GMAT Club Question Collection 2

5/28/2018 GMAT Club Question Collection 2

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Question Collection 2.0Updated & Corrected: 01/11/2005

TABLE OF CONTENTS:

1. ARITHMETIC AND ALGEBRA ........................................................................................... 2

2. COMBINATIONS AND PROBABILITY............................................................................... 5

3. COORDINATE GEOMETRY .............................................................................................. 7

4. WORD PROBLEMS.......................................................................................................... 10

5. NUMBER PROPERTIES .................................................................................................. 13

6. MISCELLANEOUS ........................................................................................................... 16

1.ARITHMETIC ANDALGEBRAANSWERS.............................................................................. 21

2.COMBINATIONS AND PROBABILITYANSWERS ................................................................... 23

3.COORDINATE GEOMETRYANSWERS ................................................................................ 25

4.WORD PROBLEMSANSWERS........................................................................................... 27

5.NUMBER PROPERTIESANSWERS..................................................................................... 29

6.MISCELLANEOUSANSWERS............................................................................................. 31

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1. Arithmetic and Algebra

1. The price of a certain commodity increased at a rate of X% per year between 2000 and 2004. If theprice was M dollars in 2001 and N dollars in 2003 what was the price in 2002 in terms of M and N?

(A) NM

(B)N M

N

(C)N M

(D)N N

M

(E)N 23

2. Which of the following numbers is the biggest?

(A)

1876455

1876452

(B)

1883459

1883456

(C)

1883494

1883491

(D)

1883449

1883446

(E)

1883456

1883453

3. a, b, c, d are positive numbers. Is

db

ca

b

a

+

+< ?

(1)d

c

b

a<

(2) a + c < b + d

(A) Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient(B) Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient(D) EACH statement ALONE is sufficient(E) Statements (1) and (2) TOGETHER are NOT sufficient

4. Is |x - 1| < 1 ?(1) (x - 1)21(2) x2- 1 > 0

(A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient(B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient(D) EACH statement ALONE is sufficient(E) Statements (1) and (2) TOGETHER are NOT sufficient


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5 Which of the following can be a remainder ofY

Xif both X and Y are positive integers and

Y

X=

2.625 ?

(A) 2(B) 4

(C) 5(D) 8(E) 9

6. XX

X 1(B) X > -1(C) |X| < 1

(D) |X| = 1(E) |X|2> 1

7. What is 987 987 ?

(A) 974,169(B) 974,219(C) 974,549(D) 975,019(E) 975,369

8. What is 9)x6-(x2 + + 3)-(xx)-(2 + if each term in this expression is well defined?

(A) x)-(2

(B) 2x - 6 + x)-(2

(C) x)-(2 + (x - 3)

(D) 2x - 6 + x)-(2

(E) x + x)-(2

9. If operation $ is defined as$X = X + 2 if X is even$X = X - 1 if X is odd,what is $(...$($($(15)))...) 99 times?

(A) 120(B) 180(C) 210(D) 225(E) 250


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10. If X and Y are positive, is 2>+X

Y

Y

X

(1) X does not equal Y(2) X and Y are integers which do not have common divisors except 1

(A) Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient

(B) Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient(D) EACH statement ALONE is sufficient(E) Statements (1) and (2) TOGETHER are NOT sufficient


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2. Combinations and Probability

11. 4 women and 6 men work in the accounting department. In how many ways can a committee of 3be formed if it has to include at least one woman?

(A) 36(B) 60(C) 72(D) 80(E) 100

12. There are 3 red chips and 2 blue chips. When arranged in a row, they form a certain color pattern,for example RBRRB. How many color patterns are possible?

(A) 10

(B) 12(C) 24(D) 60(E) 100

13. Carl has 3 movies to watch during the weekend: 1 action, 1 comedy, and 1 drama. He wants towatch all the movies but decides to watch one of the movies twice. Assuming Carl does not havetime for more than four movies, in how many ways can Carl arrange his watching schedule?

(A) 6

(B) 20(C) 24(D) 36(E) 54

14. 10 sergeants and 7 lieutenants met at the training camp. Each sergeant shook hands with othersergeants and lieutenants once and each lieutenant shook hands with all sergeants but did not shakehands with other lieutenants. How many handshakes took place?

(A) 45(B) 90(C) 115(D) 130(E) 144

15. Among 5 children there are 2 siblings. In how many ways can the children be seated in a row sothat the siblings do not sit together?

(A) 38(B) 46

(C) 72(D) 86(E) 102


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16. Two sets are defined as follows:A = {2, 3, 4, 4, 4}B = {0, 1, 2}.If a number is taken from set A at random and another number is taken from set B atrandom what is the probability that the sum of these numbers is a prime integer?

(A)15

1 (B)

15

2 (C)

3

1 (D)

15

7 (E)

5

3

17. Mary and Joe are to throw three dice each. The score is the sum of points on all three dice. IfMary scores 10 in her attempt, what is the probability that Joe will outscore Mary in his?

(A)

64

24 (B)

64

32 (C)

64

36 (D)

64

40 (E)

64

42

18. Kate and David each have $10. A fair coin is flipped five times. Every time the coin lands onheads, Kate gives David $1. Every time the coin lands on tails, David gives Kate $1. After the coin isflipped 5 times, what is the probability that Kate will have more than $10 but less than $15?

(A)16

5 (B)

32

15 (C)

2

1 (D)

32

21 (E)

16

11

19. 6 married couples are present at the party. If 4 people are selected out of these 12, what is theprobability that none of these people will be married to each other?

(A)33

1 (B)

33

2 (C)

3

1 (D)

33

16 (E)

12

11

20. The bowl contains green and blue chips. What is the probability of drawing a blue chip in twosuccessive trials if the chip drawn in the first trial is not returned to the bowl before the second trial?(1) The ratio of blue chips to green chips is 3:4(2) There are 5 more green chips than blue chips



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3. Coordinate Geometry

21. The vertices of a triangle have coordinates (x, 1), (5, 1), and (5, y) where x < 5 and y > 1. What isthe area of the triangle?

(1) x = y(2) angle at the vertex (x, 1) = angle at the vertex (5, y)


22. At what angle do the lines y = Kx + L and x = y + KL intersect?(1) K = 2

(2) K = L


23. Equation |x| + |y| = 5 encloses a certain region on the coordinate plane. What is the area of thisregion?

(A) 5(B) 10(C) 25(D) 50(E) 100

24. Which of the following lines is parallel to line x = 2.66 - 2y ?

(A) y = 2x + 25(B) y = 2.66x - 2(C) x + 3y = 2

(D) y = 33.12

1+ x

(E) 132

1=+ yx


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25. On the picture below, is the area of triangle ABC > 1 ?(1) angle ABC < 90 degrees

(2) perimeter of the triangle ABC >a

4


26. Does the curve (x - a)2+ (y - b)2= 16 intersect the Y-axis?(1) a2+ b2> 16(2) a = |b| + 5


27. A, B, and C are points on the plane. Is AB > 15 ?(1) BC + AC > 14(2) area of triangle ABC < 1


28. A circle is inscribed in a square. If the squares diagonal is 4 cm long, what is the area of thesquare that is not occupied by the circle (approximately)?

(A) 1.7 cm2(B) 2.7 cm2(C) 12 cm2(D) 24 cm2

(E) 25 cm

2


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29. Assuming the Earth's orbit around the Sun is a circle, by how much will the length of the Earth's

orbit increase if the radius of this orbit grows by2

meters?

(A) 1 meter

(B) 2 meters(C) meters(D) 2meters(E) 2meters

30. What is the angle between the minute and the hour hands of the clock which shows 12:24 ?

(A) 115(B) 120(C) 124(D) 130(E) 132


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4. Word Problems

31. Machine A can produce 50 components a day while machine B only 40. The monthly

maintenance cost for machine A is $1500 while that for machine B is $550. If each componentgenerates an income of $10 what is the least number of days per month that the plant has to work tojustify the usage of machine A instead of machine B?

(A) 6(B) 7(C) 9(D) 10(E) 11

32. If the farmer sells 75 of his chickens, his stock of feed will last for 20 more days than planned,but if he buys 100 more chickens, he will run out of feed 15 days earlier than planned. If no chickensare sold or bought, the farmer will be exactly on schedule. How many chickens does the farmerhave?

(A) 60(B) 120(C) 240(D) 275(E) 300

33. Two payment schemes are available for customers in the N&K store. The first scheme implies adownpayment of 20% of the purchase price and 10 monthly payments of 10% each. The secondimplies a downpayment of 10% and 20 monthly payments of 8% each. If a customer buys a TV for$216, by what percent will he find the first scheme cheaper than the second (approximately)?

(A) 14%(B) 27%(C) 30%(D) 34%(E) 35%

34. A man cycling along the road noticed that every 12 minutes a bus overtakes him while every 4minutes he meets an oncoming bus. If all buses and the cyclist move at constant speed, what is thetime interval between consecutive buses?

(A) 5 minutes(B) 6 minutes(C) 8 minutes(D) 9 minutes

(E) 10 minutes


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35. A plane takes off from the hill at 750 meters above the sea level and lands some time later in atown located at 50 meters below the sea level. During the first part of its flight the plane gainedheight at a rate of 50 meters per minute but then it started to descend at a rate of 20 meters perminute. The duration of the first part of the flight was what percent of the total flight time?

(1) The duration of the descent is known(2) The total flight time is known


36. Richard is 3 years younger than his sister. How old will Richard be in 5 years?(1) Two years ago Richard was twice as young as his sister(2) If Richard's sister were born 2 years earlier she would now be twice as old asRichard


37. It takes Jack 2 more hours than Tom to type 20 pages. Working together, Jack and Tom can type

25 pages in 3 hours. How long will it take Jack to type 40 pages?

(A) 5 hours(B) 6 hours(C) 8 hours(D) 10 hours(E) 12 hours

38. Each shelf of the bookcase contained 12 books. The librarian took out 21 books and rearrangedthe remaining books so that all shelves but one contained 8 books and the last shelf contained 11

books. How many shelves does the bookcase have?

(A) 5(B) 6(C) 7(D) 8(E) 9


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39. The power of the engine grows by 10% when the number of its cylinders is increased by one.Which of the following is closest to the ratio of the power of a 9-cyllinder engine to that of a 12-cyllinder engine?

(A) 0.69(B) 0.71(C) 0.72(D) 0.75(E) 0.78

40. Two sacks together contained 43.25 kg of sugar. After 4.75 kg of sugar was taken from the firstsack and poured into the second, the weight of the sugar in the first sack became 73% the weight ofthe sugar in the second. What was the original difference in weights of the sacks?

(A) 1.25 kg

(B) 2.00 kg(C) 2.75 kg(D) 3.25 kg(E) 3.50 kg


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5. Number Properties

41. If X is an odd integer which of the following numbers must be even?

(A)3

3X

(B)2

12 X

(C) Xp + 1 where p is a prime number

(D)2

2)-(X7)(X +

(E) XX - 1

42. If X and Y are positive integers, is3

Yx +10 an integer?

(1) X > 5(2) Y = 2


43. @ represents the tens digit in integer N = 1947@6. What is @?(1) N is divisible by 4(2) N is divisible by 3


44. If XY is divisible by 4 which of the following must be true?

(A) If X is even then Y is odd

(B) If X = 2 then Y is not an integer(C) If X is 0 then X + Y is not 0(D) XYis even

(E)Y

Xis not an integer


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45. If x is a positive integer, is the number of its divisors smaller than 2 1x ?

(1) x is not a square of an integer(2) x is prime

(A) Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient

(B) Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient(D) EACH statement ALONE is sufficient(E) Statements (1) and (2) TOGETHER are NOT sufficient

46. Is 10n+ 8 divisible by 18 ?(1) n is prime(2) n is even

(A) Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient(B) Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient(D) EACH statement ALONE is sufficient(E) Statements (1) and (2) TOGETHER are NOT sufficient

47. If X, Y, and Z are positive integers, is XYZ even?(1) X + Y is odd(2) YZ is odd

(A) Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient(B) Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

(D) EACH statement ALONE is sufficient(E) Statements (1) and (2) TOGETHER are NOT sufficient

48. A and B are two-digit positive integers. What is A?(1) A + B = 168(2) Both A and B are divisible by 24

(A) Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient(B) Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient(D) EACH statement ALONE is sufficient

(E) Statements (1) and (2) TOGETHER are NOT sufficient

49. If a and b are positive integers (a > b), is a2- b2divisible by 4 ?(1) a = b + 2(2) a and b are odd



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50. What is 1! + 2! + ... + 10! ?

(A) 4037910(B) 4037913(C) 4037915

(D) 4037916(E) 4037918


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6. Miscellaneous

51. Comet A is seen near the Earth every 12 years while comet B every 20 years. If both cometswere observed in 1979, for how many years do we have to wait to see the two comets together again?

(Assume now is 2004)

(A) 16(B) 20(C) 25(D) 32(E) 35

52. Out of 20 surveyed students 8 study math and 7 study both math and physics. If 10 students donot study either of these subjects, how many students study physics but not math?

(A) 1(B) 2(C) 4(D) 5(E) 6

53. Set T consists of odd integers divisible by 5. Is standard deviation of T positive?(1) All members of T are positive(2) T consists of only one element


54. The 19th of September 1987 was Saturday. What day was the 21st of September 1990 if 1988was a leap-year?

(A) Monday(B) Tuesday(C) Wednesday(D) Thursday(E) Friday


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55. At 12:00 in the afternoon Paul stopped at the gas station and filled his car's 50-liter tank tocapacity. After Paul drove 75 miles away from the station, the tank developed a leakage and the car

started to lose5

1liters of fuel per minute. If Paul is traveling at a constant speed of 50 miles per hour

and his car consumes 10 liters per every 100 miles, at what time of the day will Paul stop as a resultof the fuel depletion?

(A) 4:00 pm(B) 4:15 pm(C) 4:30 pm(D) 5:00 pm(E) 5:20 pm

56. The safety distance between cars moving on a motorway depends on their speed by the following

formula: 9100

2

+=V

D

where D is the distance between the cars in meters and V is the speed of the cars in meters persecond (ms). If two cars want to maintain a time interval of 1 second between each other, which ofthe following is a speed they can travel at without breaching the safety regulation?

(A) 10 ms(B) 14 ms(C) 18 ms(D) 24 ms

(E) 30 ms

57. Mike, Tom, and Walt work as sales agents for an insurance company. The commissions Mike,

Tom, and Walt got last month are bound by the formula: MT =6

W. If Mikes commission this

month increased by 60% and Toms commission dropped by 50%, how should Walt do to maintainthe above relationship intact? (Assume commission is directly proportional to salesmansperformance)

(A) sell 12.5% less(B) sell 20% less(C) sell 22.5% less(D) sell 12.5% more(E) sell 15% more

58. The train consists of 6 carriages, 20 meters long each. The gap between the carriages is 1 meter.If the train is moving at a constant speed of 60 kmh, how much time will it take the train to runthrough a 1 kilometer tunnel?

(A)8

1

1 (B) 2

1

1 (C) 4

3

1

(D) 2

(E) 4

1

2


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59. Among 200 people 56% like strawberry jam, 44% like apple jam, and 40% like raspberry jam. If30% of people like both strawberry and apple jam, what is the largest possible number of people wholike raspberry jam but do not like either strawberry or apple jam?

(A) 20(B) 60(C) 80(D) 86(E) 92

60. The average of four distinct positive integers is 60. How many integers of these four are smallerthan 50?(1) One of the integers is 200(2) The median of the four integers is 50


61. Is the mean of a non-empty set S bigger than its median?(1) All members of S are consecutive multiples of 3(2) The sum of all members of S equals 75


62. It took the bus three hours to get from town A to town B. What was the average speed of the busfor the entire trip?

(1) After one hour the bus finished3

1of the distance going at 60 kmh

(2) During the second hour the average speed of the bus was 120 kmh, twice its speedduring the third hour



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63. A wholesaler sells computers on following conditions: first 100 computers of the lot cost $500each and any computer above that amount costs $400. If the wholesaler received $57,600 as a resultof the deal, how many computers did he sell?

(A) 109(B) 117(C) 119(D) 121(E) 123

64. Jim bought several books and paid $216. If all books cost the same, which of the followingcannot be the book price?

(A) $3

(B) $12(C) $24(D) $40(E) $72

65. How many zeros does 100! end with?

(A) 20(B) 24

(C) 25(D) 30(E) 32

66. Function F(x) satisfies F(x) = F(x2) for all x. Which of the following must be true?

(A) F(4) = F(2)F(2)(B) F(16) - F(-2) = 0(C) F(-2) + F(4) = 0(D) F(3) = 3F(3)

(E) F(0) = 0

67. What is the last digit of (9)1+ (99)2+ (999)3+ ... + (10n- 1)n?(1) n is even(2) n is prime



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68. Because of the weight limitations, the lorry can carry a maximum of 20 crates of apples. The

weight of a crate of oranges is4

3the weight of a crate of apples. If 10 crates of apples were loaded

into the lorry, what is the maximum number of crates of oranges that the lorry can stillaccommodate?

(A) 8(B) 9(C) 12(D) 13(E) 14

69. A certain Internet provider charges its customers 3 cents per every minute spent online between 6pm and 9 pm, 2 cents per minute between 9 pm and midnight, and 1 cent per minute at other times. IfJohn was online from 4 pm till 1 am, how much will he pay the provider?

(A) $3.60(B) $5.40(C) $7.20(D) $9.00(E) $10.80

70. Deborah and Mike visit the university library at regular intervals every 3 and 4 days respectively.If both of them were in the library on Monday, what day of the week will it be when they meet in the

library again?

(A) Wednesday(B) Thursday(C) Friday(D) Saturday(E) Sunday


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Answers

1. Arithmetic and Algebra Answers

1. Answer A

Price in 2003 was 2)100

1( X

M + = N.

From this equationM

NX=+

1001 .

Price in 2002 was =+ )100

1( X

M

M

N

M

NM = .

Let's check the answer: price in 2003 =(price in

2002)(1+ NM

NMN

X==)

100as

in the stem.

2. Answer C

It is important to notice that in eachfraction the denominator is by 3 bigger

than the numerator. The greatest fraction isthe one with the biggest numerator:

1883494

1883491

3. Answer A

From S1 it follows that ad < bc or ab +ad < ab + bc or a(b + d) < b(a + c)

ordb

ca

b

a

+

+< .

S2 is not sufficient. If a = c = 1 and b = d =2, the answer is NO. If a = c = d = 1 and b

= 2, the answer is YES )3

2

2

1 0, the inequality turns into X > 1. If X < 0,the inequality turns into X > -1. In each of thecases X > -1, therefore X > -1 is true for allpossible X. Counter-examples that show that otherchoices are not necessarily true:

(A),(D),(E) X =2

1 ;

(C) X = 5.

7. Answer A

987987 = (987 + 13)(987 - 13) + 132=1000974 + 169 = 974,169. We used (a + b)(a -b) + b2= a2.

8. Answer A

This is a logic+algebra question. First of all, 2 - x

has to be 0 for the expression x2 to be

defined (it is important to remember that GMAT

deals only with real numbers). Therefore, x 2.=+ 962x xx = 3)3( 2 (because x 2

and thus x - 3 < 0).

Summing up: =+++ 32962 xxxx

xxxx =++ 2)3(2)3( .

9. Answer C

$(15) = 14. We have to compute$(...$($($(14)))...) 98 times. Each $ just adds 2 tothe previous result. Therefore, $(...$($($(14)))...)98 times = 14 + 298 = 210.


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10. Answer A

When X and Y are different,X

Y

Y

X+ is

always bigger than 2. This is because (X -

Y)2> 0, from where X2+ Y2> 2XY. After

dividing by XY we get 2>+X

Y

Y

X

S2 alone is not sufficient. The answer is NO if X= Y = 1.


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2. Combinations and Probability Answers

11. Answer E 15. Answer C

The best way to approach this problem isto consider an unconstrained version of thequestion first: in how many ways can acommittee of 3 be formed? The answer is10C3. From this figure we have to subtractthe number of committees that consistentirely of men i.e. 6C3. The final answeris 10C3 6C3 = 120 20 = 100.

Without limitations 5 children can be seated in 5!= 120 ways. What is the number of ways to seatthe 5 children so that the siblings DO sit together?The siblings can be regarded as one unit so thereare 4! Combinations. But within this unit thesiblings can sit in two different ways. So, thenumber of ways to seat the 5 children so that thesiblings DO sit together is 4!2 = 48. Thus, thenumber of combinations in which the siblings DONOT sit together is 120 48 = 72.12. Answer A

Imagine that there are 5 cells for the chipsto be put in. Each possible color pattern

will correspond to a certain choice of cellsin which blue chips are to be placed. Note,that once these two cells are selected, thered chips will automatically go into theremaining three cells. So, the number ofpossible color patterns equals the numberof ways we can select 2 cells out of 5 i.e.5C2 = 10.

16. Answer D

There is no way how we can obtain a prime biggerthan 5. In all, there are three possibilities for thesum to be a prime number (P denotes probability):P(the sum is 2) = P(2 from A, 0 from B)

=15

1

3

1

5

1=

P(the sum is 3) = P(2 from A, 1 from B) + P(3

from A, 0 from B) =15

2

3

1

5

1

3

1

5

1=+

13. Answer DP(the sum is 5) = P(3 from A, 2 from B) + P(4

from A, 1 from B) = 15

4

3

1

5

3

3

1

5

1=+ .

First of all, Carl has to choose the movie

he will watch twice. There are 3possibilities for such a selection. Withineach of the selections the movies can be

arranged in 122

!4= different ways the

general formula for permutations of 4elements of which 2 are identical. Thetotal number of ways to arrange theschedule = 312 = 36.

Summing up: P(the sum is prime) = P(the sum is

2) + P(the sum is 3) + P(the sum is 5) =15

7.

17. Answer B

This is a-look-for-shortcuts problem. To outscoreMary Joe has to score in the range of 11-18. Theprobability to score 3 is the same as theprobability to score 18 (1-1-1 combination against

6-6-6; if 1-1-1 is on the tops of the dice, the 6-6-6is on the bottoms). By the same logic, theprobability to score x is the same as theprobability to score 21 x. Therefore, theprobability to score in the range 11-18 equals theprobability to score in the range of 3-10. As range3-18 covers all possible outcomes, the probability

to score in the range 11-18 is2

1or

64

32.

14. Answer C

Handshakes between sergeants = 10C2 =

45!2!8

!10=

.

Each lieutenant made 10 handshakes (onewith every sergeant). In all, lieutenantsmade 710 = 70 handshakes.The total number of handshakes = 45 + 70= 115.


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18. Answer B

The total number of outcomes after 5tosses = 25= 32. In half of these outcomesKate wins and increases her initial $10capital. But in one particular outcome

(namely, when the coin lands on tail fivetimes in a row) Kate will increase hercapital to $15. This variant should beregarded as unfavorable. Thus, the

required probability =32

15

32

1=

16.

19. Answer D

Lets select one person at a time. There isno limitation on the first selection. Theprobability that the second person will not

be married to the first =11

10(among 11

people left there are 10 who are not thespouse of the first person selected). Theprobability that the third person will not be

married to any of the first two =10

8. The

probability that the fourth person will not be

married to any of the first three =9

6.

The overall probability to select four peopleamong which there is no married couple =

33

16

9

6

10

8

11

10= .

20. Answer C

S1 is not sufficient. Although we know theprobability that the first chip will be blue, wecannot compute the probability that the secondchip will be blue. We need to evaluate the ratio ofblue chips to green chips after the first trial and S1does not supply information to do so. S2 is not

sufficient either. We dont even know if thisdifference of 5 is significant or not.From S1 + S2 the exact number of green and bluechips in the bowl can be determined:

B + 5 = G;4

3=

G

B.

From this system G = 20 and B = 15. Thisinformation completely defines the contents of thebowl and thus allows to answer the question.


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3. Coordinate Geometry Answers25. Answer A

21. Answer CThe area of the triangle ABC is a

a

aa

=+

2

)11 2

( .It is obvious that this is a right triangle and

its area =2

)1()5( yx. S1 is sufficient. If angle ABC equals 90 degrees,

then the triangle BOC is isosceles and thus a2=

from where a = 1. If angle ABC is smaller than 90degrees, then a2> 1 and thus a = area of thetriangle ABC > 1.

a

1S1 is not sufficient. Can be x = y = 2 or x= y = 3 with different values for the area.S2 is not sufficient. It says that the triangleis isosceles but there is no limitation on itssize. S2 is not sufficient. As long as a > 0, the

perimeter of the triangle ABC is always bigger

thana

4(the base of the triangle is

a

2+ the length

of the other two sides). So, S2 doesn't provide anyvaluable information.

S1 + S2 gives that 5 - x = x - 1 from where

x = 3. Thus, the area = 22

2= 2.

22. Answer A

What matters in this question are thecoefficients in front of x and y. Theydefine the slope of the line and areimportant to find the angle of theintersection. L in the first equation andKL in the second can only shift the linesleft-right or up-down but they cannotchange the slopes of the lines.

26. Answer B

(x - a)2+ (y - b)2= 16 is the equation of a circlewith center (a, b) and radius 4.S1 says that the center of the circle is further than4 units away from the origin but it doesn't specifywhether the circle is far enough from the Y-axisnot to intersect it.

The slope of the second line is 1 (theequation can be rewritten as y = x - KL).The slope of the first line can bedetermined from S1. S2 gives irrelevantinformation.

From S2 it follows that a 5 and thus the centerof the circle is at least 5 units away from the Y-axis. As the radius of the circle is only 4 units, wecan conclude that the circle does not intersect theY-axis.

23. Answer D27. Answer E

The figure its going about in the stem is asquare. You just have to build one side ofit in case of positive x and y. Then you cannotice that this segment is mirrored into

the other three quadrants to form a squarewith the length of the side 25 . Thus,the area of the enclosed region is

50225)25( 2 == = 252 = 50.

S1 allows AB to be of any length. S2 also allowsAB to be of any length: if AB is long, angle ACBhas to be close to 180 degrees to make the area ofthe triangle smaller than 1.

S1 + S2 is not sufficient either. AB can be long (inthis case angle ACB has to be close to 180) orshort (in this case angle ACB has to be close to 0).

28. Answer A

If the diagonal of the square is 4 cm then the sides

are 228 = . So, the radius of the circle is

2 . The area of the circle is 2and that of thesquare is 8. The area of the region not occupied bythe circle = 8 - 2= approximately 1.7.

24. Answer D

x = 2.66 - 2y can be rewritten as y = 1.33

-2

1x. For lines to be parallel the

coefficients in front of x have to be the

same. Choice D is the right answer.


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29. Answer E 30. Answer E

Let R be the radius of the orbit. Thedifference in length between the actualorbit and the hypothetical one is 2(R +

2

) - 2R = 2

2

=

2

.

Each minute division of the dial makes an angle of

660

360= degrees. There is an angle of 246 = 144

degrees between the minute hand and the vertical

axis. There is an angle of 12612

24= degrees

between the hour hand and the vertical axis. So,there is an angle of 144 - 12 = 132 degreesbetween the hour and the minute hand of theclock.


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4. Word Problems Answers

31. Answer D 34. Answer B

Let x denote this number of days. Then themonthly income from machine A =5010x - 1500 and the monthly incomefrom machine B = 4010x - 550. Thequestion is what is the least integer x suchthat the income from A exceeds incomefrom B.

Let Vb denote the speed of the bus and Vc thespeed of the cyclist. Then the distance between thebuses = 4(Vb + Vc) = 12(Vb - Vc) from whereVb = 2Vc.Interval between the buses =

bustheofspeed

busesebetween thdistance=

=+

Vb

VcVb4 62

3

4 =

Vb

Vbminutes.

5010x - 1500 > 4010x 550 or100x > 950 or x > 9.Thus, x must be 10.

32. Answer E35. Answer D

Let V denote the volume of feed onechicken consumes per day. Then the totalvolume of feed in stock will be VDCwhere D is the number of days the feedwill last if the number of chickens does notchange and C is the current number ofchickens. From the stem it follows that

A for time spent in ascent, D for time spent indescent.From the stem it is clear that 20D - 50A = 750

+ 50 = 800. The question asks for %100+DA

A.

S1 says that D is known. From the equation above

A can be found and thus %100+DA

Acan be

determined.

V(D + 20)(C - 75) = VDC; V(D -15)(C + 100) = VDC.The first equation simplifies to 20C -75D = 1500. The second equation

simplifies to (-15)C + 100D = 1500.After dividing everything by 5 we get thelinear system:

S2 says that A + D is known. This allows to

express A in terms of D and find D from the

equation above. %100+DA

Acan then be

calculated.4C - 15D = 300; (-3)C + 20D =300.After solving it we receive C = 300, D =60.

36. Answer D

R for Richard's age, S for his sister's age.From S1 it follows that 2(R - 2) = S - 2. As S =R + 3, this equation can be solved for R and thusR + 5 can be found. R = 5 and R + 5 = 10.

33. Answer CPlease note that the price of $216 isirrelevant to this question and is only cited

to trap you into lengthy calculations. From S2 it follows that S + 2 = 2R or R + 5 =2R from where R = 5 and R + 5 = 10.For any price X the customer would pay:- by the first scheme: 0.2X + 100.1X =1.2X;

37. Answer E

Denote J the time it takes Jack to type 20 pages.Because working together Jack and Tom can type25 pages in 3 hours, they can type 20 pages in

5

123

5

4= hours. Now, it follows from the stem

that12

5

2

1111=

+=+JJTj

. From this equation J

= 6. So, Jack will type 20 pages in 6 hours,therefore he will type 40 pages in 12 hours.

- by the second scheme: 0.1X +200.08X = 1.7X.The percentage difference = 100%

7.1

5.0%100

7.1

2.17.1=

X

XX=

approximately 100% %30

7.1

51.0= .


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38. Answer B

Denoting X the number of the shelves, wecan compose an equation: 12X = 21 + (X- 1)8 + 11.

Solving it, we get 4X = 24 or X = 6.

39. Answer D

Let X denote the power of the 9-cyllinderengine and Y the power of the 12-cyllinderengine. It follows from the stem that Y =(1.1)3X = 1.211.1X = 1.331X or

about X3

11 . The required ratio =

75.0

4

3

311

1===

Y

X.

40. Answer C

If X is the initial weight of the first sackthen the following equation can becomposed:X - 4.75 = 0.73((43.25 - X) + 4.75) or1.73X = 480.73 + 4.75 or 173X =3504 + 475.From this equation X = 23. The weight of

the second sack was 43.25 - 23 = 20.25.The difference in weights was 23 - 20.25 =2.75.

This is an arithmetic toughie, you have tospot that 3979 = 23173.


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5. Number Properties Answers

41. Answer B

21

2

X can be written as =+2

)1()1( XX

=

2

eveneveneveninteger = even. To see

that the other choices are not necessarilyeven consider X = 3 (in C consider p = 2).

42. Answer B

S1 is not sufficient. Whatever X is, we can

manipulate Y to make

3

YX +10either

divisible or indivisible by 3.S2 is sufficient. The key thing to notice isthat if S2 holds, the sum of digits of

3

10 YX +is divisible by 3: 12,102,1002,...

43. Answer E

If N is divisible by 4 then @6 is divisible by4. We have five candidates 194716, 194736,

194756, 194776, 194796. Not sufficient.If N is divisible by 3 then 1+9+4+7+@+6 isdivisible by 3. We have four candidates194706, 194736, 194766, 194796. Notsufficient.S1 + S2 leaves us with two choices 194736and 194796 but whether @ = 3 or @ = 9cannot be determined.

44. Answer B

Only choice B must be true. If X is 2 then

Y cannot be an integer; otherwise XYwould not be an integer. Counter-examplesto other answer choices:A: X = 2, Y = 2; C: X = 0, Y = 0; D: X =1, Y = 4; E: X = 4, Y = 4.

45. Answer D

The number of divisors of any integer x

cannot exceed 12 x . This is because

the divisors form pairs: (1, x), )

2

,2 x

( all the

way to the last pair ), xx( . The maximum

number of pairs is x the maximum number of

divisors is 12 x (because the last pair

contains identical numbers). But if x is not a

perfect square, x is not a divisor of x.

Therefore, from S1 it follows that the number of

divisors of x is smaller than 12 x . If x is

prime then it is definitely not a perfect square.

Therefore, S2 is sufficient as well.

46. Answer A

S2 is not sufficient. If n = 0, the answer is NO;if n is 1, the answer is YES.However, if n is a positive integer, 10n+ 8 isalways divisible by 18. This is because 10n+ 8is even and is divisible by 9 (the sum of itsdigits is divisible by 9). As all prime numbersare positive integers, S1 is sufficient to answerthe question.

47. Answer A

From S1 it follows that X and Y cannot both beodd or even. One of them must be even and theother must be odd. This information is enoughto state that XYZ is even.From S2 it follows that both Y and Z are odd.The answer to the question will depend onwhether X is even but S2 does not provide thisinformation.

48. Answer E

Although the question seems difficult, it allowsa simple approach. If there is a pair (A, B) thatsatisfies S1 and S2 then pair (B, A) also satisfiesS1 and S2. So, we cannot determine A unless A= B which is not the case here (84 is notdivisible by 24).

From S1 and S2 it can be established that

either A = 96 and B = 72 or A = 72 and B =96 but the statements do not provide

information to decide between these two

variants. All we know is that A is either 72 or96.

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49. Answer D

S1: a2- b2= (b + 2)2- b2= 4b + 4, which isdivisible by 4.S2: a2- b2= (2n + 1)2- (2k + 1)2= 4n2+

4n - 4k2+ 4k, which is divisible by 4.

50. Answer B

1! + 2! + ... + 10! = 1 + (2! + ... + 10!) = odd +even = odd.1! + 2! + ... + 10! = 3 + (3! + ... + 10!) = integerdivisible by 3.

Among the listed choices we are looking for oddintegers divisible by 3. Only choice B fits. Thisis a backsolving problem.


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6. Miscellaneous Answers

51. Answer E

The least multiple of both 12 and 20 is 60. Itmeans that the two comets are seen togetherevery 60 years. Therefore, the next year thecomets will be observed together will be1979 + 60 = 2039. We have to wait for 2039- 2004 = 35 years.

52. Answer B

P for the number of students who studyphysics, M for the number of students whostudy math, PM for the number of studentswho study both math and physics:

P + M - PM = 20 - 10 = 10P + 8 - 7 = 10P = 9.The number of students who study physicsbut not math = P - PM = 9 - 7 = 2.

53. Answer B

This question tests the basics. Standarddeviation of a set is always non-negative andequals 0 only if all members of the set areequal. S1 doesn't help to answer the

question. S2 says that there is only oneelement in T and therefore standarddeviation of T is 0 (not positive).

54. Answer E

If it were not for the leap-year, the 19th ofSeptember in 1988 would be Sunday, in1989 it would be Monday, and in 1990 itwould be Tuesday (because there are 365days in a year and 365 = 527 + 1).Allowing for the leap-year, the 19th of

September in 1990 should be Wednesday.Therefore, the 21st of September 1990should be Friday.

55. Answer A

Because Paul's speed was 50 miles per hour,it was 1:30 when the tank started to leak. Atthat time, the tank contained 50 - 100.75 =42.5 liters of fuel. From then on the car waslosing a total of 17 liters per hour:

12605

1= liters as result of the leakage and

510100

50= liters to run the engine. Thus, after

1:30 the fuel will only last for 5.217

5.42 = hours.

Therefore, Paul will stop at 1:30 + 2:30 = 4:00pm.

56. Answer A

Let V denote the speed the cars are traveling at.Because the time interval between the cars isalways 1 second, the distance between the carswill depend on their speed as follows D = V1.This distance must not exceed the safety

distance:9100

2

+ V

V or V2- 100V + 900 0

or (V - 10)(V - 90) 0.Among the listed choices only choice A satisfiesthis inequality.

57. Answer B

Lets pick numbers: M = 5, T = 4, W = 120.Increase M by 60%: M = 8. Decrease T by 50%:T = 2. W has to be 96, which is 20% less than

120.

58. Answer A

The head of the train will run through the tunnelin 1 minute because it is moving at 60 kmhwhich is 1 km per minute. It will take additional

8

1minutes for the rest of the train to leave the

tunnel:the length of the train is 206 + 5 = 125 meters= 0.125 kilometers;

the time it will take the whole train to exit the

tunnel =kmm

km

1

125.0=

8

1minutes.


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59. Answer B 62. Answer A

The general equation is S + A + R - SA - SR- AR + SAR + N = 100% where N denotesthe number of people who like none of thethree jams (draw the Veinn diagram). To

maximize the number of people who likeraspberry but not strawberry or apple jam Nhas to be 0. The new equation can berewritten as follows:

From S1 it follows that3

1of the distance

amounts to 60 km. Therefore, the entire distanceis 180 km and the average speed for the entire

trip ish3

km180= 60 kmh.

S2 gives no information about the first hour.The bus could be nearly flying during the firsthour (and then its average speed for the entiretrip would be high) or it could be crawling (andthen its average speed for the entire trip wouldbe low).

R - AR - SR + SAR = 100% + SA - A - S.The left side of this equation is preciselywhat the question asks for - the number ofpeople who like raspberry jam but do notlike strawberry or apple jam (Venn diagramwill help to understand why it is so).Plucking the numbers:

63. Answer C

It is obvious that more than 100 computers weresold because more than $50,000 was received.For all computers above the first 100 thewholesaler received $7,600. Each of these

computers cost $400, therefore400

7600= 19

computers were sold at $400 each. Adding the100 that were sold at $500 each, we concludethat in total 119 computers were sold.

R - AR - SR + SAR = 100% + 30% - 44% -56% = 30%. Finally, 30% of 200 is 60.

60. Answer D

If one of the integers is 200 then the sum ofthe other three has to be 40. It is clear thateach of these three integers is less than 50.S1 is sufficient.From S2 it follows that two of the integersare less than 50 and two of the integers are

more than 50. If the integers are arranged inascending order then the median =

2

integerthirdintegersecond +. As all the

integers are different, no number can equal50.

64. Answer D

216 = (number of books)(price of one book).216 has to be divisible by the book price. All thelisted choices are divisors of 216 except 40.

65. Answer B

We have to find how many times factor 5 iscontained in 100!. That is, we have to find thelargest n such that 100! is divisible by 5n. Thereare 20 multiples of 5 in the first hundred but 25,50, 75, and 100 have to be counted twice

because they are divisible by 25 = 5

2

. So, theanswer is 24.

61. Answer A

S1 says that members of S are evenlydistributed in the set. In this case, the mean

= the median = 2

memberlastmemberfirst +

.This holds even if S contains only oneelement. 66. Answer BS2 gives no valuable information. S can be{75} with the answer to the question NO orit could be {1, 2, 72} with the answer to thequestion YES.

F(-2) = F(2) = F(4) = F(16). Thus, (B) must betrue.The other choices are not necessarily true,consider F(x) = 3.


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67. Answer A

In fact, we have the sum of n integers:(..9) + (..1) + (..9) + (..1) + ...If n is even then this sum ends with 0, if it'sodd then this sum ends with 9. S1 is

obviously sufficient, S2 is not (2 is primeand even, 3 is prime and odd).

68. Answer D

There is still room for 10 crates of apples.The weight of 10 crates of apples is the same

as the weight of3

113

3

4=10 crates of

oranges. Because the number of crates has to

be an integer, the lorry can accommodate amaximum of 13 crates of oranges.

69. Answer E

John will pay $1.20 for two hours between 4 pm

to 6 pm, $5.40 for three hours between 6 pm and9 pm, $3.60 for three hours between 9 pm andmidnight, and $0.60 for the hour after midnight.The total payment will constitute $1.20 + $5.40+ $3.60 + $0.60 = $10.80.

70. Answer D

Deborah and Mike will meet in the library in34 = 12 days on Saturday.

Thank you very much for your support of GMAT ClubPlease send your comments and questions to [email protected]


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