GMAT Problem solving Challenge Questions

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