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2 3 4 28 31 1 1 Introduction 2 Difficulty Levels 3 Problem Solving 4 Answer Key 5 Explanations Contents February 2008 GMAT Problem Solving: Challenge 3 Very Difficult (7) 7, 10, 11, 14, 18, 19, 22, 33, 35, 38, 47, 53, 56, 57, 59, 64, 67, 69, 70, 74, 79, 90, 92, 95 Difficult (6) 2, 4, 5, 6, 8, 9, 12, 13, 15, 16, 20, 21, 24, 25, 26, 27, 28, 32, 34, 36, 37, 39, 41, 44, 46, 50, 52, 54, 55, 58, 60, 62, 63, 65, 66, 68, 71, 72, 73, 75, 76, 77, 80, 81, 85, 86, 87, 88, 89, 91, 93, 94, 97, 98, 99, 100 Moderately Difficult (5) 1, 3, 17, 23, 29, 30, 31, 40, 42, 43,45, 48, 49, 51, 61, 78, 82, 83, 84, 96 In general, the level 5 questions in this guide are 560- to 620-level questions. The level 6 questions represent a broad range of difficulty from about 620 to 720, while the level 7 questions are higher still. 2 Difficulty Levels 2. DIFFICULTY LEVELS 4 7 18 5 9 11 18 2 3 5 6 (A) (B) (C) (D) (E) 4. Tank A is k full of water and tank B, which has three times the capacity of tank A, is ! full of water. If all of the water in tank A is poured into tank B, then tank B will be filled to what fraction of its capacity? 3. Which of the following is the value of J J0.00000001 ? (A) 0.1 (B) 0.01 (C) 0.001 (D) 0.0001 (E) 0.00001 2. In a certain company, the ratio of the number of salespeople to total employees is 1 to 3. If three of the salespeople were to leave the company and the company did not replace them, the ratio would be 1 to 4. How many total employees does the company have? (A) 12 (B) 18 (C) 24 (D) 27 (E) 36 1. For which of the following values of m is m--;;08 NOT an integer? (A) 6 (B) 8 (C) 9 (D) 12 (E) 18 Note: this guide contains both an answer key (so you can quickly check your answers) and full explanations. 3 Problem Solving 3. PROBLEM SOLVING 5 8. The table above shows the number of specialty cocktails at a certain restaurant that include each ingredient. Although no cocktail includes all three ingredients, 3 cocktails include both tequila and rum, and 2 cocktails include both tequila and vodka. What is the maximum number of cocktails that could include both rum and vodka? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 vodka: 6 tequila: 5 rum: 5 7. Twin primes are defined as prime numbers that can be expressed asp and (p + 2), and any number p that is a member of such a pair is considered to "have" a twin. For example, 3 and 5 are twin primes, and 3 has a twin. Each of the following prime numbers has a twin EXCEPT (A) 7 (B) 13 (C) 17 (D) 23 (E) 29 6. If ~ of the air in a tank is removed with each stroke of a vacuum pump, how many strokes does it take before less than 1 % of the original amount of air in the tank remains? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 5. What is the least positive integer that is divisible by each of the integers 2 through 7, inclusive? (A) 210 (B) 420 (C) 840 (D) 1,260 (E) 5,040 3. PROBLEM SOLVING 6 1 8 3 8 1 2 3 4 7 8 (A) (B) (C) (D) (E) 12. The probability is ! that a certain coin will turn up heads on any given toss. If the coin is to be tossed three times, what is the probability that on at least two of the tosses the coin will turn up tails? 11. If pis a positive integer less than 75 and ~~ is an integer, then p has how many different positive prime factors? (A) One (B) Two (C) Three (D) Four (E) Five 10. A rectangular box has dimensions of 8 feet, 8 feet, and z feet. In terms of z, what is the greatest possible (straight-line) distance, in feet, between any two points on the box? (A) 8 + z (B) 8./2 + z (C) 8z./2 (D) ../~64_+_z~2 (E) Vl28 + z2 9. How many different positive integers are factors of 484 ? (A) 6 (B) 8 (C) 9 (D) 11 (E) 12 3. PROBLEM SOLVING 7 16. Five drainage pipes, each draining water from a pool at the same constant rate, together can drain a certain pool in 12 days. How many additional pipes, each draining water at the same constant rate, will be needed to drain the pool in 4 days? (A) 6 (B) 9 (C) 10 (D) 12 (E) 15 (A) .s: ym (B) km y (C) 60ky m (D) 60km y (E) kmy 60 15. At the rate of k knots per m minutes, how many knots does a ship travel in y hours? 14. All of the bonds on a certain exchange are designated by a 3-letter, a 4-letter, or a 5-letter code that is created by using the 26 letters of the alphabet. Which of the following gives the maximum number of different bonds that can be designated with these codes? (A) 26(263 + 264) (B) 26(263 + 265) (C) 27(263 + 265) (D) 27(263) + 265 (E) 263 + 27(265) I. 150 - 2k II. 150- l k III. 150 _ k k2 (A) I only (B) II only (C) I and II (D) I and III (E) II and III 13. As k increases from 149 to 151, which of the following must decrease? 3. PROBLEM SOLVING 8 21. If n = 8p, where pis a prime number greater than 2, how many different positive even divisors does n have, including n ? (A) Two (B) Three (C) Four (D) Six (E) Eight 20. S is a set containing 8 different numbers. T is a set containing 6 different numbers, all of which are members of S. Which of the following statements CANNOT be true? (A) The mean of Sis greater than the mean of T. (B) The range of Sis equal to the range of T. (C) The median of Sis equal to the median of T. (D) The mean of Sis equal to the mean of T. (E) The range of S is less than the range of T. y +4z 3y + 1.5z 4y +2z 10y+7z 4 20y+I4z 3 (A) (B) (C) (D) (E) 19. In a certain school district, 5 percent of the x students at School A are honor students, 20 percent of the y students at School B are honor students, and 14 percent of the z students at School C are honor students. If 8 percent of the total x + y + z students are honor students, what is x in terms of y and z? 1 8 1 4 1 2 3 4 7 8 (A) (B) (C) (D) (E) 18. If xis to be chosen at random from the set {1, 2, 3, 4} and y is to be chosen at random from the set {4, 5, 6, 7}, what is the probability that xy will be even? 17. (2 + J2)( J3 - 2)( v12 - 2)(2 + J3) = (A) -v'6 (B) -2 (C) -1 (D) J3 (E) 2 3. PROBLEM SOLVING 9 (A) a+c ~ (B) b+c -2- (C) a+d -2- (D) b+d -2- (E) c+d -2- 25. Which of the following CANNOT be the median of the four positive integers a, b, c, and d, where a < b < c < d ? 24. A certain die has 10 sides, and each side has a positive integer written on it. In a board game, a player's number of points for each turn is determined by rolling the die, then multiplying the resulting integer by the next greatest integer. If the possible number of points for any turn is between 10 and 180, then the least and greatest integers on the die could be (A) 2 and 10 (B) 3 and 12 (C) 3 and 13 (D) 4 and 13 (E) 4 and 14 23. If set S consists of the first 10 positive multiples of 5, what is the positive difference between the average (arithmetic mean) of S and the median of S ? (A) 0 (B) 2.5 (C) 5 (D) 25 (E) 27.5 22. For every integer m from 1 to 100, inclusive, the mth term of a certain sequence is given by (-1)=(2-=). If N is the sum of the first 100 terms in the sequence, then N is (A) less than -1 (B) between -1 and -! (C) between-! and O (D) between O and ! (E) greater than ! 3. PROBLEM SOLVING 1() 29. How many positive integers less than 30 are either an even prime number, a multiple of 3, or the sum of an even prime and a positive multiple of 3 ? (A) 22 (B) 21 (C) 20 (D) 19 (E) 18 (A) n > -p (B) n > -q (C) n < =P (D) -q < n < =P (E) None of the above 28. If it is true that n < q and n > p, which of the following must be true? 27. In a certain company, the ratio of the number of managers to the number of non-managers in any department must always be greater than 5: 24. In the company, what is the maximum number of non-managers in a department that has 8 managers? (A) 36 (B) 37 (C) 38 (D) 39 (E) 40 (A) pn - mn - mp (B) p(n-1)-mn (C) mn-pn+mp (D) mn-p(n-1) (E)n(m + p) 26. The average (arithmetic mean) of n numbers ism. When one number is discarded, the average of the remaining numbers becomes p. In terms of m, n, and p, what is the discarded number? 3. PROBLEM SOLVING 11 34. If (s - 5) is a factor of s2 - js + 25, then j = (A) -10 (B) -5 (C) 0 (D) 5 (E) 10 33. If p is a positive integer, and if the units' digit of p2 is 1 and the units' digit of (p + 1 )2 is 4, what is the units' digit of (p + 2)2 ? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 32. If x2 = 9y2, which of the following could be the value of 1i ? (A) -3 (B) -! (C) -! (D) 1 (E) ! (A) $180 (B) $216 (C) $240 (D) $450 (E) $540 31. J66I is between (A) 21 and 22 (B) 22 and 23 (C) 23 and 24 (D) 24 and 25 (E) 25 and 26 30. An art dealer purchased a painting for $360 and then offered the painting for sale for a price equal to his purchase price plus a markup that was 40% of his offered price. If the dealer sold the painting for 10% less than the price at which he offered it for sale, what was the dealer's profit? 3. PROBLEM SOLVING 12 3 4 4 3 3 2 2 8 3 (A) (B) (C) (D) (E) 39. If ( 7~) n = 49, what is the value of n ? 38. In how many arrangements can a teacher seat 2 girls and 4 boys in a row of 6 seats if the girls must occupy the second and fifth seats? (A) 720 (B) 48 (C) 36 (D) 24 (E) 8 7Y + ~ 7 Zy 2 7Y +2 7(Y + 2) 37. When X is divided by Y, the quotient is 7 and the remainder is 2. Which of the following, in terms of Y, is the value of X ? (A) 7Y (B) (C) (D) (E) 36. It takes Elvys 25 minutes to drive from home to work at an average rate of 20 miles per hour. If Elvys drove the same route in 10 minutes, what would his average rate be, in miles per hour? (A) 25 (B) 35 (C) 40 (D) 50 (E) 60 35. For any integer p greater than 1, *P* denotes the product of all the integers from 1 top, inclusive. How many prime numbers are there between *5* and *5 * + 7, inclusive? (A) None (B) One (C) Two (D) Three (E) Four 3. PROBLEM SOLVING 13 (A) (x4)3 (B) (x6)6 (C) (x3)9 (D) x'' x16 (E) x3 + x9 44. Which of the following is equal to x12 for all positive values of x? 43. The value of :;/ -101 is (A) between -9 and -10 (B) between -8 and -9 (C) between -4 and -5 (D) between -3 and -4 (E) undefined 1 1.3 1.32 0.13 1.3 13 1.3 (A) (B) (C) (D) (E) 42. Of the following, which is greatest? 41. Which of the following is equal to the average (arithmetic mean) of (2x + 1)2 and (2x - 1)2? (A) 4x2 (B) 8x2 (C) 4x2 + 1 (D) 8x2 + 1 (E) 4x2 + 2 40. If x and y are positive integers and x3 + y3 < 1, 000, then the greatest possible value of x is between (A) 0 and 2 (B) 2 and 4 (C) 4 and 6 (D) 6 and 8 (E) 8 and 10 3. PROBLEl'vi SOLVING 14 48. For how many integers n is l " = n1 ? (A) None (B) One (C) Two (D) Three (E) More than three 47. In the addition table above, each number in the table is the sum of the terms at the top of its column and the left of its row. What is the value of p + q? (A) -6 (B) 1 (C) 5 (D) 6 (E) 7 .-f- l o: b C, .. x -l -'t Lf y to p 5" 7 i -l 46. If (2x)(8Y) = 32 and (3x)(9Y) = 81, then (x, y) = (A) (1,2) (B) (2,1) (C) (1, 1) (D) (2, 2) (E) (1, 3) (A) -2 (B) -1 (C) 0 (D) 1 (E) 2 45. If m = 2 and m;n = 1, which of the following is NOT a possible value of n? 3. PROBLEM SOLVING 15 53. (:i:+y)2 If xy = 1, what is the value of ~"''+":l ? (A) 2 (B) 3 (C) 6 (D) 9 (E) 27 10, 3, k, m, 1, 7 52. The arithmetic mean of the list of numbers above is 5. If k and m are consecutive integers, what is the median of the list? (A) 3 (B) 3.5 (C) 4 (D) 4.5 (E) 5 O and z30 3 d 1 20 an 4 land l 4 3 land~ 3 5 ~ and l 5 2 (A) (B) (C) (D) (E) 51. If a number between O and ! is selected at random, which of the following will the number most likely be between? 50. A traveler drives J miles in H hours, then rides a train K miles in half the number of hours. Which of the following represents the traveler's average speed, in miles per hour, for the entire trip? ( A) J-K 3H ( B) 3(J-K) =itr: (C) 2(J+K) =str: ( D) 3(J+K) =ur: (E) J21-iK 5 6 1 3 0 (A) (B) (C) (D) (E) If E = ~ then 'JL - E = y 3, x y 49. 3. PROBLEM SOLVING 16 57. A certain commodities exchange designates each commodity with a two- or three-character code, where each character is selected from the digits O through 9, inclusive, and the capital letters A through F, inclusive. If the characters may be repeated and if the same characters used in a different order constitute a different code, how many different commodities is it possible to uniquely designate with these codes? (A) 256 (B) 3360 (C) 3600 (D) 4096 (E) 4352 56. The positive integer p is divisible by 11. If jp is an integer divisible by 3, which of the following must be a factor of fr ? (A) 9 (B) 14 (C) 15 (D) 33 (E) 99 55. Logan and Hayley are planning to walk toward each other on the same road, Logan starting from A, Hayley starting from B. If the distance from A to B is 36 miles, Logan's walking speed is 4 miles per hour, and Hayley's walking speed is 3 miles per hour, how much longer, in hours, will it take Hayley than Logan to get to the midpoint between A and B? (A) 1 (B) 1.5 (C) 2.5 (D) 3 (E) 4.5 54. Which of the following lists the number of points at which a circle can intersect a parallelogram? (A) 2, 4, and 8 only (B) 2, 4, 6, and 8 only (C) 1, 2, 3, 4, 6, and 8 only (D) 2, 3, 4, 5, 6, 7, and 8 only (E) 1, 2, 3, 4, 5, 6, 7, and 8 3. PROBLEM SOLVING 17 62. If aAAb = ab - a(b - a) for all integers a and b, then (-2)AA(-3) = (A) -8 (B) -4 (C) 3 (D) 4 (E) 8 (A) 6 (B) 12 (C) 24 (D) 36 (E) 144 61. (92)(4")(2'1) 72' (A) 2-1 (B) 2-2 (C) 21 (D) 22 (E) 23 60. If ti is divisible by 6, then the largest positive integer that must divide n2 is (A) between 2 and 10 (B) between 10 and 20 (C) between 20 and 30 (D) between 30 and 40 (E) greater than 40 59. For every positive integer n, the function f(n) is defined to be the product of all the prime numbers from 2 to n, inclusive. If p is the smallest prime factor of f(50) + 1, then pis (A) P (B) q (C) P - q (D) pq (E) p - pq 58. If p and q are integers are pq - p2 is even, which of the following must also be even? 3. PROBLEM SOLVING 66. In a certain game, amounts of money are represented by differently colored chips. If 2 blue chips equal 10 yellow chips and 3 yellow chips equal 20 red chips, how many blue chips are equivalent to 100 red chips? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 65. The formula F = tC + 32 gives the relationship between the temperature in degrees Fahrenheit, F, and the termperature given in degrees Celsius, C. What value of Fis double the equivalent value of C ? (A) -80 (B) 0 (C) 80 (D) 160 (E) 320 3 4 1 2 3 10 1 4 1 8 (A) (B) (C) (D) (E) 64. A certain machine produces toy balls in an infinitely repeating cycle of blue, green, red, and yellow. If 10 consecutively produced balls are selected at random, what is the probability that exactly 3 of the balls selected are blue? 5 66 1 IT 1 9 4 33 1 6 (A) (B) (C) (D) (E) 63. A certain club has 66 members, each of which is participating in exactly one of the six projects in which the club is currently involved. If the numbers of club members participating in the projects are consecutive even numbers, what is the probability that a given club member is participating in the project that the fewest club members are participating in? 3. PROBLEM SOLVING 11 18 11 36 1 4 2 9 7 36 (A) (B) (C) (D) (E) 70. A certain dice game can only be won if, when a player throws two fair six-sided dice, the number showing on one of the dice is a multiple of the number showing on the other. What is the probability that a player wins this game? 11 36 1 9 1 18 1 36 0 (A) (B) (C) (D)(E) 69. If two fair six-sided dice are thrown, what is the probability that the sum of the numbers showing on the dice is 11 ? 68. There are ten players in a tennis league, and a pair of players is to be selected to play a match. At most, how many different pairs of players are possible? (A) 10 (B) 45 (C) 50 (D) 90 (E) 100 19 60 3 10 1 5 11 60 1 6 (A) (B) (C) (D) (E) 67. What is the probability that, twelve seconds from a randomly selected starting time, a certain digital clock will display a number of seconds x such that x < 10 ? 3. PROBLEM SOLVING 20 73. If the price of a certain bond on May 1st was ~ the price of the bond on June 1st and the price of the bond on July 1st was 25% greater than the price of the bond on May l st, then the price of the bond on June 1st was what percent of the average (arithmetic mean) price of the bond on May 1st and July 1st? (A) 50% (B) 75% (C) 120% (D) 133!% (E) 150% 72. A researcher computed the difference between the predicted numerical result and the actual numerical result for five different predictions, then calculated the mean, median, and standard deviation for that set of differences. (Each of the differences was greater than one.) If each of the differences were to be squared, which of these three statistics would change? (A) The mean only (B) The standard deviation only (C) The mean and the median (D) The mean and the standard deviation (E) The mean, median, and standard deviation (A) 18.0% (B) 62.0% (C) 79.2% (D) 80.0% (E) 82.0% 71. At a certain conference, 72% of the attendees registered at least two weeks in advance and paid their conference fee in full. If 10% of the attendees who paid their conference fee in full did not register at least two weeks in advance, what percent of conference attendees registered at least two weeks in advance? 3. PROBLEM SOLVING 21 (A) 37f (B) 67f (C) 91r (D) 127f (E) 277f 76. A closed cylindrical tank contains 277f cubic feet of water and is filled to half its capacity. When the tank is placed upright on its circular base on level ground, the height of the water in the tank is 3 feet. What is the circumference of the tank's circular base? (A) 75 (B) 100 (C) 135 (D) 150 (E) 200 75. A marketing firm found that, of 800 computer users surveyed, 280 were not familiar with either Website A or Website B, 220 were familiar only with Website A, and for every 3 computer users who were familiar only with Website B, one was familiar with both websites. How many of the 800 computer users were familiar with both websites? 1 30 I 20 1 12 1 6 1 3 (A) (B) (C) (D) (E) 74. A certain team of salespeople has 6 members, including Larry. One of the 6 members is to be chosen at random to be assigned Territory A, one of the remaining 5 members is to be chosen at random to be assigned Territory B, and one of the remaining 4 members is to be chosen at random to be assigned Territory C. What is the probability that Larry will be selected to be assigned either Territory B or Territory C? 3. PROBLEM SOLVING 22 80. Positive integer m is 25 percent of 25 percent of positive integer n, and m percent of n equals 25. What is the value of n ? (A) 25 (B) 50 (C) 100 (D) 200 (E) 500 (A) 47 (B) 60 (C) 92 (D) 95 (E) 96 79. Of the three-digit integers greater than 660, how many have two digits that are equal to each other and the remaining digit different from the other two? 78. The average (arithmetic mean) of the even integers from 200 to 300, inclusive, is how much greater than the average of the even integers from 60 to 100, inclusive? (A) 140 (B) 150 (C) 160 (D) 170 (E) 200 1 20 9 100 1 4 3 5 16 25 (A) (B) (C) (D) (E) 77. Among a class of 400 students, 80 percent studied at least 10 hours for the final exam, 25 percent received an 'A' on the final exam, and 20 percent of those who studied at least 10 hours received an 'A' on the final exam. If 1 student is to be randomly selected from the 400 students, what is the probability that the student selected will be one who studied at least 10 hours but did NOT received an 'A' on the final exam? 3. PROBLEM SOLVING 23 84. Last year Elaine spent 20% of her annual earnings on rent. This year she earned 15% more than last year and she spent 30% of her annual earnings on rent. The amount she spent on rent this year is what percent of the amount spent on rent last year? (A) 152.5 (B) 164.5 (C) 167.5 (D) 172.5 (E) 177.5 83. If the positive integer x is a multiple of 4 and the positive integer y is a multiple of 8, then xy must be a multiple of which of the following? I. 4 II. 8 III. 12 (A) I only (B) III only (C) I and II only (D) I and III only (E) I, II, and III 1 2 1 2v'2 1 4 1 4v'2 l 8 (A) (B) (C) (D) (E) 82. If n = ~' what is the value of fa ? 81. The product of the eight smallest two-digit integers is closest to which of the following powers of 10? (A) 109 (B) 108 (C) 107 (D) 106 (E) 105 3. PROBLEM SOLVING 24 (A) z-y x (B) z-x y (C) ~ x (D) x-z y (E) y-x z 89, If x, y, and z are nonzero integers and x - y = z, which of the following is equal to 1? 88. Which of the following could be the greatest common divisor of two prime numbers a and b, where 2 < a < b? (A) 1 (B) a - b (C) a (D) b (E) a+ b x-y = x-z (A) -2 (B) 1 -2 (C) 1 2 (D) 1 (E) 2 87, On the number line, if x > y, if y is halfway between x and z, then 5 2232 11 2:13:l 17 2:l3:l 31 2a3,, 67 2'13" (A) (B) (C) (D) (E) 86. Which of these fractions has the greatest value? 85. If 32x+l = 27x-3, then x = (A) -4 (B) -1 (C) 3 (D) 4 (E) 10 3. PROBLEM SOLVING 25 94. Last year a certain bond yielded 5 percent of its face value in interest. If that interest was approximately 4 percent of the bond's selling price of $7,500, what is the bond's face value? (A) $6,000 (B) $6,750 (C) $7,425 (D) $7,500 (E) $9,375 93. A class consists of 24 students. If a student is to be selected at random from the class, the probability that a woman will be selected is three times the probability that a man will be selected. How many women are in the class? (A) 6 (B) 8 (C) 16 (D) 18 (E) 20 92. When a is a multiple of 3, (a) = l Otherwise, (a) = 2a. Which of the following is equal to (5) x (6)? (A) (60) (B) (48) (C) (36) (D) (30) (E) (24) I. b is negative II. c is positive III. d is positive (A) I only (B) II only (C) III only (D) I and II (E) II and III 91. If a > d, c > a, d > b, and a > 0, which of the following must be true? 90. ( J7 + y'48 + J7 ~ J48)2 (A) 14 (B) 16 (C) 2V55 (D) 14 + 2v'50 (E) 16 + 2J55 3. PROBLEM SOLVING 26 8,x, y, 13,3, 8 99. The arithmetic mean of the list of numbers above is 8. If x and y are integers and the range of the list is 10, all of the following could be the value of x - y EXCEPT (A) 0 (B) 2 (C) 6 (D) 10 (E) 12 x 3 2x 3 x 2x 3x (A) (B) (C) (D) (E) 98. If x - y = -~, then which of the following represents the average (arithmetic mean) of x, y, and z, in terms of x? 97. What is the sum of the different positive prime factors of 720? (A) 7 (B) 8 (C) 10 (D) 16 (E) 17 96. If a= 3 and b = -2, then (b2 - a)(x - y) - (a+ b)(x + y) = (A) -2x (B) -2y (C) 2x (D) 2y (E) -2(x+y) Five Four Two Three 95. For a finite sequence of nonzero integers, the number of variations in parity is defined as the number of pairs of consecutive terms of the sequence for which the sum of the two consecutive terms is odd. What is the number of variations in parity for the sequence 1, 4, 3, 5, 8, 6 ? (A) One (B) (C) (D) (E) 3. PROBLEM SOLVING 27 100. For all numbers j and k, the operation * is defined by j" 1 j * k = p . If c * 2 = 8, then c = (A) -8 (B) -1 (C) 1 (D) 2 (E) 4 3. PROBLEM SOLVING
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