1,296 ACT Practice Questions, 3rd Edition (2013)
Math Practice Section 2
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. Which of the following expressions is equivalent to 2a + 4b + 6c?
A. 8(a + b + c)
B. 2(a + 2b) + 3c
C. 2(a + 4b + 6c)
D. 2(a + 2b + 6c)
E. 2(a + 2b + 3c)
2. When written in symbols, “The square of the product of a and b” is represented as:
3. Every week, Donald records the amount of mileage that has accumulated on his truck. On Monday, Donald recorded that he had driven 16,450 kilometers. After a week of deliveries, his new recording was 18,130 kilometers. He drove for thirty hours during that week. What was his average driving speed during that week to the nearest kilometer per hour?
4. The dimensions of a block of cheese are 12 inches by 3 inches by 3 inches. What is the volume, in cubic inches, of the block of cheese?
5. If x is a real number and 3x = 81, then 3 × 2x =?
6. For the songs on Charlie’s mp3 player, the ratio of folk songs to rock songs is 3:11. Which of the following statements about the songs on his mp3 player is(are) true?
I. There are fewer folk songs than rock songs.
II. For every 11 rock songs, there are 3 folk songs.
III. Folk songs comprise of the songs on Charlie’s mp3 player.
F. I only
G. II only
H. I and II only
J. II and III only
K. I, II, and III
7. Matt needs gallons of hydrochloric acid for an experiment. He has gallons already. How many more gallons of hydrochloric acid does Matt need?
8. As shown below, the diagonals of rectangle EFGH intersect at the point (−2,−4) in the standard (x,y) coordinate plane. Point F is at (−7,−2). Which of the following are the coordinates of H?
9. Which of the following expressions is equivalent to ?
A. 9x + 5
B. x + 45
C. x + 5
10. The expression 23fg − 6f(5f +3g) is equivalent to:
F. 5fg − 30f2
H. 3g − 7fg
J. 41fg − 30f2
K. 30f2 − 5fg
11. A farmer sells strawberries at a market in both pint-sized containers and quart-sized containers. The farmer charges $3 for each pint, $5 for each quart, is always paid on the day of purchase, and sells no other goods. On a recent day, the farmer sold as many pint containers as quart containers and received $120 in sales. How many pints of strawberries did the farmer sell?
12. A rectangular piece of cloth has a length of 6 feet and a width of 1.5 feet. Brad estimates that the area is 12 square feet. His estimate is approximately what percent greater than the actual area?
13. The geometric mean of 3 positive numbers is the cube root of the product of the 3 numbers. What is the geometric mean of 2, 4, and 27?
14. A model for the number of questions on an assignment, when the assignment is worth p points, is q = . According to this model, what is the number of questions, q, for an assignment worth 80 points?
15. The expression x2 + 2x − 15 can be written as the product of 2 binomials with integer coefficients. One of the binomials is (x + 5). Which of the following is the other binomial?
A. (x2 − 3)
B. (x2 + 3)
C. (x − 3)
D. (x + 3)
E. (x − 5)
16. The production cost of x computers for a company over one year is $175x + $150,000. To minimize production costs in a given year to $465,000, how many computers can the company make in that year?
17. Given g(x) = , what is g()?
18. Hannah is 5 years younger than Nora, who is x years old. Which of the following is an expression for Hannah’s age in 2 years?
F. x − 3
G. x + 3
H. x + 7
J. 2x − 3
19. A rectangle is 5 times as wide as it is long. The area of the rectangle is 320 square feet. What is the perimeter of the rectangle, in feet?
20. In the figure below, C and D are both on the measure of BAC is equal to the measure of DAE and the measure of ACD is equal to the measure of ADC. Which of the following statements must be true?
(Note: The symbol ≅ means “is congruent to.”)
F. ABC is similar to AED.
G. The areas of triangles ACD and ADE are equal.
K. and are perpendicular.
21. Which of the following is equivalent to 8?
22. Admission to the martial arts tournament is $30, but participants must purchase separate tickets for each event they wish to participate in once inside. Each event is the same price as any other event. The graph below shows the total cost for a person, for admission and events, as a function of the number of events paid for. One of the following is the price of a single event. Which one is it?
23. A right triangle, shown below, has a longer leg measuring 16 centimeters. How long is the hypotenuse of the triangle, in centimeters?
24. If you add up 5 consecutive odd integers that are each greater than 15, what is the smallest possible sum?
25. A department store escalator is 25 feet long and forms an angle of 43° with the floor, which is horizontal. Which of the follow is an expression for the horizontal distance of the escalator from beginning to end?
A. 25 sin 43°
B. 25 cos 43°
C. 25 tan 43°
D. 25 csc 43°
E. 25 sec 43°
26. If x − 15 = |−5|, then x =?
27. A grocery store is running a sale on seasonal berries. During the sale, the store sells packages of blueberries for $4 each and packages of strawberries for $6 each. Kate purchased nine packages of fruit for her mother’s dinner party for $40. How many packages of blueberries did she purchase?
28. In VWY below, X lies on ; Z lies on ; and a, b, c, and d are angle measures, in degrees. The measure of Y is 45˚. What is a + b + c + d?
29. Triangle ACE, shown in the figure below, is isosceles with base . B lies on and D lies on . Segments and bisect AEC and CAE respectively. Which one of the following angle congruences is necessarily true?
A. CAE ≅ BEC
B. CAD ≅ AEC
C. CAE ≅ ACE
D. AEC ≅ ACE
E. BEC ≅ DAE
30. A trapezoid has parallel bases that measure 3 inches and 9 inches and a height that measures 6 inches. What is the area, in square inches, of the trapezoid?
31. The table below lists the number (to the nearest 1,000) of book club members in the United States for 2001 through 2004. Of the following expressions with x representing the number of years after 2001, which best models the number of book club members (in thousands) in the United States?
Number of members (in thousands)
A. 539x + 2,001
B. x + 2,001
C. x + 539
D. 547x + 2,004
E. 2,001x + 539
Use the following information to answer questions 32−34.
The table below shows the percents of U.S. citizens who had ever consumed a certain brand-name soda, out of all soda consumers, for each year from 1986 through 2006.
32. Which of the following years had the LEAST increase in the percent of U.S. citizens who had consumed the brand-name soda from the previous year?
33. The figure below shows a scatterplot of the data in the table and solid lines that are possible models for the data. Which of the 5 lines appears to be the best representation of the data?
34. By 2002 there were 74,672,120 U.S. citizens who had consumed the brand-name soda. According to this information, approximately how many people were soda consumers, of the brand-name soda or other sodas, in 2002?
35. For all nonzero y and z, =?
36. The function g(x) = x4− 2x3− 6x2– x + 5 and line h are shown in the standard (x,y) coordinate plane below. Which of the following is an equation of line h, which passes through (−1,3) and (2,−21)?
F. −8x − 5
G. −8x + 5
H. −9x + 5
J. −9x − 5
K. −6x − 5
37. Which of the following degree measures is equivalent to 2.25π radians?
Use the following information to answer questions 38−40.
Greg is making a triangular sail for a boat, shaped like a right triangle and shown below.
38. Sail material costs $8.99 for 150 square feet. If the material can be purchased in any quantity, which of the following is closest to the cost in dollars of the material needed to fill the area of the sail as shown?
39. To determine how much trim to buy for the sail, Greg calculated the sail’s perimeter. What is the sail’s perimeter, in feet?
40. The angle opposite the 120-foot side measures about 65.2°. Greg would like to make a second sail. This one will still be a right triangle with a 50-foot side as one leg, but the 120-foot side will be shortened until the angle opposite that side is about 10°. By about how many feet will Greg need to shorten the 120-foot side?
(Note: sin 10° ≈ .17, cos 10° ≈ .98, tan 10° ≈ .18)
41. Rectangle AKLD consists of 5 congruent rectangles as shown in the figure below. Which of the following is the ratio of the length of to the length of ?
42. Jackson High School’s basketball team scored an average of 90 points in each of the first 10 games of the season. If it scored 102 points in each of the next 2 games, which of the following is closest to its average for all 12 games?
43. A ferry boat travels from a dock on the mainland toward an island, stops to discharge and load passengers, then returns to the mainland dock. Among the following graphs, which one best represents the relationship between the distance, in kilometers, of the ferry from the island and the time, in minutes, from when the ferry leaves the mainland dock until it returns?
44. A right triangle has sides measuring 12 inches, 35 inches, and 37 inches. What is the cosine of the angle that lies opposite the 35-inch side?
45. The noncommon rays of 2 adjacent angles form a straight angle. The measure of one angle is 4 times the measure of the other angle. What is the measure of the smaller angle?
46. A rectangular solid consisting of 18 smaller cubes that are identical is positioned in the standard (x,y,z) coordinate system, as shown below. Vertex M has coordinates of (−1,3,0) and point O on the y-axis has coordinates of (0,3,0). What are the coordinates of vertex N?
F. (3,0, 2)
G. (2,2, 0)
J. (0,2, 2)
K. (2,0, 2)
47. What is the median of the data given below?
18, 25, 19, 41, 23, 29, 35, 19
48. Let ab = (−2a − b)2 for all integers a and b. Which of the following is the value of ?
49. For all negative even integers x, which of the following is a correct ordering of the terms x, xx, ((–x)!)x, and ((–x)!)(–x)!?
A. ((−x)!)(−x)! ≥ ((−x)!)x ≥ x ≥ xx
B. ((−x)!)(−x)! ≥ xx ≥ ((−x)!)x ≥ x
C. xx ≥ x ≥ ((−x)!)(−x)! ≥ ((−x)!)x
D. xx ≥ ((−x)!)x ≥ x ≥ ((−x)!)(−x)!
E. x ≥ ((−x)!)x ≥ xx ≥ ((−x)!)(−x)!
50. What is the perimeter of quadrilateral STUR if it has vertices with (x,y) coordinates S(0,0), T(2,−4), U(6,−6), R(4,−2) ?
G. 2 + 2
51. The line with equation 5y − 4x = 20 does NOT lie in which quadrant(s) of the standard (x, y) coordinate plane below?
A. Quadrant I only
B. Quadrant II only
C. Quadrant III only
D. Quadrant IV only
E. Quadrants I and III only
52. The figure below shows representations of the first 4 triangular numbers, through . What is the value of ?
53. The four midpoints of the sides of a square represent four points on a circle. Line segments connect the opposing corners of the square. This circle and these line segments divide the square into how many individual, non-overlapping regions of nonzero area?
54. The circumference of a circle is 50 inches. How many inches long is its radius?
55. In the (x,y) coordinate plane, what is the diameter of the circle having its center at (−6,1) and (0,9) as one of the endpoints of a radius?
56. The graph of the function f(x) = is shown in the standard (x,y) coordinate plane below. Which of the following, if any, is a list of each of the vertical asymptotes of f(x)?
F. This function has no vertical asymptote.
G. y = x + 1
H. y = 2x – 1
J. x = −1 and x = 2
K. x = 1
57. The product of 2 distinct positive prime numbers is an even number, and one less than the product is a prime number. All of the following prime numbers could be one of the original prime numbers EXCEPT:
58. Connecting the midpoints of opposite sides of any quadrilateral to the midpoints of the adjacent sides must always create which of the following?
59. If a(x) = b(x) + c(x), where b(x) = 3x2 − 8x + 113 and c(x) = − 3x2 + 18x + 7 and x is an integer, then a(x) is always divisible by which of the following?
60. Isosceles triangle T1 has a base of 12 meters and a height of 20 meters. The vertices of a second triangle T2 are the midpoints of the sides of T1. The vertices of a third triangle, T3, are the midpoints of the sides of T2. Assume the process continues indefinitely, with the vertices of Tk+1 being the midpoints of the sides of Tk for every positive integer k. What is the sum of the areas, in square meters, of T1, T2, T3, …?
Answers and Explanations
MATH PRACTICE 2 ANSWERS
MATH PRACTICE 2 EXPLANATIONS
1. E Simplify the expression by taking a factor of 2 out of each term. The distributive property guarantees that 2(a + 2b + 3c) = 2a + 4b + 6c.
2. H “The square of the product” means you must multiply a by b before squaring the result. Choice (H) is the only choice that represents this. If you chose (G), be careful—by the order of operations, in ab2, the b will be squared before it is multiplied together with a, thus precluding the “product of a and b.”
3. D In order to get the total distance traveled, subtract the original mileage from the final mileage to find a difference of 18,130 km − 16,450 km = 1,680 km. Then, divide the change in mileage over the number of hours traveled to get the average driving speed during that week. 1,680 km ÷ 30 hr = 56 km/hr.
4. K To solve, use the formula: Volume = length × width × height. Here, our dimensions are 12 by 3 by 3, so the volume is 12 × 3 × 3, which equals 108. Choice (F) is the sum of the three dimensions. Choice (G) is the area of one of the sides. Choices (H) and (J) are the results of adding two of the sides before you multiply.
5. D 34 = 81, so x = 4. Plug x = 4 into the given expression to get 3 × 24 = 3 × 16 = 48. Choice (B) gives one 24, and choice (C) miscalculates x as 3. Choices (A) and (E) are distracting numbers from the problem and do not answer the question.
6. H The ratio of folk songs to rock songs is 3 to 11, which means that for every 3 folk songs, there are 11 rock songs. Hence, there are more rock songs on the mp3 player than folk songs. I is true, which eliminates choices (G) and (J). II is also true, which eliminates choice (F). Ratios compare parts to parts, however, not parts to wholes as do fractions. The fraction of the songs that are folk songs is actually . So, III is false. Eliminate choice (K).
7. E Subtract gallons − gallons. Convert each mixed number into a fraction: The equation becomes . To subtract, both fractions need to have common denominators, so make into .
8. J The point of intersection represents the midpoint between points F and H. You can use the midpoint formula, to determine the coordinates of point H. Solving for the x-coordinate gives . The x-coordinate of point H equals 3, eliminating choices (F), (G), and (K). Solving for the y-coordinate gives . The y-coordinate of point H equals −6, eliminating choice (H).
9. C First, factor out the numerator to get . The nines cancel out, and the x + 5 remains. Choices (A) and (B) divide only one of the terms in the numerator by the denominator. Choice (D) adds the integers in the numerator to get .
10. F To solve this question, remember to follow the order of operations. First, distribute the −6f to the (5f + 3g) and get −30f2 − 18fg. Now simplify 23fg − 30f2 − 18fg and get 5fg − 30f2. Choice (G) is the result of adding 5f and 3gand then distributing the 6. Choice (J) is the result of adding 23fg and 18fg instead of subtracting. Choice (K) is the result of switching the signs.
11. B Set a variable for the number of pints of strawberries, which is equal to the number of quarts. The number of pints at $3 each plus the number of quarts at $5 each is equal to the total of $120, so you can write an equation: 3x+ 5x = 120. Solve for x to find that 8x = 120 and x = 15. Choice (E) gives the total sales in dollars of the pints of strawberries. Choice (C) assumes all sales were at $5, and choice (D) assumes all the sales were at $3. Choice (A) makes a calculation error in solving for x. You can also test the answer choices to see which fits the requirements of the problem. When there are 15 pints sold at $3, there are also 15 quarts sold at $5: 15(3) + 15(5) =120.
12. H The formula for the area of a rectangle is A = lw. So, the actual area of the cloth is A = (6)(1.5) = 9 ft2. To find the percent greater for the estimate use the percent change formula: %change=×100. So, %change=×100≈33%. If you chose (F), be careful—you may have calculated what percent of 12 is 9. If you chose (J), you may have found the percent change from 12 to 9, rather than the percent change from 9 to 12.
13. A Use the definition provided by the problem: . Choice (B) is the arithmetic mean or average of the three numbers. Choice (C) is 27 − 4 − 2. Choice (D) is the product of the three numbers, 216, divided by 3 rather than the cube root of the product. Choice (E) is the product of the three numbers.
14. F Plug the given values for p and q into the equation. questions.
15. C You can eliminate choices (A) and (B) immediately because these are not binomial expressions. The fastest way to do this problem is to factor the quadratic equation x2 + 2x − 15 = (x + 5)(x − 3); thus, the other binomial will be (x − 3). If you’re not sure how to factor, FOIL the answer choices with (x + 5) and see which gives you the quadratic expression from the problem. When you multiply (x + 5) by choice (C), (x − 3), you find it equals x2 + 2x − 15. Eliminate choices (D) and (E).
16. H The equation for production cost is P(x) = 175x + 150,000. After plugging in 465,000 for P(x), solve for x. First subtract 150,000 and then divide by 175 to obtain 1,800 for x = 1,800 computers. If you chose (K), be careful—you may have added 150,000 + 465,000 rather than subtracting 465,000 − 150,000.
17. D To solve this question, substitute the value of x, in this case , wherever x is in the equation. This results in which equals 2. Choice (A) is the result of multiplying the terms instead of dividing. Choice (B) is the result of dividing the by and ignoring the x terms. Choices (C) and (E) are the results of calculation mistakes.
18. F Substitute an easy number, like 10, for x. Hannah would be 5 years old. In two years, Hannah will be 7. Substitute 10 into the answer choices to find a match for Hannah’s age in two years, 7. The other answer choices all make calculation errors in simplifying this expression.
19. E The formula for the area of a rectangle is A = lw. This rectangle has a length of x, a width of 5x, and an area of 320. So, 320 = (x)(5x) = 5x2. Therefore, the length is 8, the width is 40, and the perimeter is 2 × (8 + 40) = 96. Choice (A) is the length of the rectangle. Choice (B) is the width of the rectangle. Choice (C) is the sum of the length and the width. Choice (D) is the area divided by 5.
20. F ACD ≅ ADC, so their supplements, ACB and ADE, are also congruent. Since the question tells you that BAC ≅ DAE, you can see that all three angles in triangles ABC and AED are congruent to one another. Similar triangles are defined as triangles that have three congruent angles and three proportional sides. Because all the angles in these two triangles are congruent, you can determine that ABC is similar to AED, choice (F). Because ACD ≅ ADC, you know that ≅, but you cannot conclude anything about the relationship of either of these sides to . You don’t have enough information to compare ACD and ADE, eliminating choices (G) and (J). You do not know the measure of CAD, eliminating choice (K).
21. B If an exponent is a fraction, raise the base to the value of the numerator and use the value of the denominator to take the root of the base. Take these steps one at a time in either order. First take 81 = 8, then take .
22. G The table tells you that admission plus one event is $42. If you subtract the $30 admission fee from $42, you’re left with $12 as the cost of one event.
23. E If is the measurement of the longer leg, that means that it is opposite to the 60° angle. Using 30-60-90 triangles, we know that the side opposite to the 30° angle is 16, and so the hypotenuse is double that, which is 32 centimeters. Choices (B) and (D) confuse this triangle with a 45-45-90 triangle. And choice (C) finds the length of the shorter leg, not the hypotenuse.
24. K The smallest consecutive odd integers greater than 15 are 17, 19, 21, 23, and 25. These five integers add up to 105. Choice (F) is the product of 5 and 15. Choice (G) is the sum of the smallest consecutive integers, including the evens. Choice (H) includes 15. Choice (J) is the sum of the five smallest consecutive even integers.
25. B Use SOHCAHTOA. You are looking for the side adjacent to the 43° angle and you know the hypotenuse, so use C = ·cos43° = . Multiply both sides of the equation by 25 to get? = 25 cos 43°. Notice you don’t even have to solve any further than this. The other answer choices use the wrong trigonometric functions.
26. K Since x − 15 = |−5|is equivalent to x − 15 = 5, then x = 20. Choice (H) is the result of solving the equation x − 15 = −5 and attempting to divide both sides by −15. Choice (J) results from solving the equation x − 15 = −5.
27. D Let x equal the number of blueberry packages, and let y equal the number of strawberry packages. Construct two equations to solve: x + y=9, which says that the total number of packages is nine, and 4x + 6y=40, which says that the total price is $40. Solve using a system of equations. Multiply the first equation by 4 to get 4x + 4y = 36. Subtract the second equation from the result . So, y = 2. But this represents the number of strawberry packages. There are nine packages total, so there must be 7 blueberry packages. Be careful with choice (A)—it represents the number of strawberry packages. If you chose choice (B), you probably added 6 and 4 in the second equation to get |v|.
28. G To solve this question, remember that the sum of the interior angles of a triangle is 180°; therefore a + b + 45 = 180 and c + d + 45 = 180. Thus, since a + b + c + d + 90 = 360, a + b + c + d = 270. Choice (F) is the difference of 360 and 45. Choice (H) is the sum of 180 and 45. Choice (J) is the sum of either a and b or c and d.
29. E ACE is isosceles with base , so CAE≅AEC. Since both angles are bisected, the four smaller angles created are all congruent: CAD≅DAE≅AEB≅BEC. Choice (E) compares two of these congruent angles. Choices (A) and (B) compare one of the triangle’s angles to one of the bisected angles, which are half as large. Choices (C) and (D) compare the bisected angles with the third side of the isosceles triangle, a relationship it is impossible to know without any actual angle measures.
30. J To find the area of a trapezoid, multiply the average of the bases by the height. The formula for the area of a trapezoid is . Plugging in the values given, find .
31. C You can eliminate answer choices by substituting the number of years after 2001 for x. If you substitute 3 for x, you should get approximately 547 because 3 years after 2001, in 2004, there were 547,000 book club members. Choice (C) gives (3)+539=547, eliminating choices (A), (B), (D), and (E). Note that these choices all confuse years with actual values in the problem (the year 2001 becomes 2,001 in choices (A), (B), and (E), for example)—make sure you are reading carefully and not falling into these traps.
32. G From 1997 to 1998 the percent of U.S. citizens who had consumed the soda rose from 57.8% to 58.2%, 58.2 − 57.8 = 0.4%, which is the smallest increase among the years listed. If you picked choice (J), be careful—(J) gives the LARGEST increase. You’re looking for the smallest.
33. D Look at the graph. Line D roughly approximates the scatterplot, beginning and ending at the same values. Thus, Line D most closely matches what would be the ideal average line of the scatterplot data. Line E is completely above the scatterplot, and Line C is completely below it. Lines A and B don’t come close to matching the diagonal scatterplot line.
34. H The table states that by 2002 63.4% of all U.S. soda consumers had consumed the brand-name soda. Set up an equation: 63.4% of the total number of U.S. soda consumers = 74,672,120 U.S. citizens who had consumed the brand-name soda. Round the numbers since the question asks for an approximate answer: .634 × x = 74,672,120, = 119,000,000 people.
35. C You may be familiar with this format, known as scientific notation, from your science classes, but you don’t need to understand scientific notation to be able to complete this problem. All you need to know is that 105 = 100,000 and 0.0001 = 10-4. Therefore, the full expression looks like this:
36. F To solve this equation use the slope-intercept formula: y = mx + b. To find the slope, use the two points that you know: (−1,3) and (2,−21). ; thus the slope is −8. Now put the slope into the value for m and get y = −8(x) + b. Now, plug one of the points, say (−1,3), into the equation and get 3 = −8(−1) + b, which reduces to b = −5. Another way to find the y-intercept is even easier: just look on the graph. The graph clearly shows that line h crosses the y-axis at (0,−5). So either way, the equation for h is y = −8x −5. Choice (G) is using the wrong y-intercept. Choices (H), (J), and (K) miscalculated the slope.
37. C To change 2.25π radians into degrees, multiply by to get 405°. Choices (A) and (B) are fractions of 405. Choices (C) and (E) are multiples.
38. J Calculate the area of the triangular sail using the formula for the area of a triangle: A = (base)(height). Area = (50ft)(120ft) = 3,000ft2. Divide to find the number of 150 square foot pieces of material required = 20 pieces. Multiply 20 pieces × $8.99 per piece = $179.80 ≈ $180.00. Don’t be distracted by the in any quantity part of this question—the numbers divide evenly.
39. B This triangular sail happens to be a version of a 5-12-13 right triangle. If you had the dimensions of this triangle memorized already you would know that the hypotenuse must be 130 feet long. 130 + 120 + 50 = 300 feet. You also could have used the Pythagorean theorem: a2 + b2 = c2, where a and b are the two legs of the triangle and c is the hypotenuse.
40. J Since the 10° angle is across from the 120-foot side and adjacent to the 50-foot side, you can use tan 10° = . The problem states that tan 10° ≈ .18, so substitute this value for tan 10°. 0.18 = . Subtract x, which is the value by which the 120-foot side will be shortened, from 120 feet, and divide by the adjacent side, 50 feet. Solve the equation: 9 = 120 − x, x = 111 feet. If you chose choices (F) or (G), you may have found only a partial answer—make sure you read the problem closely and complete all its steps.
41. C Since all the rectangles are congruent, you can find some relationships between their lengths and widths. For example, since ≅ , is equal to three times the width of each rectangle (, , or ). represents one length (or three widths), and represents one length and two widths combined (or the equivalent of five widths). The ratio of the two sides is therefore (five widths):(three widths) or 5:3. If you’re not sure how to find the relationships in this problem, use the figure: is smaller than , so eliminate choices (A), (D), and (E). is not double the length of , so eliminate choice (B).
42. J To solve average questions remember that the average is the sum of all the members of the set divided by the total number of members in the set. Here, since there are a total of 12 games, 12 is the divisor. The sum will be (10 games × 90 points per game) + (2 games × 102 points per game), which equals 1,104 total points. Now divide 1,104 by 12 and get the average points per game of 92. Choices (F) and (K) are partial averages for the first 10 and next 2 respectively; neither includes all 12 games. Choice (H) is the average of 90 and 102.
43. C The ferry starts at its maximum distance from the island then travels toward the island. As the time moves from left to right from the start of the graph, the distance should decrease. Eliminate choices (B) and (D) in which the distance increases. The graph should dip to a distance of 0, then rise again as the ferry sails away. Choice (E) does not show the return trip. According to choice (A), the boat is at two distances from the island at the same time, which is impossible. If you chose (A), you may have confused the two axes. Choice (C) accurately represents the data given.
44. J The cosine of an angle is equal to the adjacent side over the hypotenuse, which in this triangle is . Since you know the longest side is always the hypotenuse, draw a figure and work this problem visually. Choices (F), (G), (H), and (K) represent the cotangent, sine, tangent, and secant, respectively.
45. A Don’t be thrown off by the vocabulary in this question. Adjacent simply means “next to,” and noncommon rays are simply rays that do not overlap. Accordingly, if the two noncommon rays form a straight angle (or a 180° angle), it is probably simplest to think of one of these rays pointing to the right and the other pointing to the left. That takes care of your noncommon rays; now the common ray will just be one ray set at an angle to the straight line such that it creates one large angle which is four times larger than its supplement (the angle with which it adds to form a straight or 180° angle). Since the two angles are supplementary, you can create the following equation if x is the smaller angle: 4x + x = 180°; 5x = 180°; x = 36°, choice (A). If you picked choice (E), be careful—this is the measure of the larger angle.
46. K To solve this question you have to remember the axis to which each of the numbers in the coordinates refer. Since the system is x, y, z, the coordinate (−1,3,0) means that vertex M is “back” one on the x-axis, “right” three on the y-axis, and “up” zero on the z-axis. Thus, since vertex N is “forward” two, “left or right” zero, and “up” two, the coordinates are (2,0,2). Choice (F) would be correct if vertex M were on the y-axis. Choice (G) mixes up the y and z coordinates. Choice (J) mixes up the x and y coordinates.
47. D The median of a set of numbers is the one in the middle when the data are listed in numerical order. Resort the data to read: 18, 19, 19, 23, 25, 29, 35, 41. Since there is an even number of terms, find the average of the two numbers closest to the middle to get the median. = 24. Choice (A) calculates the average of the two middle numbers in the set without reordering. Choice (B) is the mean, or average, and choice (E) is the mode, or most common number. Choice (C) chooses the median if the list of numbers includes 19 only once.
48. H Use the definition for the function provided by the problem: −5 3 = (−2(−5)−3)2 = (10−3)2 = 72 = 49. Choice (F) is −5 × 3. Be careful! Although the symbol looks a little like a multiplication sign, you still need to use the definition in the problem. Choice (G) is −5 + 3. Choice (J) is 102 + 32 and choice (K) is 102 + 32.
49. B Substitute a value for x in the given terms to find their order. For example, if
If you chose (E), be careful—note the direction of the inequality signs.
50. H Start by plotting out the figure to determine the distance between each point. You can find the distance between each point by using the distance formula: d = . From point S to point R, for example, find d = . As you compute the other sides, you’ll notice that they all equal 2, so the perimeter will be 4×2=8.
51. D First, manipulate the equation given to get y=x + 4. Then graph this line, with a slope of and a y-intercept of 4. The graph lies in the first, second, and third quadrants, and not in the fourth quadrant, Choice (D).
52. J To solve this use the formula for triangular numbers: where n represents the number you wish to find, i.e., for the first triangular number, n would equal 1, for the second, n equals 2, and so on. Here, since the question asks for the 24th triangular number, n equals 24. Using n equals 24, we get , which equals 300. Choice (F) is the result of()2. Choice (G) is ()2 + n. Choice (H) is . Choice (K) is n2 + n.
53. D Use a picture to see what’s happening here. First, draw a square and label its midpoints. Carefully draw a circle connecting these points. It should fit exactly inside the square, touching the square at the square’s four midpoints. Draw diagonals in the square to connect the opposite corners. Then count the number of regions created. Each quarter of the square is cut into one large region inside the circle and two smaller regions outside the circle, so thus 12 total. Choice (B) neglects to draw the diagonals and choice (C) miscounts the number of areas created. Choice (A) would mean that only one line ran through the square. Choice (E) includes many overlapping shapes that the question excludes with the words “non-overlapping regions.”
54. F The formula for the circumference of a circle is C = 2πr. So, 50 = 2πr and, therefore, r = . Choice (G) is the result of forgetting to divide by the 2 in the formula. Choice (H) could result by mistakenly multiplying 50 by 2 rather than dividing when using the formula. Choices (J) and (K) incorrectly solve for the radius.
55. C Use the distance formula: Note that the radius of the circle is the hypotenuse of a 6-8-10 right triangle that includes the given points as two of its vertices. So, the radius of the circle is 10 and its diameter is 20. Choice (A) is the radius of the circle. Choice (B) is the sum of the two sides of the triangle. Choice (D) is the sum of the two sides of the triangle multiplied by 2. Choice (E) is the radius squared—probably a result of forgetting to take the square root when using the Pythagorean theorem.
56. K To find the vertical asymptote, set the denominator equal to 0 and solve: x − 1 = 0, x = 1. If x is 1 the denominator is 0 and the function is undefined, therefore the graph cannot reach x = 1, and x = 1 is the graph’s vertical asymptote.
57. C In order to produce an even number using the answer choices, you must multiply by 2. Eliminate choice (A). One less than 2 × 3 = 6 is 5, which is prime. Eliminate choice (B). One less than 2 × 5 = 10 is 9, which is not prime. Eliminate choices (D) and (E).
58. K The midpoints of any quadrilateral, when connected, create a four-sided figure, which eliminates Choices (F), (G), and (H). If you chose any four points in a rectangular coordinate system, and found the midpoints, the slopes of the lines connecting the midpoints are not perpendicular, but the slopes of the opposite sides are parallel, which makes the shape a parallelogram.
59. C To solve this question you need to use simultaneous equations:
Thus, since x must be an integer, 10x + 120 must be a multiple of 10, since both terms are themselves multiples of 10.
60. K First, find the area of the original triangle T1. A = (12)(20) = 120. The sum of the triangles must be larger than 120, so eliminate choices (F), (G), and (H). Draw T1 with T2 inside it, with the vertices of T2 at the midpoints of each of the sides of T1. T2 points in the opposite direction of T1, and T1 is now split into four congruent triangles. Since congruent triangles have equal areas, T2 = (120) = 30. T1 + T2 = 120 + 30 = 150, so choice (J) is also too small. The full sum of this geometric sequence is 120 + 30 + 7.5 + 1.875 + 0.46875 +…≈ 160.