3x
H. f(x) = sin(x)
J. f(x) = 3 tan x
K. f(x) = 3 cos x
57. If A, x, and y are all distinct numbers, and A = , which of the following represents x in terms of A and y?
58. In the figure below, lines p and q are parallel and angle measures are as marked. If it can be determined, what is the value of a?
K. Cannot be determined from the information given
59. In the triangle below, the lengths of the two given sides are measured in centimeters. What is the value, in centimeters, of x?
A. 9 sin 40°
B. 9 sin 50°
C. 9 cos 50°
D. 9 tan 40°
E. 9 tan 50°
60. An angle in the standard (x,y) coordinate plane has its vertex at the origin and its initial side on the positive x-axis. If the measure of an angle in standard position is (1,314°), it has the same terminal side as an angle of each of the following measures EXCEPT:
Answers and Explanations
MATH PRACTICE 3 ANSWERS
MATH PRACTICE 3 EXPLANATIONS
1. C Divide the total number of miles the runner ran in 2005 by the number of runs she ran in that year to find the average number of miles per run: , which rounds to 3.2 miles. There’s a lot of extra information in this table—make sure you’re using only what you need.
2. G Because the angles in the polygon are 90°, find the unlabeled lengths by subtracting the shorter labeled sides from the longer labeled sides opposite them (parallel). Calculate 30 − 25 = 5, and 15 − 5 = 10, so the two missing lengths are 5 and 10, and the perimeter is 30 + 15 + 25 + 10 + 5 + 5 = 90, or (G). Choice (F) adds the given values but forgets to calculate the missing sides, while (H) assumes the missing sides are equal to the given values 5 and 15. Choice (J) calculates the area correctly, while (K) calculates the area incorrectly (30 × 15).
3. B Start by looking at the endpoints on the number line and match those up with the inequality signs in the answer choices. The left circle at 0 is an open circle, so this corresponds to < or >, eliminating choices (D) and (E). The right circle at 4 is a closed circle, so this corresponds to ≤ and ≥, eliminating choice (A). Now look at the range of values covered in the line: 0 to 4—a range that does not include −2 as choice (C) suggests.
4. J Plug In the values given into the expression, using order of operations (PEMDAS). Start with the exponent: 4 + 3(3 − (−1)) = 4 + 34 = 4 + 81 = 85. Choice (F) confuses the signs in the exponent, and choice (G) multiplies 3 × 4 instead of finding 34. Choice (H) subtracts 1 from the whole expression, instead of treating it as part of the exponent. Choice (K) adds 3 + 4 before raising it to the exponent.
5. D To determine which cannot be a value of x, try to use each of the answer choices as values for x and find a value for y for which xy = 14. In choice (A), if x = 2, y = 7. In choice (B), if x = 1, y = 14. In choice (C), if x = −7, y = −2. In choice (E), if x = −14, y = −1. Only choice (D) does not have a complementary integer value for y.
6. K The formula for the volume of a cube is as follows: V = s3. In this case, s = 6, so the volume is 216. Choice (F) finds s × 3; choice (G) finds the area of a square with sides of 6; choice (H) makes a calculation error; and choice (J) finds s2 × 3.
7. E Add the products of 80 pies sold at $25 each and B pies sold at $10 each: (80 × 25) + (B × 10) = 2,000 + 10B. Choice (D) finds the price of 50 pies instead of 80 and choice (C) gives the number of baked goods sold. Choice (A) finds the price of B loaves of bread and B apple pies. Choice (B) multiplies 10B by the sum of all other numbers in the problem.
8. G The sum of the two angles will be 180°, since together they make a straight line, and there are 180° in a straight line. To find WXY, first find the value of a. 4a° + 11a° = 180°, a = 12. WXY = 4a°, 4(12) = 48°. If you chose (F), you found the correct value for a but did not answer the question, which asks for WXY or 4a°.
9. E Probability is represented by a proper fraction, and (A), (B), (C), and (D)—even the small numbers—can be converted into proper fractions (e.g., .00004 becomes ). Choice (E) is correct because it is the only fraction here greater than one, which makes it an impossible probability (more than 100%). Something cannot happen 5 out of 4 times.
10. F Teddy’s standardized test scores do not decrease at any point, so you can eliminate choices (G) and (J). Nor do they only increase—at the end, his scores leveled off and neither increased nor decrease, so you can eliminate (K). And since these scores level off at the end, not in the middle, you can eliminate choice (H). Only choice (F) gives an accurate representation of scores that increase slowly, then increase quickly, then remain constant.
11. E First multiply the dimensions of the base of the sandbox to find its volume, then divide by the number of cubic inches per bag to get the number of bags needed. 60 × 72 × 18 = 77,760. 77,760 ÷ 3,600 = 21.6. Round up, since part of a 22nd bag will be needed. Choice (D) rounds down.
12. G When you are applying scales as the one given in this problem, ensure that you apply the same scale to both dimensions. Since the scale you are given is inch = 1 foot, simply multiply each of your values by . You don’t need to convert the values to inches because the scale is given from feet to inches. Accordingly, × 9 = 3 in, and × 16 = 5 in, answer choice (G). If you chose (F), you may have changed the scale value to in, and if you chose (H), you may have applied different scale values to each of the dimensions.
13. D First find what percentage of students are seniors. Since the percentage of students who are not seniors is 25 + 35 + 20 = 80, the remaining 20% of the students are seniors. Since there are 150 seniors, 150 is 20% of the total number of students. Now put the numbers into an equation: 150 = .20 × total. Divide both sides by .20 to find that the total is 750. Choice (C) miscalculates 150 to be 25% of the total, and choice (B) miscalculates it to be 30% of the total. Choice (A) adds all numbers in the problem, and choice (E) resembles the number of students but does not use the information given.
14. H For every single revolution, the tire will travel a horizontal distance equivalent to that tire’s circumference. To find how many revolutions this tire makes, simply divide 2,700 in ÷ 75 in = 36 rev. If you chose (F), be careful—you may not have converted the horizontal distance into inches and simply divided 225 ÷ 75. If you chose (K), you may have multiplied an extra 12 to the 2,700 in—but this value is already in inches!
15. A Distribute the minus sign throughout the parentheses before combining like terms: (2 − 4t + 5t2) − (3t2 + 2t − 7) = 2 − 4t + 5t2 − 3t2 − 2t + 7 = 2t2 − 6t + 9. The other choices all confuse signs in calculating. Choices (C) and (E) also add the exponents of the terms.
16. H Set up an equation with x as the total amount of money the gift certificate is worth. Subtract of the total amount for the niece’s entrée and drink, and $6.00 for the niece’s dessert, from the total gift certificate value. Set the equation equal to the amount of money Cliff will have left over: $18.00 for his regular entrée: x −(x) − $6.00 = $18.00. Solve for x: $x = $24.00, x = $36.00.
17. D To solve, check some possible fractions of books read (out of a possible 15): = 66.6666%; = 73.33333%; = 80%; = 86.6666%; and = 93.3333%. Only choice (D) gives a possible percentage. Alternatively, you can try all the percentages in the answer choices and see which gives you an integer value.
18. K A geometric sequence is one that has a constant factor between its terms. To find this constant multiple, divide the second term by the first (or the third by the second, the fourth by the third, etc.). In this case, the constant factor between all terms is 1 ÷ (−0.125) = (−8) ÷ 1 = −8. To find the 5th term, simply multiply the fourth term by −8: 64 × (−8) = −512. If you forget what a geometric sequence is, look at what the numbers are doing. First and foremost, they’re alternating between negative and positive, so the fifth term must be negative—this enables you to eliminate choices (F) and (G). Next, notice that the magnitude of each number (ignoring its sign) is getting larger, so eliminate choice (H). From there, make sure that you’ve got a relationship of multiplication (geometric sequence) rather than addition (arithmetic sequence). If you chose (J), be careful: you added the second through third terms of the sequence—making them all negative—to get (−1) + (−8) + (−64) = −73.
19. C The square refers to everything inside the parentheses, so use FOIL (First Outer Inner Last). (a − 5b) (a − 5b) = a2 − 5ab − 5ab + 25b2 = a2 − 10ab + 25b2. Choices (D) and (E) miscalculate the middle term, and choice (B) squares each term individually. Choice (A) adds the binomials instead of multiplying them.
20. J Use similar triangles. Since FGK is congruent to HGJ, and angles F and H are both right angles, you know the remaining angles are equal as well. Because they share three congruent angles, FKG and HJG are similar. You know, therefore, that the sides are proportional. Set up the following proportion to find the length of the longer ramp :
21. C First distribute the minus sign through the parentheses to get 9x − 3x + 1 = 3. Combine the terms on the left and subtract 1 from both sides: 6x = 2. Divide both sides by three to find that x = . Choice (B) and (D) do not distribute the parentheses and choices (A) and (E) divide incorrectly.
22. J Since you have the full area of the triangle and its height, or altitude, , you can find its base by rearranging the area formula A = bh to be b = . Simply enter the numbers from the problem to find the base: b = = 12. If you chose (G), be careful—you may have forgotten the .
23. B Plug −5 into the expression given for g(x) and watch your signs. g(−5) = 4(−5)2 − 8(−5) + 2 = 4(25) − (−40) + 2 = 100 + 40 + 2 = 142. Choice (A) squares the value of 4x and choice (C) omits the 4 in calculating. Choices (D) and (E) confuse signs.
24. G For the first $100.00 spent, multiply $100.00 × $0.80 = $80.00 that the company will reimburse. For the next $200.00 spent, multiply $200.00 × $0.70 = $140.00. So far, for $300.00 spent, the company will have reimbursed $80.00 + $140.00 = $220.00. Subtract $400.00 − $220.00 = $180.00 that the employee was reimbursed. To find the additional amount of money the employee must have spent, set up an equation with x as the additional number of dollars. $0.60(x) = $180.00, x = $300.00. Finally, add all of the dollars spent: $100.00 + $200.00 + $300.00 = $600.00.
25. E The campers who are at least 11 years old include the 11-, 12-, and 13-year-olds. Because all the values in the chart represent percents of the same number, you can simply add them together to get 21 + 37 + 8 = 66%, or choice (E). Choice (A) counts only the 9 and 10 year old percentages; choice (B) counts the percentage of everyone older than 11 (12 and 13 year olds); and choice (D) counts ages up to and including 11. Choice (C) incorrectly guesses 50% because 11 is the median of the 5 numbers.
26. G Although this question is short, it gives the terms in an order that might be unfamiliar to you. It might help to rearrange the terms in the question to this: is what percent of ? To find the answer simply divide = 0.2 or 20%. Choice (F) is converted to a percent, and choice (J) is converted to a percent. If you chose (K), be careful—you switched your terms to find that is 500% of .
27. D Multiply the number of residences times the fraction affected by the outage to find the total number of people affected. (63,000)() = 42,000. Choice (B) tells the number of residences not affected. Choice (C) finds of the residences. Choice (A) divides the number of residences by the product of 2 and 3, while choice (E) subtracts 2 and 3 from the number of residences.
28. K Since you know that the sides of a square are always equal, pay close attention to the numbers given in the problem. Use them as you draw square X and rectangle Y. In this case, that means the sides of square X are both 3, and the sides of rectangle Y are 2 and 4. Since you know that the area of a square is A = s2 and the area of a rectangle is A = lw, find the area of each figure with the numbers from the problem. The area of square X is 9, and the area of rectangle Z is 8, so the (area of square X):(area of rectangle Z) is 9:8. Note, you can choose different numbers for the sides of square X and rectangle Z as long as these numbers are in the proportion outlined in the problem. As long as they are, you’ll see that whatever ratio you find between the areas will reduce to 9:8.
29. A Take the statement in small pieces, and don’t worry, you don’t need to know how the constant of proportionality works to answer this question correctly. Since a varies inversely as the product of b2 and c, make sure that these values are both in the denominator of the right side of the equation. This eliminates choices (B), (C), (D), and (E). In case you are interested, the standard forms of direct and inverse variation with k as the constant of proportionality are y = kx (direct) and y = (inverse).
30. J The base angles of any isosceles triangle are equal, the sum of the angles of any triangle is 180°, and the vertex angle of this triangle is 4 times its base angles, so 4x + x + x = 180. 6x = 180, or x = 30. The question asks for the vertex angle, which is 4x = 4(30) = 120. Choice (F) gives the base angles (or the partial answer x you found in your equation), choice (H) gives the sum of the base angles, and choice (K) only subtracts the value of one base angle from 180. Choice (G) finds the value of x when 4x is equal to 180, omitting the base angles.
31. A Set N = 0 and factor the equation into (x − 800)(x + 200), because −800 × 200 = −160,000 and −800 + 200 = −600. Set each factor equal to 0 and solve for x. Either x = 800 ounces or x = −200 ounces. Since −200 ounces is not possible, x = 800 ounces. You could also try substituting the numbers in the answer choices for the x values in the given equation. You would have found that only 800 ounces used yields 0 new ounces that need to be purchased.
32. G If Fresco finally broke even, his cost should be the same as the revenue. The point where the C(x) and R(x) functions cross represents 5 sculptures sold, or choice (G). Choice (F) shows a revenue that’s still less than the cost, while (H) and (J) show a profit for the company because the revenue is greater than the cost. If you’re reading the y-axis numbers instead of the x-axis, you might be misled to choice (K).
33. C The cost function, or C(x), begins at the $10,000 mark on the y-axis, so this is its fixed cost, or choice (C). Choice (A) calculates the cost per sculpture, or fee paid each celebrity ($1,000). If you’re mistakenly looking at the x-axis you might guess choice (D) or (E).
34. J To determine the cost of each trash sculpture, consider a point on the revenue function line, R(x), and divide the money by the number of sculptures sold (). A convenient point shows 5 sculptures sold for $15,000, which is $3,000 per sculpture, or choice (J). Choice (F) calculates the cost per sculpture, or fee paid each celebrity ($1,000). If you’re mistakenly reading the C(x) line, you might calculate that 10 sculptures cost $20,000, or choice (H) $2,000 per sculpture; or that 15 sculptures cost $25,000, or roughly choice (G), $1,667.
35. C As you factor, take out one piece at a time, and remember that if you factor something out of the equation, you must be able to factor it out of every term: 12b2c + 6bc + 3b = b(12bc + 6c + 3) = 3b(4bc + 2c + 1). Choice (A) omits the 3b you’ve factored out of the other terms; choice (B) erroneously calculates 12 − 3 = 9 instead of 12 ÷ 3 = 4 for the first term in the parentheses; choice (D) omits the 3b ÷ 3b = 1 term (the last in the parentheses); and choice (E) is factored correctly but it is not completely factored as the problem asks.
36. G Use the two points given to find the slope of the line make a: . Manipulate the answer choices to match the slope-intercept form of a line, where y = mx + b. For choice (G), add y and subtract −5 from both sides to get 6x + 5 = y, or y = 6x + 5. m = 6, which is the slope you need. None of the other lines have this slope, so you don’t need to find the value of b. Choice (H) confuses signs in calculating the slope. Choice (K) confuses the slope and y-intercept. You can also answer this question by plugging the two given points into answer choices, but make sure both points work. Choice (F) works only for the first point given, and choice (J) works only for the second point.
37. E The perimeter of a square is equal to the sum of its sides. For this square, 64 = 4s, s = 16. The radius of the circle is thus also 16. To find the area of the circle, use A = πr2 = π (16)2 = 256π. Choice (A) gives the radius of the circle and choice (C) gives its circumference. Choice (D) mistakes the perimeter of the square for its area, and choice (B) uses that calculation to find the circumference of the circle.
38. J You can solve this problem at least three different ways. The fastest way, if you know how to do it, is to set up simultaneous equations like this:
8x + 4y = 96
8x + y = 30 −1(8x + y = 30) −8x − y = −30, and when you’ve got opposite coefficients for one of your variables, add the two equations together like this: , and with 3y = 66, find y = 22. If you’re not sure how to use simultaneous equations, you can also solve for one variable in one equation: 8x + y = 30 8x = 30 − y x = , and then substitute it into the other equation: 8x + 4y = 96 + 4y = 96 3y = 66 and again, y = 22. Finally, since you know that your x-coordinate and y-coordinate must be the same in each equation, you can substitute possible y-coordinates and see which one produces the same x-coordinate in both equations. If you chose (F), be careful—this is your x-coordinate, and the problem asks for the y-coordinate.
39. C Substitute the numbers in the diagram for the variables in the formula given for the area of a trapezoid: A = (40)(50 + 30) = 1,600 ft2. If you chose (A), be careful—you may have forgotten the .
40. F Make a right triangle by drawing in a perpendicular line from R to . This new line will be the same height as . Since is 30 feet, the base of the new triangle will be 50 − 30 = 20 feet. To find , use the Pythagorean theorem: a2 + b2 = c2, where a and b are the two legs of a right triangle, and c is the hypotenuse. 202 + 402 = , 2,000 = , = .
41. B , which is 40 feet, limits the diameter of the circle. If you chose (A), be careful—this is the radius.
42. J To find the height of the ramp you’ll need to use the tangent of 10°, since tangent is equal to length of side opposite the angle (unknown ramp height) divided by length of side adjacent to the angle (length of the base, or 25 feet). Calculate 0.176 × 25, and you’ll get approximately 4.4, or choice (J). If you use the sine of 10°, you’ll get 4.3, or (H); cosine will give you 24.6, or (K). If you multiply all three given numbers (cos, sin, tan) by 25, you’ll get 2.3, or (F).
43. D Draw the circular clock and mark twelve evenly spaced points on it, and start trying out the numbers for n in the given problem. Choose any starting point—let’s say two for this example. If you start at 2 and every two numbers are painted, as in choice (A), then in the first revolution, the numbers painted will be 4, 6, 8, 10, 12; in the second revolution, they will be 2, 4, 6, 8, 10, 12. In other words, if the integer n is 2, then there is no way that all of the numbers on the face of the clock will be painted. Choices (B), (C), and (E) all create the same issue. Only choice (D), n = 7, will fill in all of the numbers on successive revolutions.
44. F Since (x + 1) will never be a negative exponent, increasing this exponent will always increase the value of a number greater than 1 and decrease the value of a real number between 0 and 1. The problem stipulates that p must be positive, so if y decreases as x increases, p must be a fractional constant less than 1.
45. D Plug In the x and y values of the two given points into the distance formula. You can start with either point as long as you start with the same point each time. distance = . Choice (A) does not square the differences of the x and y values. Choice (B) confuses signs in subtracting in the first step. Choice (E) makes this same error and omits the square root sign as well. Choice (C) gives the amount of change along the x axis, which neglects the vertical distance.
46. F To solve this problem, use the distance formula d = rt. If both runners start from the point at which Carl had to stop to tie his shoes, and both d represent the distance at which they meet, then Carl will run d = 9.2s and Melissa will run d = 8s + 20 because she has a 20 foot head start. Because the d is the same for each equation, simply set these equations equal to each other to find 8s + 20 = 9.2s. If you chose (G), be careful—you may have given Carl the head start instead of Melissa. Choices (J) and (K) give the time it takes Carl and Melissa to run 20 feet, respectively.
47. E Manipulate the first given inequality to get x alone on one side. −3x > −6, or x < 2. Don’t forget to flip the inequality sign when dividing both sides by a negative number. Combine the first inequality with the second to get −4 < x < 2. Choice (B) gives the solution to only the first inequality and choice (A) repeats the second inequality. Choices (C) and (D) make errors in manipulating the first inequality.
48. K To solve this problem, use the group formula: Total = Group1 + Group2 – Both + Neither. In this problem, that means 110 employees = (35 sales) + (50 operations) – (15 both) + Neither. Solve for Neither = 40, choice (K). Choice (F) erroneously subtracts all the smaller values from the total number of employees; choice (G) subtracts sales employees from operations employees; choice (H) subtracts employees working in both departments from the number of employees in sales; and choice (J) subtracts employees working in both departments from the number of employees in operations.
49. C The slope of a perpendicular line is the negative reciprocal of the slope of the line to which it is perpendicular.
50. H If (24,3) is the midpoint of the other two points, then the average of the two x values should be 24; calculate = 24, and z = 3, or (H). Choice (G) mistakenly calculates z + 15z = 24 (where z = 1.5). The other choices mistakenly set point values equal to each other. Choice (F) calculates 2z + 1 = 3 (where z = 1); (J) calculates z − 4 = 3 (where z = 7); and (K) simply sets z equal to 24.
51. D The word BADGERS has 7 letters, and you are looking for combinations of four letters with no repeats. There are four letter-slots to fill, so you need to figure out how many letters can go in each of these four slots. Since there are 7 letters in BADGERS, you can choose any of them for the first slot. Since there are no repeats, and one of your letters will be in the first slot, you have 6 options for the second slot, 5 options for the third, and 4 options for the fourth. Multiply these numbers together to get (7) (6) (5) (4) = 840. Choice (A) gives only the number of slots; choice (B) gives only the number of letters in the word BADGERS; choice (C) gives 44; and choice (E) gives 74.
52. H Compare point O to the answer choices, watching the order of the three axes. Point O shifts neither to the left nor right in the y direction, so the y value will be 0, eliminating choices (G) and (K). The height of the cube is 2 units and side starts where z is 0, so the z coordinate of O is 2, eliminating choice (J). The distance along the x axis from Q to S is the diagonal of square PQRS. The diagonal of a square is the length of its side x , or 2, so the x value of O is 2, eliminating choice (F).
53. A To combine logarithms (logs), the bases must be the same, so eliminate choice (C). Group the first and last terms together, since they have common bases, to get log4 a + log4 c − 2log8b. The laws of logarithms state that . Eliminate choices (B) and (D) which multiply b by 2 instead of squaring it. A fractional power represents a root, so . Eliminate choice (E), which does not include . You are left with choice (A). The laws of logs also state that .
54. J Since you’re dealing with absolute value, all solutions of the problem will be positive, so work with the extremes of each inequality until you find the greatest value. The greatest value results when a = −6 and b = 7:|a − 2b| = |−6 − 14| = |−20| = 20. Choice (H) gives the value if a = −4 and b = 7. Choice (K) gives the absolute value of ab if a = −6 and b = 7.
55. D Octagons have 8 sides, so using the formula in the question, the sum of the interior angles measures (8 − 2)180 = 1,080. The angles of regular polygons are equal, so divide by 8 to find the measure of each angle: = 135°. The designated angle in the figure is an exterior angle, and there are 360° in a circle, so subtract the interior angle from 360° to find the measure of the designated angle: 360° − 135° = 225°. If you chose (A), the measure of one of the interior angles, be careful—make sure you’re answering the right question.
56. K First, tan x does not have an amplitude: Choice (J) is out. Second, multiplying the entire function by a constant stretches the graph vertically and changes the amplitude. Both sin x and cos x alone have amplitudes of 1. Multiplying the functions sin x or cos x by a constant will make the amplitude equal to that constant. Altering the angle, as in choices (G) and (H) does not change the amplitude of the function—it changes the period.
57. A To solve this algebraically, you need to work with the original equation, A = , and try to isolate the x variable. First multiply both sides by (x – y), and you’ll get A(x – y) = xy − 2. If you distribute the left side of the equation, you get Ax – Ay = xy − 2. Now subtract and add terms from each side so that all the terms with x are on one side; you should get Ax – xy = Ay − 2. Pull the x from each term on the left side and you have x(A – y) = Ay − 2. Divide both sides by (A – y) and you have the answer: x = , or choice (A). If you try to solve by substituting numbers and you pick tricky numbers such as x = 1 and y = 0 (for which A = −2), you’ll be misled to choices (D) and (E), since both will seem correct (x = 1). If you mistakenly set the answer choices equal to the original equation (and again try to solve using 0 and 1 for the variables), you may be misled to choices (B) and (C).
58. H To complete this problem, extend the left transversal to form a triangle with a as one of its angles:
First, find that the supplement of 100° will be 80°. Note that because lines p and q are parallel, the uppermost angle of this triangle will be 45°. Now you can find a by subtracting your two known angles from the total angle measure of this triangle: 180° − 45° − 80° = 55°. Choice (F) erroneously subtracts the 45° and 100° angles from 180°. Choice (G) cannot work because a does not lie along the same transversal as 45°; choice (J) cannot work because the 100° angle and a do not share any parallel lines; and choice (K) cannot work because the value can be determined.
59. B Use SOHCAHTOA, eliminating answer choices that are not supported by the figure. Notice that the third angle of the triangle is 50° to reveal that sin 50° = , or x = 9 sin 50°. The remaining choices confuse the sides of the figure in finding sine, cosine, and tangent of the two angles.
60. G There are 360° in a circle, so each time you add or subtract 360° from an angle measure, you get another angle with the same terminal side. In the case of this problem, every time you find an angle with same the terminal side as 1,314°, you can eliminate it because this problem asks for the angle that does NOT share a terminal side with 1,314°. 1,314° − 360° = 954°. This is not an answer choice, so keep going. 954° − 360° = 594°, so eliminate choice (F). 594° − 360° = 234°, eliminating choice (H). 234° − 360° = −126°, so eliminate choice (J). −126° − 360 = −486, eliminating choice (K). Choice (G) resembles the angle in the problem, but does not land at the same point.