Contents
Contents
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1 IInnttrroodduuccttiioon n 22
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2 DDii¢¢ccuulltty y LLeevveells s 33
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3 PPrroobblleem m SSoollvviinng g 44 4
4 DDaatta a SSuu¢¢cciieennccy y 2200 5
5 An Ansswweerr KKeeyy 2266
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1
1 In
Intr
trod
oduc
ucti
tion
on
This document contains nothing but GMAT questions on the topics of Statistics This document contains nothing but GMAT questions on the topics of Statistics and Sets–10
and Sets–100 0 of them, to of them, to be be exactexact. . The GMAThe GMAT T lovloves these types of es these types of questquestions,ions, and has stead
and has steadily incrily increased theased the nume number of them on the testber of them on the test. . It’s almIt’s almost guar-ost guar-anteed that, on test day, you’ll see items that focus on subjects such as mean, anteed that, on test day, you’ll see items that focus on subjects such as mean, median, standard deviation, and overlapping sets.
median, standard deviation, and overlapping sets.
As in all of my GMAT preparation resources, you’ll …nd these questions 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 indexed by di¢culty. That doesn’t mean you should skip straight to the hardest quest
questions, or even that you ions, or even that you shoulshould d start with the easier ones. start with the easier ones. On the On the GMAGMATT itself, questions won’t come labeled with their di¢culty level, and despite the 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 intent of the adaptive algorithm, they won’t be precisely consistent in terms of di¢cult
of di¢culty y eitheeither. r. EacEach h questquestion presention presents s its own its own uniqunique ue chachallenllenges, and ges, and thethe sooner you get accustomed to changing gears with every single question, 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. 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 For additional practice, I have produced several other resources that may help you.
help you. YYou’lou’ll …nd l …nd the most Statisthe most Statistics and Sets-retics and Sets-relatelated questions in "Wd questions in "Wordord Probl
Problems: ems: ChallChallenge" and "Wenge" and "Word Probleord Problems: ms: FFundamundamententals.als." " There are alsoThere are also plen
plenty ty in in the the "Ari"Arithmethmetic: Challengtic: Challenge" e" and and "Ari"Arithmethmetic: tic: FFundamundamententals" als" sets,sets, especially those dealing with averages and overlapping sets.
especially those dealing with averages and overlapping sets. Even
Eventualltuallyy, , youyou’ll start seeing questio’ll start seeing questions that ns that look famililook familiar. ar. That’That’s s a a goodgood thing
thing: : there are only so there are only so manmany ways the GMAy ways the GMAT T can test these concepcan test these concepts, and if ts, and if you’ve done a few hundred Rates, Ratios, and Percents questions, you’ve seen you’ve done a few hundred Rates, Ratios, and Percents questions, you’ve seen just
just about about every every permutation permutation they they can can throw throw your your wayway..
Also, The GMAT Math Bible has several chapters (along with focused Also, The GMAT Math Bible has several chapters (along with focused prac-tice) on these topics, including individual chapters on averages, weighted tice) on these topics, including individual chapters on averages, weighted av-erages, statistics such as mean, median, mode, range, and standard deviation, erages, statistics such as mean, median, mode, range, and standard deviation, and overlapping sets. If you …nd you are struggling with the mechanics of these 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 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 in doing dozens and dozens of practice problems, hoping to pick up those skills along the way.
along the way. As far as
As far as strategy is concerned, there strategy is concerned, there are dozens of are dozens of articles at GMAarticles at GMAT HACKST HACKS to help you with your strategic approach to Arithmetic questions. Most to help you with your strategic approach to Arithmetic questions. Most impor-tantly, you should make sure you understand every practice problem you do. It tantly, 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 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 get it right the next time you see it, because the next time you see it could be on the GMAT.
on the GMAT.
With that in mind, carefully analyze the explanations. Redo questions that With that in mind, carefully analyze the explanations. Redo questions that took you too long the …rst time around. Review questions over multiple sessions, 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 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 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. between those people who reach their score goals and those who never do.
Enough talking; there are 100 Statistics and Sets questions waiting inside. Enough talking; there are 100 Statistics and Sets questions waiting inside. Get to work!
1
1 In
Intr
trod
oduc
ucti
tion
on
This document contains nothing but GMAT questions on the topics of Statistics This document contains nothing but GMAT questions on the topics of Statistics and Sets–10
and Sets–100 0 of them, to of them, to be be exactexact. . The GMAThe GMAT T lovloves these types of es these types of questquestions,ions, and has stead
and has steadily incrily increased theased the nume number of them on the testber of them on the test. . It’s almIt’s almost guar-ost guar-anteed that, on test day, you’ll see items that focus on subjects such as mean, anteed that, on test day, you’ll see items that focus on subjects such as mean, median, standard deviation, and overlapping sets.
median, standard deviation, and overlapping sets.
As in all of my GMAT preparation resources, you’ll …nd these questions 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 indexed by di¢culty. That doesn’t mean you should skip straight to the hardest quest
questions, or even that you ions, or even that you shoulshould d start with the easier ones. start with the easier ones. On the On the GMAGMATT itself, questions won’t come labeled with their di¢culty level, and despite the 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 intent of the adaptive algorithm, they won’t be precisely consistent in terms of di¢cult
of di¢culty y eitheeither. r. EacEach h questquestion presention presents s its own its own uniqunique ue chachallenllenges, and ges, and thethe sooner you get accustomed to changing gears with every single question, 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. 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 For additional practice, I have produced several other resources that may help you.
help you. YYou’lou’ll …nd l …nd the most Statisthe most Statistics and Sets-retics and Sets-relatelated questions in "Wd questions in "Wordord Probl
Problems: ems: ChallChallenge" and "Wenge" and "Word Probleord Problems: ms: FFundamundamententals.als." " There are alsoThere are also plen
plenty ty in in the the "Ari"Arithmethmetic: Challengtic: Challenge" e" and and "Ari"Arithmethmetic: tic: FFundamundamententals" als" sets,sets, especially those dealing with averages and overlapping sets.
especially those dealing with averages and overlapping sets. Even
Eventualltuallyy, , youyou’ll start seeing questio’ll start seeing questions that ns that look famililook familiar. ar. That’That’s s a a goodgood thing
thing: : there are only so there are only so manmany ways the GMAy ways the GMAT T can test these concepcan test these concepts, and if ts, and if you’ve done a few hundred Rates, Ratios, and Percents questions, you’ve seen you’ve done a few hundred Rates, Ratios, and Percents questions, you’ve seen just
just about about every every permutation permutation they they can can throw throw your your wayway..
Also, The GMAT Math Bible has several chapters (along with focused Also, The GMAT Math Bible has several chapters (along with focused prac-tice) on these topics, including individual chapters on averages, weighted tice) on these topics, including individual chapters on averages, weighted av-erages, statistics such as mean, median, mode, range, and standard deviation, erages, statistics such as mean, median, mode, range, and standard deviation, and overlapping sets. If you …nd you are struggling with the mechanics of these 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 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 in doing dozens and dozens of practice problems, hoping to pick up those skills along the way.
along the way. As far as
As far as strategy is concerned, there strategy is concerned, there are dozens of are dozens of articles at GMAarticles at GMAT HACKST HACKS to help you with your strategic approach to Arithmetic questions. Most to help you with your strategic approach to Arithmetic questions. Most impor-tantly, you should make sure you understand every practice problem you do. It tantly, 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 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 get it right the next time you see it, because the next time you see it could be on the GMAT.
on the GMAT.
With that in mind, carefully analyze the explanations. Redo questions that With that in mind, carefully analyze the explanations. Redo questions that took you too long the …rst time around. Review questions over multiple sessions, 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 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 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. between those people who reach their score goals and those who never do.
Enough talking; there are 100 Statistics and Sets questions waiting inside. Enough talking; there are 100 Statistics and Sets questions waiting inside. Get to work!
2
2 Di
Di¢c
¢cul
ultty
y Le
Lev
vel
elss
In general, the level 5 questions in this guide are 560- to 620-level questions. 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 The level 6 questions represent a broad range of di¢culty from about 620 to 720, while the level 7 questions are higher still.
720, while the level 7 questions are higher still. Easy (4) Easy (4) PS PS 5, 12, 14, 35, 37, 45, 52, 54 5, 12, 14, 35, 37, 45, 52, 54 DS DS 61, 66, 68, 71, 74, 82, 83, 84, 88, 89, 90 61, 66, 68, 71, 74, 82, 83, 84, 88, 89, 90 Moderate (5) Moderate (5) PS PS 1, 4, 6, 7, 8, 9, 11, 16, 18, 20, 21, 24, 27, 28, 29, 30, 33, 34, 36, 39, 40, 42, 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 43, 46, 48, 49, 51, 53, 55, 57, 60 DS DS 63, 65, 69, 72, 73, 77, 79, 80, 85, 86, 87, 92, 94, 96, 97, 98, 100 63, 65, 69, 72, 73, 77, 79, 80, 85, 86, 87, 92, 94, 96, 97, 98, 100 Di¢cult (6) Di¢cult (6) PS PS 2, 10, 13, 15, 17, 19, 22, 23, 26, 31, 32, 38, 44, 47, 50, 56, 58, 59 2, 10, 13, 15, 17, 19, 22, 23, 26, 31, 32, 38, 44, 47, 50, 56, 58, 59 DS DS 62, 64, 70, 75, 76, 81, 91, 93, 95, 99 62, 64, 70, 75, 76, 81, 91, 93, 95, 99 Very Di¢cult (7) Very Di¢cult (7) PS PS 3, 25, 41 3, 25, 41 DS DS 67, 78 67, 78
3
3 Pr
Prob
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g
Note
Note: : this guide contathis guide contains both ins both an answer key (so an answer key (so you can quicklyou can quickly y checheck yourck your answers) and full explanations.
answers) and full explanations. 1.
1. A laA law o¢w o¢ce ece empmploloys sys seveven sen secrecretaetarieries. It pas. It pays ays annnnual ual salsalariarieses of $17,000 to 2 of the secretaries, $19,000 to 2 of the
of $17,000 to 2 of the secretaries, $19,000 to 2 of the secre
secretaritaries, and es, and $23,0$23,000 00 to to the remainithe remaining ng 3 3 secresecretaritaries. es. TheThe average (arithmetic mean) annual salary of these employees is average (arithmetic mean) annual salary of these employees is closest to which of the following?
closest to which of the following? ((AA) ) $$1199,,000000 ((BB) ) $$1199,,770000 ((CC) ) $$2200,,000000 ((DD) ) $$2200,,110000 ((EE) ) $$2211,,000000 2.
2. A frA freeleelancance wrie writer ter foufound thnd that lat last ast yeyear har his ais aveveragrage (are (arithithmetmeticic mean
mean) reven) revenue from Januaue from January to October $4,800. ry to October $4,800. In NoveIn Novembermber and December, his monthly revenues were 2 and 3 times, and December, his monthly revenues were 2 and 3 times, respec
respectivtivelyely, the avera, the average for the ge for the other 10 montother 10 months. hs. What wasWhat was the writer’s average monthly revenue last year?
the writer’s average monthly revenue last year? ((AA) ) $$55,,440000
((BB) ) $$66,,000000 ((CC) ) $$66,,440000 ((DD) ) $$66,,880000 ((EE) ) $$77,,220000
33. . IIf f xx is to be chosen at random from the set {1, 2, 3, 4} andis to be chosen at random from the set {1, 2, 3, 4} and yy
is to be chosen at random from the set {4, 5, 6, 7}, what is to be chosen at random from the set {4, 5, 6, 7}, what is the probability that
is the probability that xyxy will be odd?will be odd? (A) (A) 1188 (B) (B) 11 4 4 (C) (C) 1122 (D) (D) 3344 (E) (E) 7788
I. I. 880, 0, 8181, , 8282, , 8383, , 8844 II II. . 8282, , 882, 2, 882, 2, 8282, , 8282 II III. I. 6969, , 8282, , 8282, , 882, 2, 9966 4.
4. The The datdata sea sets Its I, II, II, an, and III d III aboabove ve are are ordordereered fd from rom gregreateatestst standard deviation to least standard deviation in which standard deviation to least standard deviation in which of the following?
of the following? ((AA) ) II, , IIII, , IIIIII ((BB) ) II, , IIIIII, , IIII ((CC) ) IIII, , IIIIII, , II ((DD) ) IIIIII, , II, , IIII ((EE) ) IIIIII, , IIII, , II 5.
5. The aThe amoumountnts of ras of rainfinfall iall in incn inches rhes recoecorderded in sid in six di¤x di¤ereerent cnt citiities ines in a certain state last month were 12.5", 11.5", 10.8", 17", 18.2", a certain state last month were 12.5", 11.5", 10.8", 17", 18.2", and 15". What is the median of those amounts?
and 15". What is the median of those amounts? ((AA) ) 1144"" ((BB) ) 1133..7755"" ((CC) ) 1133..22"" ((DD) ) 1133"" ((EE) ) 1122..55"" 6.
6. The The ariarithmthmetietic mec mean aan and snd stantandardard ded deviaviatiotion of n of a cea certartain nin normormalal distribution are 11.5 and 2.0, respectively. What value is exactly distribution are 11.5 and 2.0, respectively. What value is exactly 2 standard deviations less than the mean?
2 standard deviations less than the mean? ((AA) ) 77..55
((BB) ) 88..00 ((CC) ) 88..55 ((DD) ) 99..00 ((EE) ) 99..55
77. . IIf f S S = = {{1133,, 3355, , 0,0, 2121, , 1,1, 1166}, }, what is what is the positive di¤erethe positive di¤erence bnce betwetweeneen the median of the numbers in S and the mean of the numbers the median of the numbers in S and the mean of the numbers in S? in S? (A) (A) 121211 (B) (B) 151511 (C) (C) 303011 (D) (D) 606011 (E) (E) 12012011
{–7, –6, –2,
14, 0, 14, 12, 1, 2, 4, 6, 7}
8. 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? (A) 121 (B) 16 (C) 14 (D) 13 (E) 129. 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
$15, $18, $20, $24, $25, $30, $50, $50, $64, $80
12. 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
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
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?(A) 13m
(B) m
(C) 5m
(D) m
5(E) m
1520. 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
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,p
z, z2, and z3 ? (A) z (B) z
1 (C)p
z (D) z2 (E) z327. 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) ? (A) x2 (B) x2+ 2 (C) x2+ 4 (D) x2+ 2x (E) x2 + 4x31. W represents the sum of the weights of s steaks in pounds. Which of the following represents the average (arithmetic mean) of the sweights in ounces? (1 pound = 16 ounces) (A) 16W s (B) 16W s (C) W s 16 (D) 16W s (E) s 16W
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
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) $0:2536. 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 69 78 78 79 80 82 84 90
37. 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
38. 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
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 ?
(A) z
(B) 3z
(C) z+ 3 (D) z+ 9
(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
35, 37.5, 40, 42.5, 45, 52.5, 52.5, 55, 60, 60
43. 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) Two (B) Three (C) Four (D) Five (E) 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?
(A) 18 (B) 15329 (C) 11625 (D) 23251 (E) 1 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
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
List I: 4, 8, 10, 15 List II: k, 4, 8, 10, 15
51. 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
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 bis 60 and the average (arithmetic mean) of b andc 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 both 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) None
(B) Two (C) Four (D) Five (E) Six
60. 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
4 Data Su¢ciency
For all Data Su¢ciency questions, the answer choices are as follows: (A) Statement (1) ALONE is su¢cient, but statement (2) alone
is not su¢cient.
(B) Statement (2) ALONE is su¢cient, but statement (1) alone is not su¢cient.
(C) BOTH statements TOGETHER are su¢cient, but NEITHER statement ALONE is su¢cient.
(D) EACH statement ALONE is su¢cient.
(E) 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) The n numbers are positive, consecutive integers. (2) The smallest integer in the set is odd.
63. If a, b, c, d, and eare di¤erent positive integers, which of the …ve integers is the median?
(1) a, b, c, d, and e are consecutive integers.
(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.
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?
(1) p < q
(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 ? (1) x+y2+z = 9
(2) x+y3+z = 6
69. If z is a positive integer, is z <16 ?
(1) z is less than the average (arithmetic mean) of the …rst ten positive even integers.
(2) z is the square of an integer.
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 eare di¤erent positive integers, which of the …ve integers is the median?
(1) b+c < e
(2) a+d < e
72. Is q equal to the median of the three positive integers n, p, and q ? (1) p= 4n= 3q
(2) q = 12
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.
74. What is the average (arithmetic mean) of a, b, and c ? (1) a+b= 7
(2) b+c= 10
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?
(1) a < b
79. If the average (arithmetic mean) of seven numbers is 72, how many of the numbers are equal to 72?
(1) Of the seven numbers, six are odd integers, and the average of those odd integers is 72.
(2) At least one of the numbers is equal to 72. 80. How many of the 56 people in a group are blonde males?
(1) 36 of the 56 people are blonde. (2) 32 of the 56 people are male.
81. Of the 32 speakers managed by a certain talent agency, some specialize in marketing and some are willing to travel to City X. How many of the speakers specialize in marketing but are not willing to travel to City X?
(1) All of the speakers who are willing to travel to City X specialize in marketing.
(2) 5 of the speakers are not willing to travel to City and do not specialize in marketing.
82. If v+z = 32, what is the value of vz ? (1) The average of v and z is 16.
(2) The positive di¤erence between v and z is 16.
83. If set S consists of the numbers w, x, y, and z, is the range of the numbers in S greater than 6 ?
(1) w
x= 4 (2) x
z = 484. If x is a positive number less than 10, is z less than the average (arithmetic mean) of x and 10 ?
(1) z= 6x
(2) On the number line, z is closer to 10 than it is to x. 85. What is the average (arithmetic mean) of j and k ?
(1) The average (arithmetic mean) of j+ 3 and k+ 3 is 12. (2) The average (arithmetic mean) of j, k, and 6 is 8. 86. Of the 200 belts at a certain store, 115 are made of leather.
How many of the leather belts are designed for men?
(1) 65 of the belts at the store are neither made of leather or designed for men.
(2) 20 of the belts at the store that are designed for men are not made of leather.
87. Is the range of the integers 7, 4, x, 5, 6, and y greater than 9 ? (1) y >4x
(2) 0< x < y
88. If m, n, p, q , and r are di¤erent positive integers, which of the …ve integers is the median?
(1) q +r < n
(2) m < p
89. If the average (arithmetic mean) of 4 numbers is 100, how many of the numbers are greater than 100?
(1) One of the numbers is equal to 100. (2) One of the numbers is equal to 50.
90. If John’s average (arithmetic mean) score for three games of pinball was 112, what was his median score?
(1) The average (arithmetic mean) of John’s highest and lowest scores was 118.
(2) John’s highest score was 154.
91. Are at least 80 percent of the people in Country X who are 25 years old or younger full-time students?
(1) In Country X, 32 percent of the population is 25 years old or younger.
(2) In Country X, of the population 25 years old or younger, 90 percent of the women and 75 percent of the men are full-time students.
92. A certain credit rating …rm assigned ratings of AAA, AA, or A to each of several countries. What percent of the rated countries have a population that is greater than 20 million people? (1) Of the countries that received a rating of AAA, 25
percent have a population that is greater than 20 million people.
(2) Of the countries that received a rating of AA or A, 70 percent have a population that is 20 million people or fewer.
93. Each of the terms in sets R, S, and T are positive integers. If 9 of the integers in R are also in S, 12 of the integers that are in S are also T, and 4 of the integers that are in R are also in T, how many unique positive integers are contained in the three sets? (1) 1 of the integers is in R, S, and T.
94. Of the people who attended a conference, 75 percent were employed by biotechnology …rms, and some of those employed by biotechnology …rms were executives of those …rms. What percent of the people who attended the conference were executives of biotechnology …rms?
(1) 60 percent of the employees of biotechnology …rms who attended the conference were executives of those …rms.
(2) 300 people attended the conference.
95. Set X consists of seven consecutive integers, and set Y consists of nine consecutive integers. Is the median of the numbers in set X equal to the median of the numbers in set Y ?
(1) The sum of the numbers in set X is equal to the sum of the numbers in set Y.
(2) The median of the numbers in set Y is 0. 96. If set S consists of the numbers 3, -4, -9, 0, and m, is
6< m <2?(1) The median of the numbers in set S is greater than -4. (2) The median of the numbers in set S is negative.
97. Of the six hundred children in a certain county, how many have been immunized against neither disease D or disease E?
(1) Of the children in the county, 580 have been immunized against disease E.
(2) Of the children in the county, 570 have been immunized against both disease D and disease E.
98. If k is a positive odd integer, what is the average (arithmetic mean) of a certain set of k integers?
(1) The median of the set of integers is 44.
(2) The integers in the set are consecutive multiples of 4. 99. Is s equal to the median of the four positive integers r, s, t, and u?
(1) r is the smallest of the four integers r, s, t, and u. (2) s, t, and u are consecutive even integers.
100. A and B are sets of positive integers. Is the least integer in A greater than the least integer in B ?
(1) A is a set of 5 consecutive even integers, each less than 30.
(2) B is a set of 3 consecutive odd integers, each less than 25.
5 Answer Key
For full explanations, see the next section.
1. D 2. B 3. B 4. D 5. B 6. A 7. D 8. B 9. D 10. D 11. D 12. B 13. A 14. E 15. E 16. B 17. D 18. B 19. B 20. B 21. B 22. D 23. D 24. C 25. D 26. D 27. D 28. C 29. A 30. D 31. B 32. B 33. A 34. B 35. C 36. D 37. D 38. B 39. E 40. A 41. B
42. D 43. C 44. A 45. C 46. B 47. D 48. B 49. B 50. E 51. C 52. D 53. E 54. D 55. B 56. B 57. D 58. C 59. C 60. D 61. D 62. A 63. C 64. E 65. D 66. E 67. E 68. D 69. A 70. C 71. E 72. A 73. D 74. E 75. E 76. B 77. E 78. C 79. A 80. E 81. E 82. E 83. C 84. B 85. D 86. E 87. E
88. E 89. E 90. A 91. E 92. E 93. C 94. A 95. C 96. C 97. E 98. C 99. C 100. E
6 Explanations
For a quick-reference answer key, see the previous section.
1. D
Explanation: Explanation: Technically, you can set up this weighted average problem as follows:
2(17;000)+2(19;000)+3(23;000) 7
However, that’s time-consuming and error-prone. Instead, subtract 17,000 from all the numbers (and remember to add it to your answer when you’re done!).
2(0)+2(2;000)+3(6;000)
7 =
22;000
7
3;000Add 17,000 back to the total, and the result is a little more than 20,000: choice (D), 20,100.
2. B
Explanation: To avoid doing more math than necessary, call the average monthly revenue for the …rst 10 months x. (x = 4800, but this way we aren’t doing a lot of unnecessary arithmetic.) The revenue in November, then, was 2x, and the revenue in December was 3x. The total revenue for the entire year was 10x+ 2x+ 3x = 15x.
The average revenue is 15x
12 = 1:25x. That’s the answer we’re looking for,
once we plug x = 4800 back into the expression: 1:25x= 1:25(4800) = 6000, choice (B).
3. B
Explanation: The product of two integers will be odd if and only if both of the integers are odd. 2 of the 4 possible values of xare odd, so the probability that x is odd is 12. 2 of the 4 possible values of y are odd, so the probability that y is odd is also 12. The probability that both x and y are odd, then, is
1
2
12 = 14, choice (B).4. D
Explanation: Standard deviation is, approximately, the average distance from the mean of each term in the set. II clearly has the smallest standard devation: if each term is the same, the standard deviation is zero. That limits our choices to (B) and (D).
While III has three terms that are equal, the two outliers make the standard deviation much larger. You can probably evaluate the approximate standard deviations without doing the math, but the average di¤erence from the mean in III is 13+0+0+0+14
5 = 275 , while the average di¤erence from the mean in I is 2+1+0+1+2
5 = 65. It’s not even close. Choice (D) is correct.
Explanation: To …nd the median, line up the measurements in ascending or descending order:
10.8, 11.5, 12.5, 15, 17, 18.2
Since there is an even number of terms in the set, the median is the mean of the two middle terms, in this case 12.5 and 15:
m= 12:5+15 2 =
27:5
2 = 13:75, choice (B).
6. A
Explanation: The mean is 11.5 and 1 standard deviation is 2. Two standard deviations, then, is 2(2) = 4, so the value that is 2 standard deviations less than the mean is 11:5
4 = 7:5, choice (A).7. D
Explanation: To …nd the median, arrange the terms in ascending order: 0, 16, 13, 12, 35, 1
The median is the mean of the two middle terms, since there is an even number of terms in the set:
median= 1 3+ 1 2 2 = 5 6 2 = 125
Next, …nd the mean:
mean = 0+16+ 1 3+ 1 2+ 3 5+1 6 = 5 30+ 10 30+ 15 30+ 18 30+ 30 30 6 = 78 30 6 = 18078
For ease of comparison, …nd a common denominator:
5 12
15 15 = 75 180The di¤erence is 18078
18075 = 1803 = 601 , choice (D).8. B
Explanation: There are three possible solutions to the equation; each one satis…es one of the following:
y+ 4 = 0
y
7 = 0 4y+ 1 = 0That means the possible values of y are -4, 7, and
14. The latter two are in the set, but the …rst is not. Therefore, of the 12 numbers in the set, two of them would be a solution to the equation. Since probability is desiredpossible, the
answer is 122 = 16, choice (B).
9. D
Explanation: If the standard deviation is 2.5, 2.5 standard deviations is the 2:5(2:5) = 6:25. So, we want to know which of the numbers is NOT more than 6.25 away from the mean of 30.0. That would be any number not less than 30
6:25 = 23:75 or any number greater than 30 + 6:25 = 36:25. The only number that is within that range is (D).10. D
Explanation: To …nd one of the endpoints, set up the overlapping sets formula as if 40 of the students (the minimum number) were neither born nor attended high school in the state:
T =G1 + G2
B+N300 = 210 + 250
x+ 40 300 = 500
xx = 200
That limits our choices to (D) and (E). The number that …t into both categories is constrained by the size of each category: since there are only 210 students who were born in the state, only 210 students can both be born in the state and have attended high school in the state. Thus, (D) is correct.
11. D
Explanation: If the average price of three frames is $18, the total price for the three frames is 3(18) = 54. If the average price of 4 frames is $20, the total cost of the four frames is 4(20) = 80. The fourth frame must cost the di¤erence: 80
54 = 26, choice (D).12. B
Explanation: The numbers are already arranged in ascending order, so to …nd the median, …nd whether the number of terms is even or odd (it’s even) and then …nd the mean of the two middle terms:
25+30 2 =
55
2 = 27:5
The mean requires more work. Add the terms and divide by 10:
15+18+20+24+25+30+50+50+64+80
10 =
376
10 = 37:6
Only one term, $30, is less than 37.6 and more than 27.5. The correct choice is (B).
13. A
Explanation: When only the most and least expensive amounts are changed, the middle term is una¤ected, so the median stays the same. That leaves us (A) and (C).
To determine whether the mean changes, see if the total price of the homes changes. Since the most expensive house increases by the same amount that the least expensive house decreases, the total price stays the same. Thus the mean does, as well. (A) is the correct answer.
14. E
Explanation: The mean of 40, 60, and 80 is:
40+60+80 3 = 60
60 is 5 more than 55, so 55 is the average of 30, 50, and the answer, which we can call x: 30+50+x 3 = 55 30 + 50 +x= 165 x = 85, choice (E). 15. E
Explanation: 8 standard deviations is $0:04(8) = $0:32, while 9 standard deviations is $0:04(9) = $0:36. 8 and 9 standard deviations above the mean, then is:
$0:16 + $0:32 = $0:48 $0:16 + $0:36 = $0:52
The only choice between those two values is (E), $0.49. 16. B
Explanation: The 100-page document would cost $50 for the …rst 20 pages, leaving 80 more pages. That’s 8 10-page segments, each of which cost $15, for a total of 8(15) = 120. The total, then, is 50+120 = 170. The average charge per page is the total charge divided by the number of pages:
average= 170100 = 1:70, choice (B). 17. D
Explanation: Since the median is 50, an equal number of terms must be greater than 50 and less than 50. That means one of the terms is 50, and 12 are less than 50. Since none of the terms are the same and we want the largest possible integer in the set, the numbers less than 50 should be consecutive, so that the smallest number in the set is as large as possible.
If all the numbers are consecutive from 50 down to the least integer, the bottom 13 numbers are as follows:
38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50
If the lowest number is 38 and the range is 50, the greatest number is 38 + 50 = 88, choice (D). There is no way for the set to have a larger smallest number, so there’s no way for the greatest number to be any larger, either.
18. B
Explanation: To …nd the median, arrange the terms in order: 72, 72, 72, 76, 76, 76, 78, 89, 89, 93, 93
Since there is an odd number of terms, the middle number is the median. There are …ve terms below or equal to 76, and …ve terms greater than 76, so that’s our median, and choice (B) is correct.
19. B
Explanation: If every term is increased or decreased by the same amount, the dispersal of the set is unchanged. The standard deviation depends on how far each term is from the mean, and subtracting 5 from every term doesn’t change that. Thus, m is unchanged, and (B) is the correct answer.
20. B
Explanation: If 4 is the greatest integer in a series of 15 consecutive integers, the smallest number is 4
14 =
10. (Subtract 14 because the di¤erence between the least and greatest of x consecutive integers is x
1.) The negative integers in the set, then, run from
10 up to
1. The range of those is given by:
1
(
10) = 9, choice (B).21. B
Explanation: Technically, a weighted average problem should be set up like this:
15;000(3)+17;000(1)+18;000(2) 6
However, it’s easier to subtract 15,000 from each. (Actually, you can do even better, and forget about the thousands.) The result looks like this:
0(3)+2(1)+3(2)
6 = 86 = 113
Add 15 back to the result: 113 + 15 = 1613
Since we’re measuring in thousands, that’s about the same asa $16,300, choice (B).
22. D
Explanation: Since the list is in increasing order, we can …nd the median. There is an even number of terms, so the median is the mean of the middle two:
5+7
2 = 122 = 6
The median, 6, is 23 of the mean: 6 = 23(mean)
mean = 6(32) = 9
Now that we know the mean, we can solve for x:
2+4+5+7+15+x
6 = 9
33 +x = 54
x = 21, choice (D). 23. D
Explanation: If a GPA of 2.8 is 1.5 standard deviations below the mean, we can devise an equation, in which m is the mean and dis standard deviation:
2:8 = m
1:5d3.6 is 2.5 standard deviations above the mean: 3:6 = m+ 2:5d
The easiest way to solve for one of the variables is to subtract the …rst equation from the second:
3:6
2:8 = (m
m) + (2:5d
(
1:5d)) 0:8 = 0 + 4dd= 0:2
We can use the value of dto …nd the mean in the …rst equation: 2:8 = m
1:5(0:2)2:8 = m
0:3m= 3:1, choice (D). 24. C
Explanation: Add the terms together, divide by 4, and set the result equal to 17:
k+(2k5)+(3k+2)+(6k1)
12k4 4 = 17 12k
4 = 68 12k= 72 k= 6, choice (C). 25. DExplanation: Since we don’t know the standard deviation, the only way to add terms and guarantee that the standard deviation will decrease is to add terms that are equal to the mean.
The standard deviation is approximately the average di¤erence of terms from the mean. If a term is equal to the mean, its variance from the mean is 0. So if we add 85 and 85, choice (D), the standard deviation must get smaller, since we know that the deviation is currently positive.
26. D
Explanation: Since the question suggests that the answer is always the same, it is probably most e¢cient to choose a value for z and evaluate each of the expressions. 12 is usually a good bet for values between 0 and 1, so let’s say
z= 12: z = 12 z
1 =
12p
z=q
1 2 = p 12
11:4 = 17 5 = 5 7 z2 = (12)2 = 1 4 z3 = (12)3 = 18In order, we have
12, 18, 14, 12, and 57. The middle term is 14, which is equal to z2, choice (D).27. D
Explanation: In order to maximize the smallest weight, we need to min-imize the other two weights. Since the median is 180, one of the weights must be 180. The weight that is to the right of 180 in an ascending list can be equal to 180, so to minimize it, let’s say it is 180 as well.
Since the mean is 160, the total weight is 3(160) = 480. If both of the heavier weights are 180, the lightest of the 3 must be:
480
180
180 = 120, choice (D). 28. CExplanation: Technically, a weighted average should be set up like this:
26:5(30)+25:5(20) 50
However, the math is much simpler if we subtract 25.5 from each of the terms, remembering to add it back to the result later on:
1(30)+0(20) 50 =
30 50 = 0:6
Our answer, then, is 0:6 + 25:5 = 26:10, choice (C). 29. A
Explanation: A value two standard deviations from the mean is 2(1:75) = 3:5 away from the mean. We don’t know whether our answer is 3.5 less or 3.5 greater, so let’s …nd both:
9:5 + 3:5 = 13 (not a choice) 9:5
3:5 = 6, choice (A). 30. DExplanation: First, multiply out each of the expressions, since we’ll have to add them to …nd the average:
(x+ 2)2 =x2+ 4x+ 4
(x+ 2)(x
2) = x2
4 Now …nd the average:(x2+4x+4)+(x24) 2 = 2x 2 +4x 2 = x2+ 2x, choice (D). 31. B
Explanation: Average is given by sum over number, so in pounds, the average is:
W pounds
s
However, we’re looking for the average in ounces, so we must convert given the ratio in the question:
W pounds s
16 ounces 1 pound = 16W ounces s , choice (B). 32. BExplanation: If 40% of the employees have passed the exam, 60% have not. Those 60% comprise a total of 32+16 = 48 employees, so the total number of employees at the …rm is given as follows:
0:6e= 48
6
10e= 48
e= 48(106 ) = 8(10) = 80, choice (B). 33. A
Explanation: There’s only one variable, so we can set up the average formula and solve for y:
32+(y+2)+(y+5)+30+y 5 = 24 62 + 3y+ 7 = 24(5) 69 + 3y = 120 3y = 51 y = 17, choice (A). 34. B
Explanation: To …nd the least possible number for one of the students, we must maximize the other two numbers of hours. If the upper limit on the number of hours is 10 hours, let’s say each of the other two students spent the full 10 hours. Now, we can use the average formula to …nd the number of hours that the third student must have spent in order for the three to average 7.5 hours:
7:5 = 10+10+x
3
22:5 = 20 +x x = 2:5, choice (B). 35. C
Explanation: To …nd the median, start by arranging the terms in ascend-ing order:
0:25,
0:10,
0:05, 0:05, 0:25, 0:60Since there is an even number of terms, the median is the mean of the middle two terms: median= 0:05+0:05 2 = 0 2 = 0, choice (C). 36. D
Explanation: First, …nd the total number of pieces of luggage: 50(2) + 40(1) + 10(3) + 10(4) = 100 + 40 + 30 + 40 = 210
The number of pieces of luggage per passenger is that total number of pieces divided by the number of passengers:
210 120 = 21 12 = 7 4 = 1:75, choice (D). 37. D
Explanation: Since the mean is 80, numbers within 1 standard deviation (6, in this case) of the mean must be between the following two numbers:
80 + 6 = 86 80
6 = 746 of the 8 numbers are between 74 and 86:
6
8 = 34 = 75%, choice (D).
38. B
Explanation: The total number of pilots is the sum of the number who ‡y the individual routes, minus the number of overlaps, as those indicate that those pilots were counted twice in the table:
35 + 25 + 20
(7 + 4 + 3) = 80
14 = 66, choice (B). 39. EExplanation: In April, if the range is $22,000 and the lowest price is $184,000, the highest price is $184;000 + $22;000 = $206;000. In April, the range is $25,000 and the lowest is $192,000, so the highest price is $192;000 + $25;000 = $217;000.
Overall, the lowest price is $184,000 and the highest is $217,000, for a range of
$217;000
$184;000 = $33;000, choice (E). 40. AExplanation: The greatest possible average price relies on the greatest possible prices for the …nal two machines, which are each $9,000. This average is, then:
5;000+9;000+9;000+9;000
4 =
32;000
4 = 8;000
The least possible price relies on the least possible prices for the …nal two machines, or $5,000 each:
5;000+9;000+5;000+5;000
4 =
24;000
4 = 6;000
The di¤erence between the least and greatest average prices is 8;000
6;000 = 2;000, choice (A).41. B
Explanation: If each of the terms is multiplied by the same number, the di¤erences in the terms is multiplied by the same amount. For instance, if the terms were initially 1, 2, and 3, the new terms are 3, 6, and 9 – the numbers tripled, and the di¤erences between them tripled as well. While this is only an approximation, it’s enough to know that the standard deviation changes by the same factor, so the new standard deviation is 3z, choice (B).
42. D
Explanation: In order to meet the goal, the retailer must sell 250;000
170;000 = 80;000 in the …nal 40 days. As a per day average, that is80;000 40 =
8;000
4 = 2;000, choice (D).
43. C
Explanation: Before we can determine what is 1 standard deviation away from the mean, we must calculate the mean:
35+37:5+40+42:5+45+52:5+52:5+55+60+60
10 = 48010 = 48
One standard deviation is plus or minus 9.2: 48 + 9:2 = 57:2
48
9:2 = 38:8The terms outside of that range are 35, 37.5, 60, and 60, a total of four, choice (C).
44. A
Explanation: 13 of those who own their own home (153 people) have children, so the overlap between the two sets is 13(153) = 51. Since 80 have children and 51 of those own their own home, 29 do not. If one person is randomly selected from the group of 232, there are 232 possible outcomes, and 29 desired outcomes:
probability = 23229 = 18, choice (A). 45. C
Explanation: The range is the di¤erence between the lowest and highest terms. In this case, those terms are 6 and 16, so the range is 16
6 = 10, choice (C).46. B
Explanation: A decrease is a negative increase, so the three increases are 15,000, 25,000, and -17,500. The average, then, in thousands, is
15+2517:5 3 = 22
:5
3 = 7:5
Converted back to thousands of dollars, the average is $7,500, choice (B). 47. D
Explanation: To maximize the number of classes attended by a single member, minimize the number by everyone else that you can. 16 attended 0, 1, or 2 classes, meaning that 4 attended 3 or more classes. Since one of those is the member we’re trying to maximize, say that three of those remaining members attended 3 classes each. That gives us a total number of classes of
3(0) + 8(1) + 5(2) + 3(3) + 1(x) = 8 + 10 + 9 +x = 27 + x
If the average number per member is 1.75 and there are 20 members, the total number of classes attended by this sample is 20(1:75) = 35. We can solve for x:
35 = 27 +x x = 8, choice (D). 48. B
Explanation: First, …nd the median by arranging the terms in ascending order:
28;39;40;51;52
The middle term is 40, and since the number of terms is odd, that’s the median.
Next, …nd the average:
28+39+40+51+52
5 = 2105 = 42
The di¤erence is 42
40 = 2, choice (B). 49. BExplanation: If the three people called an average of three times, the total number of calls was 3(3) = 9. If Hank called 5 times and Jelena called 2 times, that leaves 2 times unaccounted for, so that must be the number of times that Kristof called. The correct answer is (B).
50. E
Explanation: The weighted average formula allows us to set up this prob-lem like this (ignoring the thousands for simplicity):
6(14)+5(16)+4(x) 15 = 16
To make the math easier, subtract 14 from each selling price, and add it back later: 6(0)+5(2)+4(x) 15 = 2 10 + 4x = 30 4x = 20 x = 5
Add 14 back to the total: 5 + 14 = 19
Convert to thousands of dollars, and you have your answer, $19,000, choice (E).
51. C
Explanation: The median of list I is the mean of the middle two numbers:
8+10
2 = 182 = 9
Since list II has an odd number of terms, the median must be one of the terms. Since the median must be 9, and 9 isn’t one of the integers listed in II,
k must be 9, choice (C). 52. D
Explanation: The average is the sum of terms (conference calls, in this case) divided by the number:
5+7+8+5+12
5 =
37
5 = 7:4, choice (D).
53. E
Explanation: Set up the weighted average equation as follows:
16(3;000)+4(x)
20 = 4;000
Make the math simpler by subtracting 3,000 from each dollar amount and add it back to the value of x later:
16(0)+4x
20 = 1;000
4x = 20;000
x = 5;000
Add 3,000 back: 5;000 + 3;000 = 8;000, choice (E). 54. D
Explanation: The average of 10, 15, and 20 is
10+15+20
3 = 453 = 15
If 15 is the average of 12, 18, and the answer, we can solve for the variable: 15 = 12+18+x
3
45 = 30 +x
x = 15, choice (D). 55. B
Explanation: This is a perfect example for the overlapping sets equation:
T =G1 +G2
B+N18 = 10 + 12
B+ 3 18 = 25
BB = 7, choice (B). 56. B
Explanation: If the average of a andb is 60, the sum of the terms is 120. Similarly, the sum of b and c is 70(2) = 140. That gives us two equations:
b+c= 140
a+b= 120
To …nd c
a, subtract the second equation from the …rst:c+b
(a+b) = 140
12057. D
Explanation: This question is tailor-made for the overlapping sets for-mula. Remember, when dealing with percents, that the total is always 100.
T =G1 +G2
B+N100 = 65 + 65
B+ 25 100 = 155
BB = 55, choice (D). 58. C
Explanation: The total number of sales Joan made over the past x days is 15x, so after making 25 sales today, her total is 15x+ 25. If her average is 16 sales over thex+ 1 days, her total can also be expressed as 16(x+ 1). To solve for x, set the two expressions for the total equal to each other:
15x + 25 = 16(x+ 1) 15x+ 25 = 16x+ 16
x = 9, choice (C). 59. C
Explanation: Another way of phrasing this question is like so: "How many positive even integers less than 75 have square roots that are also positive even integers less than 75?"
Or, simpler still: "How many positive even integers less than 75 are perfect squares?"
There are 4 of those: 4, 16, 36, and 64, which means that choice (C) is correct.
60. D
Explanation: The remaining pieces, after removing a 16-foot piece, have a total length of 40
16 = 24 feet. There are 4 pieces left, so the average is24
4 = 6, choice (D).
61. D
Explanation: Statement (1) is su¢cient: set up the equation and sim-plify:
p+q+8 3 =p
p+q + 8 = 3 p q = 2 p
8If p is an integer, 2 p is also an integer, and 2 p
8 is an integer as well. Therefore, q must be an integer.Statement (2) is su¢cient: again, set up the equation and simplify:
p+q+3:5 3 = 3:5
p+q + 3:5 = 10:5
q = 7
pIf p is an integer, then q is equal to an integer (7) minus an integer ( p), so it must be an integer as well. Choice (D) is correct.
