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This chapter is from the book

This chapter is from the book

Answer Explanations

Multiple Choice (Questions 1–8)

  1. The correct answer is C. To solve this problem, simply count the numbers that appear both in set P and in set Q. Sets P and Q share the following numbers: 16, 17, 18, 19, and 20. Therefore, five numbers in set P are also in Set Q.

  2. The correct answer is C. You are given that John traveled 30 miles in 6 hours and that Seth traveled three times as far in half the time. This means that Seth traveled 30(3), or 90 miles in 6(), or 3 hours. Therefore, Seth traveled , or 30 miles per hour.

  3. The correct answer is D. You are given that r = -2. Therefore, the best approach to solving this problem is to replace r with -2 in each of the answer choices as follows:

    • Answer choice A: 4pr2 = 4p(-2)2, which equals (4)(4)p, or 16p.

    • Answer choice B: 2pr3 = 2p(-2)3, which equals (2)(-8)p, or -16p.

    • Answer choice C: -2pr4 = -2p(-2)4, which equals (-2)(16)p, or -32p.

    • Answer choice D: -4pr5 = -4p(-2)5, which equals (-4)(-32)p, or 128p.

    • Answer choice E: -6pr6 = -6p(-2)6, which equals (-6)(64)p, or -384p.

    The question asks for the least value. You are given that p < 0, which means that p is a negative number. If you substitute any negative number into the results obtained above, only answer choices A and D will yield a negative value. Therefore, you can eliminate answer choices B, C, and E, because they will all be larger than either answer choice A or D. When p < 0, 128p will always be less than 16p, so answer choice D is correct.

  4. The correct answer is C. The equation of the line is given in the slope-intercept form (y = mx + b), where b is the y-intercept. The y-intercept is the point at which the line intersects the y-axis. Since the graph shown has the equation y = ax + b, and the new graph has the equation y = ax + b, both graphs should have the same y-intercept. The graph shown has a y-intercept of 1. Eliminate answer choices A, B and D, since they do not have a y-intercept of 1. The slope of the graph shown is a, and the slope of the new graph (based on its equation) is a. Because both slopes are positive, you can eliminate answer choice E, which has a negative slope. This leaves you with answer choice C.

  5. The correct answer is B. To solve this problem, you must recognize that the triangle is a "special triangle." A right triangle in which the length of the longer leg is times the length of the shorter leg is a 30°–60°–90° right triangle. Another property of this type of right triangle is that the hypotenuse is 2 times the length of the shorter leg. So, this right triangle has lengths x, x, and 2x. The perimeter is the sum of the lengths of the sides. You are given that the perimeter equals 12 + 4. Plug this value into the formula for the perimeter of a triangle, as follows:

    • 12 + 4 = x + x + 2x

    • 12 + 4 = 3x + x

    For the right side of the equation to equal the left side of the equation, x must be equal to 4.

  6. The correct answer is C. The median of a list of values is the middle value when the list is in chronological order and there are an odd number of values. In this case, if we include Mike and his 75 coins, there will be 19 coin collectors. List the number of coins for each collector in chronological order as follows:

  7. As you can see, the middle number is 85, so that is the median.

  8. The correct answer is C. To solve, first translate the words into their mathematical equivalents:

    • 4 more than twice n translates to 4 + 2n. You are given that this quantity is a negative number, which translates to 4 + 2n < 0.

    • 6 more than n translates to 6 + n. You are given that this quantity is a positive number, which translates to 6 + n > 0.

    Solve each inequality as follows:

    • 4 + 2n < 0

    • n < -2

    • 6 + n > 0

    • n > -6

    So, n could be any number less than -2 and greater than -6. Only -4, answer choice C, is greater than -6 and less than -2.

  9. The correct answer is D. Since the figure is a rectangle, the adjacent sides are perpendicular. Perpendicular lines have slopes that are negative reciprocals of each other, meaning that the product of their slopes is -1. Since there are four lines and four perpendicular angles, the product of the slopes is (-1)(-1)(-1)(-1), or 1.

Student-Produced Response (Questions 9–18)

  1. Answer: . To solve this problem, convert hours into minutes. 2 hours is equivalent to 150 minutes. You are given that the movie included 30 minutes of commercials; therefore, the movie included 120 minutes that were not commercials. The fraction of the movie that was not commercials is , or .

  2. Answer: 2.66, 2.67, or .

  3. To solve this problem, set up an equation based on the information given:

    • 0.75x = 2

    Solve for x.

    • x = 2 ÷ 0.75

    • x = 2.66666 (This is a repeating decimal.)

    For the purposes of the PSAT, the answer can be simplified to either 2.66 or 2.67. If you first converted .75 to , you would arrive at as your answer, which is also correct.

  4. Answer: . You are given that 8s = 96. Therefore, s = , or 12. You are given that sp = 4, and you now know that s = 12. Therefore, 12p = 4 and p = , or .

  5. Answer: 2 or 3. The formula for the area of a rectangle is A = (l)(w). Therefore, the area of ACFD will be (AC)(AD). Because you must solve for the value of AB, let AB = x. Since ABED is a square, it follows that AB = AD = x. Therefore, since BC = 8, the area of ACFD will be (8 + x)x, or x2 + 8x. Since, according to the question, the area must be between 14 and 35, set up the inequality 14 < x2 + 8x < 35. Now, pick some easy numbers to work with to replace x in the inequality. Because x is a positive value and 14 and 35 are not particularly large numbers, start with x = 1. If x = 1, then the area is x2 + 8x, or (1)2 + 8(1), or 9. This value is not between 14 and 35, so x cannot be 1. If x = 2, then the area is x2 + 8x, or (2)2 + 8(2), or 20. This value is between 14 and 35, so x = 2 is a possible solution. If x = 3, then the area is x2 + 8x, or (3)2 + 8(3), or 33. This value is between 14 and 35, so x = 3 is another possible solution. If x = 4, then the area is x2 + 8x, or (4)2 + 8(4) or 48. This value is not between 14 and 35, so x cannot be 4, or any number greater than 4. Therefore, x, or AB, can be either 2 or 3.

  6. Answer: 100. To solve this problem, let the total number of napkins in the box equal x. According to information in the problem, there are 25 red napkins in the box. One quarter of the total number of napkins is blue, so there are ()x blue napkins in the box. There are twice as many white napkins as blue napkins, so there are 2()x, or ()x white napkins in the box. To calculate the total number of napkins in the box, set up an equation and solve for x.

    • x = 25 + x + x

    • x = 25 + x

    • x = 25

    • x = 100

  7. Answer: 20. You are given that there are 4 points on line l, and 5 points on the line parallel to line l. Therefore, from each of the 4 points on line l, T, U, V, and W, you can draw one line to each of the 5 points on the second line. This means that there can be(4)(5), or 20 different lines containing exactly 2 of the 9 points.

  8. Answer: 4. To solve this problem, rewrite 42x + 42x + 42x + 42x as 4(42x). Now you have 4(42x) = 49. When you multiply numbers with the same base, you add the exponents. The number 4 is equivalent to 41. Therefore, 4(42x) = 42x + 1. Now, because 42x + 1 = 49, set the exponents equal to each other, and solve for x.

    • 2x + 1 = 9

    • 2x = 8

    • x = 4

  9. Answer: 496. If the average of the 9 numbers is 56, then the total value of the 9 numbers is (9)(56), or 504. The question asks for the greatest possible integer on one card, so set the values of the other cards at the lowest possible integer, which is 1. If there are 8 cards with a value of 1, then the value of the ninth card must be 504 – 8(1), or 496.

  10. Answer: 3. Let j be the length of one of Jeff's steps, and let s be the length of one of Scott's steps. If they each take 12 steps away from each other, the distance between them is 12j + 12s. Since it takes Jeff 16 steps to reach Scott, then 12j + 12s = 16j. Subtract 12j from both sides to get 12s = 4j. Divide both sides by 4 to get 3s = j. The length of one of Jeff's steps (j) is 3 times the length of one of Scott's steps (s).

  11. Answer: 130. To solve this problem, you should recognize that rs is a factor of 3r2s + 2rs2. Rewrite 3r2s + 2rs2 as rs(3r + 2s). You are given that rs = 13 and 3r + 2s = 10. Substitute those values into rs(3r + 2s) to get 13(10), which equals 130.

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