GMAT Statistics and Sets Je¤ Sackmann / GMAT HACKS February 2008
Contents 1 Introduction
2
2 Di¢culty Levels
3
3 Problem Solving
4
4 Data Su¢ciency
20
5 Answer Key
26
6 Explanations
29
1
1. INTRODU INTRODUCTIO CTION N
1
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Intr Introd oduc ucti tion on
This document contains nothing but GMAT questions on the topics of Statistics and Sets–100 Sets–100 of them, to be exact. exact. The GMAT GMAT loves loves these types of questions, questions, and has steadily steadily increased increased the the number number of them on the test. test. It’s almost almost guaranteed that, on test day, you’ll see items that focus on subjects such as mean, median, standard deviation, and overlapping sets. As in all of my GMAT preparation resources, you’ll …nd these questions indexed by di¢culty. That doesn’t mean you should skip straight to the hardest questions, questions, or even that you should should start with the easier ones. On the GMA GMAT T itself, questions won’t come labeled with their di¢culty level, and despite the intent of the adaptive algorithm, they won’t be precisely consistent in terms of di¢culty di¢culty either. either. Each Each question question presents presents its own unique unique challen challenges, ges, and the sooner you get accustomed to changing gears with every single question, the more time you’ll have to prepare for that particular challenge of the exam. For additional practice, I have produced several other resources that may help you. You’ll ou’ll …nd the most Statistics Statistics and Sets-relate Sets-related d questions in "Word "Word Problems: Problems: Challenge" Challenge" and "Word "Word Problems: Problems: Fundament undamentals. als." " There are also plenty plenty in the "Arithme "Arithmetic: tic: Challenge" Challenge" and "Arithme "Arithmetic: tic: Fundament undamentals" als" sets, especially those dealing with averages and overlapping sets. Eventuall Eventually y, you’ll you’ll start seeing questions questions that look familiar. familiar. That’s That’s a good thing: thing: there are only so many many ways the GMAT GMAT can test these concepts, concepts, and if you’ve done a few hundred Rates, Ratios, and Percents questions, you’ve seen just about every permutation they can throw your way. way. Also, The GMAT Math Bible has several chapters (along with focused practice) on these topics, including individual chapters on averages, weighted averages, statistics such as mean, median, mode, range, and standard deviation, and overlapping sets. If you …nd you are struggling with the mechanics of these problems, your time is probably better spent with the GMAT Math Bible than in doing dozens and dozens of practice problems, hoping to pick up those skills along the way. As far as strategy is concerned, there are dozens of articles at GMAT GMAT HACKS to help you with your strategic approach to Arithmetic questions. Most importantly, you should make sure you understand every practice problem you do. It doesn’t matter if you get it right the …rst time–what matters is whether you’ll get it right the next time you see it, because the next time you see it could be on the GMAT. With that in mind, carefully analyze the explanations. Redo questions that took you too long the …rst time around. Review questions over multiple sessions, rather than cramming for eight hours straight each Saturday. These basic study skills may not feel like the key to GMAT preparation, but they are the di¤erence between those people who reach their score goals and those who never do. Enough talking; there are 100 Statistics and Sets questions waiting inside. Get to work! 2 Copyright 2008 Je¤ Sackmann je¤
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1. INTRODU INTRODUCTIO CTION N
1
Je¤ Sackmann GMAT Tutor je¤
[email protected] je¤
[email protected] www.gmathacks.com
Intr Introd oduc ucti tion on
This document contains nothing but GMAT questions on the topics of Statistics and Sets–100 Sets–100 of them, to be exact. exact. The GMAT GMAT loves loves these types of questions, questions, and has steadily steadily increased increased the the number number of them on the test. test. It’s almost almost guaranteed that, on test day, you’ll see items that focus on subjects such as mean, median, standard deviation, and overlapping sets. As in all of my GMAT preparation resources, you’ll …nd these questions indexed by di¢culty. That doesn’t mean you should skip straight to the hardest questions, questions, or even that you should should start with the easier ones. On the GMA GMAT T itself, questions won’t come labeled with their di¢culty level, and despite the intent of the adaptive algorithm, they won’t be precisely consistent in terms of di¢culty di¢culty either. either. Each Each question question presents presents its own unique unique challen challenges, ges, and the sooner you get accustomed to changing gears with every single question, the more time you’ll have to prepare for that particular challenge of the exam. For additional practice, I have produced several other resources that may help you. You’ll ou’ll …nd the most Statistics Statistics and Sets-relate Sets-related d questions in "Word "Word Problems: Problems: Challenge" Challenge" and "Word "Word Problems: Problems: Fundament undamentals. als." " There are also plenty plenty in the "Arithme "Arithmetic: tic: Challenge" Challenge" and "Arithme "Arithmetic: tic: Fundament undamentals" als" sets, especially those dealing with averages and overlapping sets. Eventuall Eventually y, you’ll you’ll start seeing questions questions that look familiar. familiar. That’s That’s a good thing: thing: there are only so many many ways the GMAT GMAT can test these concepts, concepts, and if you’ve done a few hundred Rates, Ratios, and Percents questions, you’ve seen just about every permutation they can throw your way. way. Also, The GMAT Math Bible has several chapters (along with focused practice) on these topics, including individual chapters on averages, weighted averages, statistics such as mean, median, mode, range, and standard deviation, and overlapping sets. If you …nd you are struggling with the mechanics of these problems, your time is probably better spent with the GMAT Math Bible than in doing dozens and dozens of practice problems, hoping to pick up those skills along the way. As far as strategy is concerned, there are dozens of articles at GMAT GMAT HACKS to help you with your strategic approach to Arithmetic questions. Most importantly, you should make sure you understand every practice problem you do. It doesn’t matter if you get it right the …rst time–what matters is whether you’ll get it right the next time you see it, because the next time you see it could be on the GMAT. With that in mind, carefully analyze the explanations. Redo questions that took you too long the …rst time around. Review questions over multiple sessions, rather than cramming for eight hours straight each Saturday. These basic study skills may not feel like the key to GMAT preparation, but they are the di¤erence between those people who reach their score goals and those who never do. Enough talking; there are 100 Statistics and Sets questions waiting inside. Get to work! 2 Copyright 2008 Je¤ Sackmann je¤
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2.
2
DIFFICUL DIFFICULTY LEVELS LEVELS
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Di¢c Di¢cul ultty Lev Levels els
In general, the level 5 questions in this guide are 560- to 620-level questions. The level 6 questions represent a broad range of di¢culty from about 620 to 720, while the level 7 questions are higher still. Easy (4) PS 5, 12, 14, 35, 37, 45, 52, 54 DS 61, 66, 68, 71, 74, 82, 83, 84, 88, 89, 90 Moderate (5) PS 1, 4, 6, 7, 8, 9, 11, 16, 18, 20, 21, 24, 27, 28, 29, 30, 33, 34, 36, 39, 40, 42, 43, 46, 48, 49, 51, 53, 55, 57, 60 DS 63, 65, 69, 72, 73, 77, 79, 80, 85, 86, 87, 92, 94, 96, 97, 98, 100 Di¢cult (6) PS 2, 10, 13, 15, 17, 19, 22, 23, 26, 31, 32, 38, 44, 47, 50, 56, 58, 59 DS 62, 64, 70, 75, 76, 81, 91, 93, 95, 99 Very Di¢cult (7) PS 3, 25, 41 DS 67, 78
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3.
PROBLEM PROBLEM SOLVING SOLVING
3
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Prob Proble lem m Solv Solvin ing g
Note: Note: this guide contains contains both an answer key (so you can quickly quickly check check your answers) and full explanations. 1.
A law law o¢ce o¢ce emp emplo loys ys sev seven en secr secreta etarie ries. s. It pays pays ann annual ual salari salaries es of $17,000 to 2 of the secretaries, $19,000 to 2 of the secretari secretaries, es, and $23,0 $23,000 00 to the remaining remaining 3 secretari secretaries. es. The average (arithmetic mean) annual salary of these employees is closest to which of the following? (A) $19,000 (B) $19,700 (C) $20,000 (D) $20,100 (E) $21,000
2.
A freel freelanc ancee writer writer found found that that last last year year his his ave averag ragee (arith (arithmet metic ic mean) mean) revenue revenue from January January to October $4,800. In November November and December, his monthly revenues were 2 and 3 times, respectiv respectively ely,, the average average for the other 10 months. months. What was the writer’s average monthly revenue last year? (A) $5,400 (B) $6,000 (C) $6,400 (D) $6,800 (E) $7,200
3.
If x is 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 odd? 1 (A) 8 1 (B) 4 1 (C) 2 3 (D) 4 7 (E) 8
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3.
PROBLEM PROBLEM SOLVING SOLVING
4.
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I. 80, 81 81,, 82 82,, 83 83,, 84 II. II. 82 82,, 82, 82, 82 82,, 82 III. III. 69 69,, 82 82,, 82 82,, 82, 96 The data data sets sets I, I, II, II, and and III above above are ordere ordered d from from greate greatest st standard deviation to least standard deviation in which of the following? (A) (B) (C) (D) (E)
I, II, III I, III, II II, III, I III, I, II III, II, I
5.
The amou amount ntss of rainf rainfall all in in inches inches reco recorde rded d in six six di¤ere di¤erent nt citi cities es in a certain state last month were 12.5", 11.5", 10.8", 17", 18.2", and 15". What is the median of those amounts? (A) 14" (B) 13.75" (C) 13.2" (D) 13" (E) 12.5"
6.
The arithm arithmeti eticc mean mean and and stan standar dard d devia deviatio tion n of a certa certain in norm normal al distribution are 11.5 and 2.0, respectively. What value is exactly 2 standard deviations less than the mean? (A) 7.5 (B) 8.0 (C) 8.5 (D) 9.0 (E) 9.5
7.
If S = { 13 , 35 , 0, 12 , 1, 16 }, what is the positive di¤erence di¤erence betw b etween een the median of the numbers in S and the mean of the numbers in S? 1 (A) 12 1 (B) 15 1 (C) 30 1 (D) 60 1 (E) 120
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3.
PROBLEM SOLVING
8.
9.
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{–7, –6, –2, 14 , 0, 14 , 12 , 1, 2, 4, 6, 7} A number is to be selected at random from the set above. What is the probability that the number selected will be a solution of the equation (y + 4)(y 7)(4y + 1) = 0? 1 (A) 12 1 (B) 6 1 (C) 4 1 (D) 3 1 (E) 2 If a certain sample of data has a mean of 30.0 and a standard deviation of 2.5, all of the following values are more than 2.5 standard deviations from the mean EXCEPT (A) 22.0 (B) 22.5 (C) 23.5 (D) 36.0 (E) 36.5
10.
Of the 300 students at a certain university in State S, 210 were born in State S and 250 attended high school in State S. If at least 40 of the students were neither born nor attended high school in State S, then the number of students who were both born and attended high school in State S could be any number from (A) 40 to 80 (B) 80 to 110 (C) 90 to 210 (D) 200 to 210 (E) 200 to 250
11.
Jaclyn purchased 3 picture frames with an average (arithmetic mean) price of $18. If, after Jaclyn purchases another picture frame, the average price of the 4 picture frames is $20, what is the price of the fourth picture frame? (A) $19 (B) $20 (C) $22 (D) $26 (E) $30
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3.
PROBLEM SOLVING
12.
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$15, $18, $20, $24, $25, $30, $50, $50, $64, $80 The total amounts paid by a series of customers at a certain store are shown above. How many amounts were greater than the median amount but less than the mean amount? (A) None (B) One (C) Two (D) Three (E) Four
13.
In a certain neighborhood, …ve houses are listed for sale at the following prices: $92,000, $98,000, $112,000, $115,000, and $128,000. If the price of the most expensive house is increased $6,000 and the price of the least expensive home is decreased by the same amount, which of the following best describes the change in the mean and the median of the house prices? (A) The mean and the median will remain unchanged. (B) The mean will remain unchanged but the median will increase. (C) The mean will increase but the median will remain unchanged. (D) The mean and the median will increase by the same amount. (E) The mean and the median will increase by di¤erent amounts.
14.
The average (arithmetic mean) of 40, 60, and 80 is 5 more than the average of 30, 50, and (A) 45 (B) 55 (C) 65 (D) 75 (E) 85
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15.
A researcher tracked the daily changes in price of all the stocks on a certain exchange for a period of one month, and found that the mean daily price change of the stocks on the exchange was $0.16, and the standard deviation of the price changes was $0.04. The largest price change the researcher observed took place on the 21st of last month, and was between 8 and 9 standard deviations above the mean. Which of the following could have been the dollar value of the largest price change observed last month? (A) $0.25 (B) $0.28 (C) $0.35 (D) $0.42 (E) $0.49
16.
A certain typing service charges $50 to type a document of up to 20 pages and $15 for each additional 10 pages, or portion thereof. If Paul employed the service to type a 100-page document, what would be the average (arithmetic mean) charge per page? (A) $0.65 (B) $1.70 (C) $1.75 (D) $1.90 (E) $2.00
17.
A set of 25 di¤erent integers has a median of 50 and a range of 50. What is the greatest possible integer that could be in this set? (A) 62 (B) 68 (C) 75 (D) 88 (E) 100
18.
Of the 11 students who took the …nal exam in a biology class, 3 scored a 72, 2 scored a 93, 3 scored a 76, 1 scored a 78, and 2 scored an 89. What was the median score on the …nal exam? (A) 72 (B) 76 (C) 78 (D) 89 (E) 93
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PROBLEM SOLVING
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19.
If m is the standard deviation of x, y , and z , what is the standard deviation of x 5, y 5, and z 5? 1 m (A) 3 m (B) (C) 5m m5 (D) m 15 (E)
20.
List M consists of 15 consecutive integers. If 4 is the greatest integer in list M, what is the range of the negative integers in list M? (A) 8 (B) 9 (C) 10 (D) 14 (E) 15
21.
A certain doughnut shop has six employees. It pays annual salaries of $15,000 to each of 3 employees, $17,000 to 1 employee, and $18,000 to each of the remaining 2 employees. The average (arithmetic mean) annual salary of these employees is closest to which of the following? (A) $15,800 (B) $16,300 (C) $16,600 (D) $16,800 (E) $17,000
22.
A list of measurements in increasing order is 2, 4, 5, 7, 15, and x. If the median of these measurements is 23 their arithmetic mean, what is the value of x ? (A) 16 (B) 18 (C) 19 (D) 21 (E) 23
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23.
In a certain graduating class, a grade-point average of 2.8 was 1.5 standard deviations below the mean, and a grade-point average of 3.6 was 2.5 standard deviations above the mean. What the mean grade-point average of the graduating class? (A) 3.4 (B) 3.3 (C) 3.2 (D) 3.1 (E) 3.0
24.
The average (arithmetic mean) of the 4 numbers k , 2k 5, 3k + 2, and 6k 1 is 17, what is the value of k ? (A) 4 (B) 5 (C) 6 (D) 8 (E) 9
25.
A certain list of 200 test scores has an average (arithmetic mean) of 85 and a standard deviation of d, where d is positive. Which of the following two test scores, when added to the list, must result in a list of 202 test scores with a standard deviation less than d ? (A) 80 and 80 (B) 80 and 85 (C) 80 and 90 (D) 85 and 85 (E) 85 and 90
26.
If 0 < z < 1, what is the median of the values z , z 1, and z 3 ? z (A) z1 (B) p z (C) z2 (D) z3 (E)
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p z, z 2 ,
3.
PROBLEM SOLVING
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27.
Three adults have an average (arithmetic mean) weight of 160 pounds and a median weight of 180 pounds. What is the maximum possible weight, in pounds, of the lightest of the three adults? (A) 109 (B) 110 (C) 119 (D) 120 (E) 140
28.
An investor purchased 30 shares of a certain stock at a price of $26.50 per share. Later this investor purchased 20 more shares at a price of $25.50 per share. What was the average (arithmetic mean) price per share that this investor paid for the 50 shares? (A) $25.90 (B) $26.00 (C) $26.10 (D) $26.25 (E) $26.30
29.
In a certain normal distribution, the arithmetic mean is 9.5 and the standard deviation is 1.75. What value is exactly 2 standard deviations from the mean? (A) 6 (B) 6.75 (C) 7 (D) 12.25 (E) 14
30.
Which of the following is equal to the average (arithmetic mean) of (x + 2)2 and (x + 2)(x 2) ? x2 (A) x2 + 2 (B) x2 + 4 (C) x2 + 2 x (D) x2 + 4 x (E)
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PROBLEM SOLVING
31.
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W represents the sum of the weights of s steaks in pounds.
Which of the following represents the average (arithmetic mean) of the s weights in ounces? (1 pound = 16 ounces) (A) 16W s 16W (B) s (C)
Ws
(D) (E)
W 16s s 16W
16
32.
Forty percent of the employees at a brokerage …rm have passed a certain licensing exam. Among the employees who have not passed the exam, 32 are exempt from the licensing requirement and 16 are not exempt. How many employees does the brokerage …rm have? (A) 60 (B) 80 (C) 96 (D) 108 (E) 120
33.
The maximum temperatures, in degrees Celcius, recorded in a city on 5 consecutive days were 32, y + 2, y + 5, 30, and y. If the average (arithmetic mean) of these temperatures was 24 degrees Celcius, what is the value of y ? (A) 17 (B) 19 (C) 21 (D) 22 (E) 23
34.
Three students reported that the amount of time they spent preparing for a certain exam was between 0 and 10 hours, inclusive. If the average (arithmetic mean) number of hours the students reported that they spent preparing was 7.5 hours, what was the least possible number of hours that one of the students spent preparing, in hours? (A) 2 (B) 2.5 (C) 4 (D) 4.5 (E) 6
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35.
The one-day price changes in dollars of several stocks on a certain exchange were $0:25, $0:05, $0:10, $0:05, $0:25, and $0:60. What is the median of these price changes? (A) $0:10 (B) $0:05 (C) $0:00 (D) $0:05 (E) $0:25
36.
Of the 120 passengers on an airplane, 50 had 2 pieces of luggage each, 40 had 1 piece of luggage each, 10 had 3 pieces of luggage each, 10 had 4 pieces of luggage each, and the remainder had no luggage. What was the average (arithmetic mean) number of pieces of luggage per passenger on the airplane? (A) 1 (B) 1.25 (C) 1.5 (D) 1.75 (E) 2
37.
38.
69 78 78 79 80 82 84 90 The mean and the approximate standard deviation of the 8 numbers shown are 80 and 6, respectively. What percent of the 8 numbers are within 1 standard deviation of the mean? (A) 87.5% (B) 82.5% (C) 80% (D) 75% (E) 62.5% Route A: 35 Route B: 25 Route C: 20 The table above shows the number of pilots who ‡y three routes for an airline. Although none of the pilots ‡ies all three routes, 7 pilots ‡y both A and B, 4 pilots ‡y both A and C, and 3 pilots ‡y both B and C. How many di¤erent pilots ‡y these three routes for the airline? (A) 64 (B) 66 (C) 67 (D) 73 (E) 75
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PROBLEM SOLVING
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39.
A realtor sold 15 of the new homes in a certain neighborhood in April, and sold the remaining 10 in May. In April, the range of the selling prices was $22,000 and the lowest selling price was $184,000. In May, the range of the selling prices was $25,000 and the lowest selling price was $192,000. What was the range of the selling prices of the 25 homes that the realtor sold in April and May? (A) $23,500 (B) $27,500 (C) $28,000 (D) $30,000 (E) $33,000
40.
A …rm purchased two machines whose prices were $5,000 and $9,000, respectively. The …rm is to purchase two more machines from a list of machines whose prices range from $5,000 to $9,000, inclusive. The greatest possible average (arithmetic mean) price of the 4 machines is how much greater than the least possible average price of the 4 machines? (A) $2,000 (B) $2,500 (C) $3,000 (D) $3,500 (E) $4,000
41.
If z is the standard deviation of p, q , and r , what is the standard deviation of 3 p, 3q , and 3r ? z (A) (B) 3z z+ 3 (C) z+ 9 (D) (E) It cannot be determined from the information given.
42.
A retailer has a goal of selling $250,000 worth of software in 100 days. If it sold $170,000 in the …rst 60 days, what is the average (arithmetic mean) number of dollars per day of software that it must sell for the last 40 days in order to achieve its goal? (A) $1,333 (B) $1,500 (C) $1,750 (D) $2,000 (E) $2,500
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PROBLEM SOLVING
43.
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35, 37.5, 40, 42.5, 45, 52.5, 52.5, 55, 60, 60 The list shown consists of the heights, in inches, of each of 10 schoolchildren. If the standard deviation of the heights is 9.2 inches, rounded to the nearest tenth of an inch, how many of the 10 heights are more than 1 standard deviation away from the mean of the 10 heights? (A) (B) (C) (D) (E)
Two Three Four Five Six
44.
In a survey of 232 people, 153 own their own home, 80 have children, and 13 of those who own their own home have children. If a person is to be randomly selected from those surveyed, what is the probability that the person selected will have children but does not own his or her own home? 1 (A) 8 29 (B) 153 25 (C) 116 51 (D) 232 1 (E) 4
45.
The number of credits being taken by each of eight students at a certain university are 16, 12, 11, 15, 8, 6, 14, and 15. What is the range in the number of credits being taken by the eight students? (A) 8 (B) 9 (C) 10 (D) 11 (E) 12
46.
An investor’s portfolio increased in value by $15,000 in 1992 and by $25,000 in 1993. In 1994 the portfolio decreased in value by $17,500. What was the portfolio’s average (arithmetic mean) increase in value for the 3 years? (A) $5,000 (B) $7,500 (C) $10,000 (D) $12,500 (E) $15,000
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47.
In a random sample of 20 members of a certain health club last week, 3 did not attend any classes, 8 each attended one class, 5 each attended 2 classes, and the remaining members in the sample attended at least 3 classes. If the average (arithmetic mean) number of classes attended by each member was 1.75, what is the maximum number of classes than any single member could have attended? (A) 4 (B) 5 (C) 6 (D) 8 (E) 10
48.
Last week, Franka recorded the time that she spent each day in a morning meeting. The times, in minutes, were 52, 28, 40, 39, and 51. How many minutes greater was the average (arithmetic mean) time than the median time? (A) 1.5 (B) 2 (C) 2.5 (D) 3 (E) 3.5
49.
Hank, Jelena, and Kristof each called customer support for a certain product. Hank called customer support 5 times and Jelena called customer support 2 times. If the 3 people called customer support an average (arithmetic mean) of 3 times per person, how many times did Kristof call customer support? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
50.
Each of the 15 cars a certain dealer sold last week were purchased for one of three di¤erent prices and the average selling price of the 15 cars was $16,000. If 6 of the cars were sold for $14,000 each and 5 of the cars were sold for $16,000 each, what was the selling price of each of the remaining 4 cars? (A) $16,000 (B) $17,500 (C) $18,000 (D) $18,500 (E) $19,000 16 Copyright 2008 Je¤ Sackmann je¤
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PROBLEM SOLVING
51.
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List I: 4, 8, 10, 15 List II: k , 4, 8, 10, 15 If the median of the numbers in list I above is equal to the median of the numbers in list II above, what is the value of k ? (A) 6 (B) 8 (C) 9 (D) 10 (E) 11
52.
Delia had 5 conference calls on Monday, 7 conference calls on Tuesday, 8 conference calls on Wednesday, 5 conference calls on Thursday, and 12 conference calls on Friday. What was the average (arithmetic mean) number of conference calls that Delia made for those 5 days? (A) 8.0 (B) 7.8 (C) 7.6 (D) 7.4 (E) 7.2
53.
A group of 20 people each received tax refunds, and that the average (arithmetic mean) tax refund was $4,000. If the average tax refund of 16 of the people in the group was $3,000, what was the average tax refund for the other four people in the group? (A) $4,000 (B) $5,000 (C) $6,000 (D) $7,000 (E) $8,000
54.
The average (arithmetic mean) of 10, 15, and 20 equals the average of 12, 18, and (A) 6 (B) 10 (C) 12 (D) 15 (E) 24
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55.
Of the 18 entrees o¤ered on a certain menu, 10 contain meat, 12 contain cheese, and 3 contain neither meat nor cheese. How many of the entrees contain both meat and cheese? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10
56.
If the average (arithmetic mean) of a and b is 60 and the average (arithmetic mean) of b and c is 70, what is the value of c a ? (A) 65 (B) 20 (C) 10 (D) 5 (E) It cannot be determined from the information given.
57.
If 65 percent of apparel retailers in City X sell Brand A clothing, 65 percent carry Brand B, and 25 percent carry neither brand, what percent carry b oth brands? (A) 35 (B) 45 (C) 50 (D) 55 (E) 65
58.
For the past x days, Joan has made an average (arithmetic mean) of 15 sales per day. If Joan makes 25 sales today and raises her daily average to 16 sales per day, what is the value of x ? (A) 4 (B) 5 (C) 9 (D) 10 (E) 12
59.
M is the set of positive even integers less than 75, and N is the set of the square roots of the integers in M . How many elements does the intersection of M and N contain?
(A) (B) (C) (D) (E)
None Two Four Five Six
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60.
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A rope 40 feet long is cut into 5 pieces. If one of the pieces is 16 feet long, what is the average (arithmetic mean) length, in feet, of the remaining pieces? (A) 3.2 (B) 4 (C) 4.8 (D) 6 (E) 8
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Data Su¢ciency
For all Data Su¢ciency questions, the answer choices are as follows: (A) (B) (C) (D) (E)
Statement (1) ALONE is su¢cient, but statement (2) alone is not su¢cient. Statement (2) ALONE is su¢cient, but statement (1) alone is not su¢cient. BOTH statements TOGETHER are su¢cient, but NEITHER statement ALONE is su¢cient. EACH statement ALONE is su¢cient. Statements (1) and (2) TOGETHER are NOT su¢cient.
61.
If p is an integer, is q an integer? (1) The average (arithmetic mean) of p, q , and 8 is p. (2) The average (arithmetic mean) of p, q , and 3.5 is 3.5.
62.
For a certain set of n numbers, where n > 2, is the average (arithmetic mean) equal to the median? (1) (2)
The n numbers are positive, consecutive integers. The smallest integer in the set is odd.
63.
If a, b, c, d, and e are di¤erent positive integers, which of the …ve integers is the median? a, b, c, d, and e are consecutive integers. (1) (2) The average (arithmetic mean) of the …ve integers is d.
64.
How many of the 250 attendees at a certain concert paid full price for their ticket and bought their ticket at least two weeks in advance? (1) Of the 250 attendees, 65 bought their ticket at least two weeks in advance but did not pay full price for their ticket. (2) Of the 250 attendees, 75 paid full price for their ticket but did not buy their ticket at least two weeks in advance.
65.
Of a physician’s clients, 120 have health insurance or prescription drug coverage or both. If 40 of the clients do not have prescription drug coverage, how many of the clients have both health insurance and prescription drug coverage? (1) A total of 92 of the clients have health insurance. (2) Of the 120 clients, 28 do not have health insurance.
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66.
If the average (arithmetic mean) of …ve numbers is 60, how many of the numbers are equal to 60? (1) At least two of the numbers are greater than 60 (2) At least two of the numbers are less than 60.
67.
Set S has a range of p and the largest number in set S is v . Set T has a range of q and the largest number in set T and the largest number in set T is w. Is the smallest number in set S greater than the smallest number in set T? p < q (1) (2) The median of set S is less than the median of set T, and the average (arithmetic mean) of set S is greater than the mean of set T.
68.
What is the average (arithmetic mean) of x, y, and z ? x+y+z (1) =9 2 (2)
x+y+z
3
=6
69.
If z is a positive integer, is z < 16 ? z is less than the average (arithmetic mean) of the …rst (1) ten positive even integers. z is the square of an integer. (2)
70.
Of the 600 employees in a certain company, 180 drive to work and are more than 30 years old. How many of the 600 employees drive to work and are 30 years old or less? (1) 480 of the employees in the company are more than 30 years old. (2) 50 employees in the company are 30 years old or less and do not drive to work.
71.
If a, b, c, d, and e are di¤erent positive integers, which of the …ve integers is the median? b+c < e (1) a+d < e (2)
72.
Is q equal to the median of the three positive integers n, p, and q ? p = 4 n = 3 q (1) q = 12 (2)
73.
If twelve consecutive even integers are listed from least greatest, what is the average (arithmetic mean) of the twelve integers? (1) The average of the …rst eight integers is 13. (2) The average of the last eight integers is 21. 21 Copyright 2008 Je¤ Sackmann je¤
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74.
What is the average (arithmetic mean) of a, b, and c ? a+b =7 (1) b + c = 10 (2)
75.
At least 100 employees in a certain company have management experience. If 15 percent of the employees in the company who have sales experience also have management experience, do more employees have sales experience than management experience? (1) 72 employees in the company have both sales experience and management experience. (2) 252 employees in the company have neither sales experience nor management experience.
76.
In the last two years, each of Jeremy’s six children grew in height by at least one inch. If the standard deviation of their heights two years ago was 4.5 inches, what is the standard deviation of their heights? (1) In the last two years, the heights of Jeremy’s six children have increased a total of 17 inches. (2) In the last two years, each child’s height has increased by 5 percent.
77.
When scientists introduced a certain chemical into 30 di¤erent ponds, the population of bacteria in some of the ponds decreased, and the the population of …sh in some of the ponds decreased. In how many of the ponds did the population of …sh decrease but the population of bacteria did not decrease? (1) The population of …sh decreased in 7 of the 30 ponds. (2) The population of both …sh and bacteria stayed constant or increased in 18 of the 30 ponds.
78.
For the members of Team A, the range of their heights is a inches and the least height is j inches. For the members of Team B, the range of their heights is b inches and the least height is k inches. Is the greatest height of the members of Team A less than the greatest height of the members of Team B? a 4x (1) (2) 0 n So, q is the median.
Statement (2) is insu¢cient: we need to know something about the other two variables, as well. 73. D Explanation: Since the integers are consecutive evens, if we can …nd out one of the numbers–the least or the greatest, or any term that gives us those– we’ll know all we need to know about the entire set. Statement (1) is su¢cient. If the average of the …rst eight integers is 13, we can …nd what those …rst 8 integers are. As it turns out, they are the evens from 6 to 20, so the twelve integers are the evens from 6 to 28. Statement (2) is also su¢cient. If the average of the last eight is 21, we can …gure out that those eight are the evens from 14 to 28. That means that entire set of 12 evens stretches from 6 to 28. Choice (D) is correct. 74. E Explanation:
To …nd the average, we need the sum of the three terms.
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Statement (1) is insu¢cient: there’s nothing about c. (2) is also insu¢cient, since we don’t know anything about a. Taken together, the statements are still insu¢cient. We have two linear equations and three variables, which isn’t enough to …nd the values of each of the variables. There’s no other way to …nd a + b + c, so choice (E) is correct. 75. E Explanation: Statement (1) is insu¢cient. We know that 72 represents 15% of the total, so we can …nd the total, but there’s no way to compare the number with sales experience to the number with management experience, since all we know is the number that have both. Statement (2) is also insu¢cient. The number who have neither is irrelevant to a comparison of how many have each type of experience. Taken together, the statements are still insu¢cient. If 72 is 15% of the total, there are 480 employees in the company, which means that 480 252 = 228 have either management or sales experience, or both. Since 72 have both, that means 156 have one or the other, but not both. However, we have no way of determining how many have each kind, so (E) is the correct answer. 76. B Explanation: Statement (1) is insu¢cient. Standard deviation is based on the exact measurements. So knowing the total increase is not enough: the standard deviation would be di¤erent if that 17 inches was entirely due to the growth of one child than if the 17 inches was evenly distributed among the six children. Statement (2) is su¢cient. If each term in a set increases by the same percent, the standard deviation increases by that percent as well. We know the standard deviation two years ago, and we know how many each individual term changes (in percent terms), so we can calculate the new standard deviation. Choice (B) is correct. 77. E Explanation: Statement (1) is insu¢cient. We know in how many ponds the population of …sh decreased, but not in how many of those the population of bacteria did not decrease. Statement (2) is also insu¢cient: If both increased or stayed constant in 18, that leaves 12 ponds in which either the population of …sh, of bacteria, or both decreased. However, that’s not enough to answer the question–in how many ponds the …sh population decreased and bacteria didn’t. Taken together, the statements are insu¢cient. 7 is the total of the subgroups "…sh decrease" and "both decrease", while 12 is the total of those two, plus "bacteria decrease." Thus, 5 ponds have "bacteria decrease," but again, that’s not what we’re looking for. We don’t have the information to separate the two subgroups of the total of 7, so the correct choice is (E). 78.
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Explanation: For each of the two teams, the height of the tallest member is the height of the least member plus the range. Team A: j + a Team B: k + b So, the question is asking: "Is j + a < k + b ?" Statements (1) and (2) both fail to provide enough information: to answer the question, we need something about all four variables, and each statement only o¤ers something about two each. Taken together, the statements are su¢cient. If j < k and a < b, then the sum j + a must be less than k + b. That’s what the question wants to know, so (C) is correct. 79. A Explanation: Statement (1) is su¢cient. If the average of six of the numbers is 72, and the average of the seven numbers is 72, the seventh number must be 72. To see why, consider that if the average of six of the numbers is 72, the total of those numbers is 6(72). So, to solve for the …nal number, use the average formula: 6(72)+x = 72 7 6(72) + x = 7(72) x = 7(72) 6(72) = 72 Since the other six are odd integers, none of those can be equal to 72. One number is equal to 72. Statement (2) is insu¢cient. We’re only given information about one of the numbers, and the possibility is left over that many more are also equal to 72. (A) is the correct choice. 80. E Explanation: The overlapping sets formula is a handy way to work with this question. If Group 1 is the number of blondes, and Group 2 is the number of males: T = G1 + G2 Both + Neither Each of statement (1) and (2) only give us one of those variables each, and the question only gives us one as well. In a …ve-variable equation, two variables isn’t going to give us enough information. Taken together, the statements are still insu¢cient. Plug in the information given: 56 = 36 + 32 Both + Neither Both = 12 + Neither Unless we know how many people in the group are neither blonde nor male, we won’t know how many are blonde and male. Choice (E) is correct. 81. E Explanation: Statement (1) is insu¢cient. It helps a little bit: it reduces the four subgroups (willing to travel only, specialize in marketing only, both,
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and neither) to three, since none are only willing to travel to City X. Still, there are three subgroups and we know nothing but the total of the three. Statement (2) is also insu¢cient. In terms of the subgroups listed in (1), it tells us that 5 are "neither." Of the four subgroups, that leaves us with three unknowns. Taken together, the statements are still insu¢cient. 32 is the total of three subgroups: those who specialize in marketing but are not willing to travel to City X (which is what we’re looking for), those who do both, and those who do neither (5). We’re left with two variables, so we can’t solve for our target. Choice (E) is correct. 82. E Explanation: Statement (1) tells you something you already know. If the average is 16, the sum of the two terms is 32. That’s redundant, so not only is (1) insu¢cient, but the statements combined won’t be su¢cient (if we get that far), because (1) doesn’t contribute any information at all. Statement (2) is also insu¢cient. The positive di¤erence means that 16 could be equal to v z or z v . A di¤erence of 16 and a sum of 32 means that the numbers are 24 and 8, but the answer to the question is di¤erent depending on which variable is which of the two numbers. As we saw with (1), there’s no purpose in combining the statements in this example, so choice (E) is correct. 83. C Explanation: Each of the statements is insu¢cient on its own. Each tells you that the gap between two of the numbers is 4. If the other two terms are between those endpoints, the range of the set could be as little as 4. But since we don’t know, in either case, about the other two terms, they could be much more dispersed, resulting in a range larger than 6. Combined, the statements are su¢cient. Add the equations together: wx =4 xz =4 wz =8 Since the di¤erence between w and z is 8, the range of the set must be at least 8. It could be larger, if y is not between the 2, but regardless of whether that’s the case, the range is at least 8. Choice (C) is correct. 84. B Explanation: Converting the question to algebra: 0 < x < 10 z < x+10 ? 2 Statement (1) is insu¢cient. Combined with the equation in the question, this means that z is between 0 and 60. Sometimes that’s less than the average of z and 10 (if z = 1, x = 16 , and the average of x and 10 is about 5), but most of the time it’s not (if z = 24, x = 4, average of x and 10 is 7). Statement (2) is su¢cient. This could mean one of two things. If z is greater than 10, this is always true, since a number greater than 10 is always 46 Copyright 2008 Je¤ Sackmann je¤
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closer to 10 than it is to a number that is less than 10, such as x. It could also mean, if z is less than 10, that 10 z < z x. In the …rst case, z is always greater than the average of x and 10. In the second case, it is also greater. The average of x and 10 is the halfway point between those two numbers. If z is closer to 10 than to x, it is greater than the halfway point, so it is greater than the average. Choice (B) is correct. 85. D Explanation: As always with average, we can …nd the answer if we can …nd the sum of the two terms. (To …nd the average, we’d divide that sum by 2.) Statement (1) is su¢cient. Set up the equation: (j +3)+(k+3) = 12 2 j + k + 6 = 24 j + k = 18 Statement (2) is also su¢cient. Again, set up the equation and try to isolate j + k : j +k+6 =8 3 j + k + 6 = 24 j + k = 18 Choice (D) is correct. 86. E Explanation: Statement (1) is insu¢cient. If 115 are made of leather, 85 are not made of leather. If 65 of those are not designed for men, 20 of them are designed for men. That’s doesn’t answer the question, though. Statement (2) is also insu¢cient. In fact, it repeats what the other statement tells us, since we solved for that data point in (1). If 20 of those designed for men are not made of leather and 85 are not made of leather, that means 65 are not made of leather and not designed for men. Taken together, we don’t gain anything. The statements tell us the same thing, and since each is insu¢cient, combined they are also insu¢cient. (E) is correct. 87. E Explanation: Aside from the two unknowns, the endpoints of the set are 4 and 7. The range will be greater than 9 only if one of three scenarios is true: - The di¤erence between x and y is greater than 9. - x or y is greater than 13. - x or y is less than -6. Statement (1) is insu¢cient. If x is 0, all it means is that y must be positive. The numbers could be very close together, and between -6 and 13. However, it’s a weak limitation: even if x = 0, y could be anything, including numbers much greater than 13. Statement (2) is also insu¢cient. Again, the variables could be very large numbers, but they could also be 1 and 2, in which case the range is less than 9. 47 Copyright 2008 Je¤ Sackmann je¤
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Taken together, the statements are still insu¢cient. In both statements, one or both of the variables could be very large, in which case the range is greater than 9. However, if x = 1, y could be as small as 5. If those are the values of x and y , then the range is 6, which is less than 9. Choice (E) is correct. 88. E Explanation: Statement (1) is insu¢cient. Since all the integers are positive, we do know that n is greater than either q or r . However, we need to know about the other two terms, as well. Statement (2) is also insu¢cient. It doesn’t give us any information about the other three terms. Taken together, the statements are still insu¢cient. Any of the …ve terms could be the median: for instance, if m and p are both greater than n, n is the median. However, if m and p are both less than q and r, either q or r coudl be the median. (E) is the correct choice. 89. E Explanation: Statement (1) is insu¢cient: all or none (or one or two) of the other numbers could also be equal to 100 and still allow the average to be 100. Statement (2) is also insu¢cient. If one is 50, at least one must be greater than 100, but that leaves the options of zero, one, or two of the numbers to be equal to 100. Taken together, the statements are insu¢cient. If the average is 100, the sum of the 4 terms is 400. That means the sum of the remaining two terms is 250. One of those terms could be 100, in which case the other is 150; also, it’s possible that both could be 125, among many other possibilities. There could be one or two total terms that are equal to 100. Choice (E) is correct. 90. A Explanation: Statement (1) is su¢cient. If John’s average score was 112, his total scores were 112 times 3. If the average of his highest and lowest were 118, the sum of those two is 2 times 118. We don’t need to do the math, but with those numbers, we could …nd his middle score – it’s the di¤erence between the total of all 3 and the total of the highest and lowest. Since the median is, by de…nition, the middle score, the third game is the median. Statement (2) is insu¢cient. Between the average score and his highest score, we can …nd the sum of his lowest and middle scores, but in order to …nd the median, we need the exact amount of his middle score. Choice (A) is correct. 91. E Explanation: Statement (1) is insu¢cient. In fact, it’s irrelevant. We don’t need to know anything about the total population or how the number 25 years olds relates to it.
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Statement (2) is also insu¢cient. This is more helpful, though. There are two subgroups of 25 year olds (men and women), and the subgroups are 90% and 75% full-time students, respectively. If there is an equal number of men and women, that means 82.5% of those 25 years old and younger are full-time students. However, we don’t know whether there are equal numbers of men and women. If there are enough men, that drags the average down below 80% and gives us a di¤erent answer to the question. Taken together, the statements are still insu¢cient. (1) doesn’t help us …gure out what fraction of those 25 and younger are men or women, which is what we need to add to (2) to answer the question. (E) is correct. 92. E Explanation: Statement (1) is insu¢cient: we don’t know anything about the populations of the AA and A rated countries. Statement (2) is also insu¢cient, as it doesn’t tell us anything about the populations of the AAA rated countries. Taken together, the statements are still insu¢cient. 25 percent of the AAA countries have populations greater than 20 million, and 30 percent of the AA and A countries have populations greater than 20 million. For an overall percentage, though, we’d need to know the numbers (or at least the ratio) of AAA and AA/A countries. While we know that the percent is between 25 and 30 percent, we can’t …nd the exact percent without that ratio. (E) is the correct choice. 93. C Explanation: Statement (1) is insu¢cient. Between the question and this statement, we know how much the sets overlap, but we don’t know anything about the actual size of the sets, which is very important when determining how many unique integers are in the three sets. Statement (2) is also insu¢cient. We know the total number in each set, and the number of overlaps in two sets, but we don’t know how many terms are in all three sets. Taken together, we have enough information. Given the size of the sets, the overlaps between each pair of sets, and the overlap of all three sets, we can …nd the total number of integers. Choice (C) is correct. 94. A Explanation: Statement (1) is su¢cient. 60% of biotech employees are biotech executives, and 75% of the total are biotech employees. So, the percent of the total that is biotech executives is 60% of 75% of the total, or (0:6)(0:75) = 0:45. Statement (2) is insu¢cient. We’re looking for a percent. We don’t need a number, and furthermore, this number isn’t relevant to the speci…c subgroup we’re looking for. Choice (A) is correct. 95.
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Explanation: Statement (1) is insu¢cient. Since the sets are di¤erently sized, they could reach the same sum in a variety of ways. If both sets are centered on zero, the sums could be zero; or, the smaller set could have a higher average, in which case the medians would be di¤erent. Statement (2) is also insu¢cient. If the median of Y is 0, Y = {-4, -3, -2, -1, 0, 1, 2„ 3, 4}, but we don’t know anything about the median of X. Taken together, the statements are su¢cient. The sum of Y, as given in (2), is 0. If the sum of X is also 0, X = {-3, -2, -1, 0, 1, 2, 3}, which has a median of 0. Thus, the medians of the two sets are equal. Choice (C) is correct. 96. C Explanation:
In order, the terms in the set, other than m, are {-9, -4, 0,
3}. Statement (1) is insu¢cient. If the median is greater than -4, the median must be either 0 (if m is greater than 0) or m, if m is between -4 and 0. m could be anywhere from -4 up to a very large number. Statement (2) is also insu¢cient. If the median is -4, the median is negative, and m would be less than -4. If m is the median, as in (1), it could be a number between -4 and 0. Taken together, the statements are su¢cient. The only way the median is between -4 and 0 is if m is the median, in which case m is between -4 and 0, which is within the range of -6 and 2 that the question is asking about. Choice (C) is correct. 97. E Explanation: Statement (1) tells us that 20 have either only been immunized against D, or immunized against neither. Statement (2) tells us that 30 have either been immunized only against D, only against E, or against neither. Taken together, we can determine that 10 have been immunized only against E, but that leaves two variables. 20 is the sum of those who have been immunized only against D or against neither. Since there’s no way of separating those groups, the correct choice is (E). 98. C Explanation: Statement (1) is insu¢cient: knowing the median doesn’t, by itelf, tell you the mean. For the mean, you need to either know all the terms, or the relationships between all the terms. Statement (2) is also insu¢cient. Knowing the integers are equally spaced tells us that the median and the mean are equal, but since we don’t know anything about the size of the numbers, we can’t answer the question. Taken together, the statements are su¢cient. (2) tells us that the mean and the median are the same, while (1) gives us the median. Between them, we know the mean is 44. Choice (C) is correct. 99.
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