﻿ ﻿Sample Questions - Problem Solving - GMAT Quantitative Review

## 4.0 Problem Solving

### 4.3 Sample Questions

Solve the problem and indicate the best of the answer choices given.

Numbers: All numbers used are real numbers.

Figures: A figure accompanying a problem solving question is intended to provide information useful in solving the problem. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.

1.  Points A, B, C, and D, in that order, lie on a line. If AB = 3 cm, AC = 4 cm, and BD = 6 cm, what is CD, in centimeters?

1. (A) 1

2. (B) 2

3. (C) 3

4. (D) 4

5. (E) 5

2.  What is the value of x2yzxyz2, if x = −2, y = 1, and z = 3?

1. (A) 20

2. (B) 24

3. (C) 30

4. (D) 32

5. (E) 48

3.  If x > y and y > z, which of the following represents the greatest number?

1. (A) xz

2. (B) xy

3. (C) yx

4. (D) zy

5. (E) zx

4.  To order certain plants from a catalog, it costs \$3.00 per plant, plus a 5 percent sales tax, plus \$6.95 for shipping and handling regardless of the number of plants ordered. If Company C ordered these plants from the catalog at the total cost of \$69.95, how many plants did Company C order?

1. (A) 22

2. (B) 21

3. (C) 20

4. (D) 19

5. (E) 18

5.  Company C produces toy trucks at a cost of \$5.00 each for the first 100 trucks and \$3.50 for each additional truck. If 500 toy trucks were produced by Company C and sold for \$10.00 each, what was Company C”s gross profit?

1. (A) \$2,250

2. (B) \$2,500

3. (C) \$3,100

4. (D) \$3,250

5. (E) \$3,500

6.  A group of store managers must assemble 280 displays for an upcoming sale. If they assemble 25 percent of the displays during the first hour and 40 percent of the remaining displays during the second hour, how many of the displays will not have been assembled by the end of the second hour?

1. (A) 70

2. (B) 98

3. (C) 126

4. (D) 168

5. (E) 182

7.  Of the following, which is least?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

8.  The maximum recommended pulse rate R, when exercising, for a person who is x years of age is given by the equation . What is the age, in years, of a person whose maximum recommended pulse rate when exercising is 140?

1. (A) 40

2. (B) 45

3. (C) 50

4. (D) 55

5. (E) 60

9.  There are five sales agents in a certain real estate office. One month Andy sold twice as many properties as Ellen, Bob sold 3 more than Ellen, Cary sold twice as many as Bob, and Dora sold as many as Bob and Ellen together. Who sold the most properties that month?

1. (A) Andy

2. (B) Bob

3. (C) Cary

4. (D) Dora

5. (E) Ellen

10. Which of the following represent positive numbers?

I. −3 − (−5)

II. (−3)(−5)

III. −5 − (−3)

1. (A) I only

2. (B) II only

3. (C) III only

4. (D) I and II

5. (E) II and III

11. If is 2 more than , then

1. (A) 4

2. (B) 8

3. (C) 16

4. (D) 32

5. (E) 64

12. If Mario was 32 years old 8 years ago, how old was he x years ago?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

13. The toll T, in dollars, for a truck using a certain bridge is given by the formula T = 1.50 + 0.50(x − 2), where x is the number of axles on the truck. What is the toll for an 18-wheel truck that has 2 wheels on its front axle and 4 wheels on each of its other axles?

1. (A) \$2.50

2. (B) \$3.00

3. (C) \$3.50

4. (D) \$4.00

5. (E) \$5.00

14. If and , then

1. (A) −8

2. (B) −2

3. (C)

4. (D)

5. (E) 2

15. For what value of x between −4 and 4, inclusive, is the value of x2 − 10x + 16 the greatest?

1. (A) −4

2. (B) −2

3. (C) 0

4. (D) 2

5. (E) 4

16. The number is how many times the number ?

1. (A) 2

2. (B) 2.5

3. (C) 3

4. (D) 3.5

5. (E) 4

17. In the figure above, if F is a point on the line that bisects angle ACD and the measure of angle DCF is , which of the following is true of x?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

18. In which of the following pairs are the two numbers reciprocals of each other?

I. 3 and

II. and

III. and

1. (A) I only

2. (B) II only

3. (C) I and II

4. (D) I and III

5. (E) II and III

19. A rope 20.6 meters long is cut into two pieces. If the length of one piece of rope is 2.8 meters shorter than the length of the other, what is the length, in meters, of the longer piece of rope?

1. (A) 7.5

2. (B) 8.9

3. (C) 9.9

4. (D) 10.3

5. (E) 11.7

20. What is the perimeter, in meters, of a rectangular garden 6 meters wide that has the same area as a rectangular playground 16 meters long and 12 meters wide?

1. (A) 48

2. (B) 56

3. (C) 60

4. (D) 76

5. (E) 192

21. Of the total amount that Jill spent on a shopping trip, excluding taxes, she spent 50 percent on clothing, 20 percent on food, and 30 percent on other items. If Jill paid a 4 percent tax on the clothing, no tax on the food, and an 8 percent tax on all other items, then the total tax that she paid was what percent of the total amount that she spent, excluding taxes?

1. (A) 2.8%

2. (B) 3.6%

3. (C) 4.4%

4. (D) 5.2%

5. (E) 6.0%

22. At the opening of a trading day at a certain stock exchange, the price per share of stock K was \$8. If the price per share of stock K was \$9 at the closing of the day, what was the percent increase in the price per share of stock K for that day?

1. (A) 1.4%

2. (B) 5.9%

3. (C) 11.1%

4. (D) 12.5%

5. (E) 23.6%

23. The price of a certain television set is discounted by 10 percent, and the reduced price is then discounted by 10 percent. This series of successive discounts is equivalent to a single discount of

1. (A) 20%

2. (B) 19%

3. (C) 18%

4. (D) 11%

5. (E) 10%

24. The number of rooms at Hotel G is 10 less than twice the number of rooms at Hotel H. If the total number of rooms at Hotel G and Hotel H is 425, what is the number of rooms at Hotel G?

1. (A) 140

2. (B) 180

3. (C) 200

4. (D) 240

5. (E) 280

25. In the figure above, the sum of the three numbers in the horizontal row equals the product of the three numbers in the vertical column. What is the value of xy?

1. (A) 6

2. (B) 15

3. (C) 35

4. (D) 75

5. (E) 90

26.

1. (A) −4

2. (B) 2

3. (C) 6

4. (D)

5. (E)

27. In the rectangular coordinate system above, the shaded region is bounded by straight lines. Which of the following is NOT an equation of one of the boundary lines?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

28. A certain population of bacteria doubles every 10 minutes. If the number of bacteria in the population initially was 104, what was the number in the population 1 hour later?

1. (A) 2(104)

2. (B) 6(104)

3. (C) (26)(104)

4. (D) (106)(104)

5. (E) (104)6

29. If the perimeter of a rectangular garden plot is 34 feet and its area is 60 square feet, what is the length of each of the longer sides?

1. (A) 5 ft

2. (B) 6 ft

3. (C) 10 ft

4. (D) 12 ft

5. (E) 15 ft

30. In a poll of 66,000 physicians, only 20 percent responded; of these, 10 percent disclosed their preference for pain reliever X. How many of the physicians who responded did not disclose a preference for pain reliever X?

1. (A) 1,320

2. (B) 5,280

3. (C) 6,600

4. (D) 10,560

5. (E) 11,880

31.

1. (A) 0.357

2. (B) 0.3507

3. (C) 0.35007

4. (D) 0.0357

5. (E) 0.03507

32. Which of the following fractions is equal to the decimal 0.0625?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

33. If r and s are positive integers such that (2r)(4s) = 16, then 2r + s =

1. (A) 2

2. (B) 3

3. (C) 4

4. (D) 5

5. (E) 6

34. If positive integers x and y are not both odd, which of the following must be even?

1. (A) xy

2. (B)

3. (C)

4. (D)

5. (E)

35. The annual budget of a certain college is to be shown on a circle graph. If the size of each sector of the graph is to be proportional to the amount of the budget it represents, how many degrees of the circle should be used to represent an item that is 15 percent of the budget?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

36. During a two-week period, the price of an ounce of silver increased by 25 percent by the end of the first week and then decreased by 20 percent of this new price by the end of the second week. If the price of silver was x dollars per ounce at the beginning of the two-week period, what was the price, in dollars per ounce, by the end of the period?

1. (A) 0.8x

2. (B) 0.95x

3. (C) x

4. (D) 1.05x

5. (E) 1.25x

37. In a certain pond, 50 fish were caught, tagged, and returned to the pond. A few days later, 50 fish were caught again, of which 2 were found to have been tagged. If the percent of tagged fish in the second catch approximates the percent of tagged fish in the pond, what is the approximate number of fish in the pond?

1. (A) 400

2. (B) 625

3. (C) 1,250

4. (D 2,500

5. (E) 10,000

38.

1. (A)

2. (B)

3. (C)

4. (D) 8

5. (E) 16

39. The organizers of a fair projected a 25 percent increase in attendance this year over that of last year, but attendance this year actually decreased by 20 percent. What percent of the projected attendance was the actual attendance?

1. (A) 45%

2. (B) 56%

3. (C) 64%

4. (D) 75%

5. (E) 80%

40. What is the ratio of to the product ?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E) 4

41. If , then

1. (A) −24

2. (B) −8

3. (C) 0

4. (D) 8

5. (E) 24

42. In the system of equations above, what is the value of x?

1. (A) −3

2. (B) −1

3. (C)

4. (D) 1

5. (E)

43. If , then the value of is closest to

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

44. In above, what is x in terms of z?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

45. What is the maximum number of foot pieces of wire that can be cut from a wire that is 24 feet long?

1. (A) 11

2. (B) 18

3. (C) 19

4. (D) 20

5. (E) 30

46. The expression above is approximately equal to

1. (A) 1

2. (B) 3

3. (C) 4

4. (D) 5

5. (E) 6

47. If the numbers , , , , and were ordered from greatest to least, the middle number of the resulting sequence would be

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

48. Last week Jack worked 70 hours and earned \$1,260. If he earned his regular hourly wage for the first 40 hours worked, times his regular hourly wage for the next 20 hours worked, and 2 times his regular hourly wage for the remaining 10 hours worked, what was his regular hourly wage?

1. (A) \$7.00

2. (B) \$14.00

3. (C) \$18.00

4. (D) \$22.00

5. (E) \$31.50

49. Last year if 97 percent of the revenues of a company came from domestic sources and the remaining revenues, totaling \$450,000, came from foreign sources, what was the total of the company”s revenues?

1. (A) \$1,350,000

2. (B) \$1,500,000

3. (C) \$4,500,000

4. (D) \$15,000,000

5. (E) \$150,000,000

50.

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

51. A certain fishing boat is chartered by 6 people who are to contribute equally to the total charter cost of \$480. If each person contributes equally to a \$150 down payment, how much of the charter cost will each person still owe?

1. (A) \$80

2. (B) \$66

3. (C) \$55

4. (D) \$50

5. (E) \$45

52. Craig sells major appliances. For each appliance he sells, Craig receives a commission of \$50 plus 10 percent of the selling price. During one particular week Craig sold 6 appliances for selling prices totaling \$3,620. What was the total of Craig”s commissions for that week?

1. (A) \$412

2. (B) \$526

3. (C) \$585

4. (D) \$605

5. (E) \$662

53. What number when multiplied by yields as the result?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

54. If 3 pounds of dried apricots that cost x dollars per pound are mixed with 2 pounds of prunes that cost y dollars per pound, what is the cost, in dollars, per pound of the mixture?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

55. Which of the following must be equal to zero for all real numbers x?

I.

II.

III. x0

1. (A) I only

2. (B) II only

3. (C) I and III only

4. (D) II and III only

5. (E) I, II, and III

56. In the table above, what is the least number of table entries that are needed to show the mileage between each city and each of the other five cities?

1. (A) 15

2. (B) 21

3. (C) 25

4. (D) 30

5. (E) 36

57. If is a factor of , then

1. (A) −6

2. (B) −2

3. (C) 2

4. (D) 6

5. (E) 14

58.

1. (A) 0.248

2. (B) 0.252

3. (C) 0.284

4. (D) 0.312

5. (E) 0.320

59. Members of a social club met to address 280 newsletters. If they addressed of the newsletters during the first hour and of the remaining newsletters during the second hour, how many newsletters did they address during the second hour?

1. (A) 28

2. (B) 42

3. (C) 63

4. (D) 84

5. (E) 112

60.

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

61. After 4,000 gallons of water were added to a large water tank that was already filled to of its capacity, the tank was then at of its capacity. How many gallons of water does the tank hold when filled to capacity?

1. (A) 5,000

2. (B) 6,200

3. (C) 20,000

4. (D) 40,000

5. (E) 80,000

62. The sum of three integers is 40. The largest integer is 3 times the middle integer, and the smallest integer is 23 less than the largest integer. What is the product of the three integers?

1. (A) 1,104

2. (B) 972

3. (C) 672

4. (D) 294

5. (E) 192

63. Five machines at a certain factory operate at the same constant rate. If four of these machines, operating simultaneously, take 30 hours to fill a certain production order, how many fewer hours does it take all five machines, operating simultaneously, to fill the same production order?

1. (A) 3

2. (B) 5

3. (C) 6

4. (D) 16

5. (E) 24

64. If Mel saved more than \$10 by purchasing a sweater at a 15 percent discount, what is the smallest amount the original price of the sweater could be, to the nearest dollar?

1. (A) 45

2. (B) 67

3. (C) 75

4. (D) 83

5. (E) 150

65. If and d∗ is the decimal obtained by rounding d to the nearest hundredth, what is the value of ?

1. (A) − 0.0053

2. (B) − 0.0003

3. (C) 0.0007

4. (D) 0.0047

5. (E) 0.0153

66. At a monthly meeting, of the attendees were males and of the male attendees arrived on time. If of the female attendees arrived on time, what fraction of the attendees at the monthly meeting did not arrive on time?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

67. The sequence a1, a2, a3, a4, a5 is such that for . If , what is the value of a1?

1. (A) 1

2. (B) 6

3. (C) 11

4. (D) 16

5. (E) 21

68. A certain bridge is 4,024 feet long. Approximately how many minutes does it take to cross this bridge at a constant speed of 20 miles per hour? (1 mile = 5,280 feet)

1. (A) 1

2. (B) 2

3. (C) 4

4. (D) 6

5. (E) 7

69. If , how much greater than the median of the numbers in S is the mean of the numbers in S?

1. (A) 0.5

2. (B) 1.0

3. (C) 1.5

4. (D) 2.0

5. (E) 2.5

70. A total of 5 liters of gasoline is to be poured into two empty containers with capacities of 2 liters and 6 liters, respectively, such that both containers will be filled to the same percent of their respective capacities. What amount of gasoline, in liters, must be poured into the 6-liter container?

1. (A)

2. (B) 4

3. (C)

4. (D) 3

5. (E)

71. When positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that is a multiple of 35?

1. (A) 3

2. (B) 4

3. (C) 12

4. (D) 32

5. (E) 35

72. List S consists of 10 consecutive odd integers, and list T consists of 5 consecutive even integers. If the least integer in S is 7 more than the least integer in T, how much greater is the average (arithmetic mean) of the integers in S than the average of the integers inT?

1. (A) 2

2. (B) 7

3. (C) 8

4. (D) 12

5. (E) 22

73. In the figure above, what is the area of triangular region BCD?

1. (A)

2. (B) 8

3. (C)

4. (D) 16

5. (E)

74. Of the goose eggs laid at a certain pond, hatched, and of the geese that hatched from those eggs survived the first month. Of the geese that survived the first month, did not survive the first year. If 120 geese survived the first year and if no more than one goose hatched from each egg, how many goose eggs were laid at the pond?

1. (A) 280

2. (B) 400

3. (C) 540

4. (D) 600

5. (E) 840

75. If and , which of the following must be equal to 0?

I.

II.

III.

1. (A) I only

2. (B) II only

3. (C) III only

4. (D) II and III only

5. (E) I, II, and III

76. If a square region has area n, what is the length of the diagonal of the square in terms of n?

1. (A)

2. (B)

3. (C)

4. (D) 2n

5. (E) 2n2

77. The “prime sum” of an integer n greater than 1 is the sum of all the prime factors of n, including repetitions. For example, the prime sum of 12 is 7, since and . For which of the following integers is the prime sum greater than 35?

1. (A) 440

2. (B) 512

3. (C) 620

4. (D) 700

5. (E) 750

78. At a garage sale, all of the prices of the items sold were different. If the price of a radio sold at the garage sale was both the 15th highest price and the 20th lowest price among the prices of the items sold, how many items were sold at the garage sale?

1. (A) 33

2. (B) 34

3. (C) 35

4. (D) 36

5. (E) 37

79. For all positive integers m and v, the expression m Θ v represents the remainder when m is divided by v. What is the value of ((98 Θ 33) Θ 17) − (98 Θ (33 Θ 17))?

1. (A) −10

2. (B) −2

3. (C) 8

4. (D) 13

5. (E) 17

80. In a certain sequence, each term after the first term is one-half the previous term. If the tenth term of the sequence is between 0.0001 and 0.001, then the twelfth term of the sequence is between

1. (A) 0.0025 and 0.025

2. (B) 0.00025 and 0.0025

3. (C) 0.000025 and 0.00025

4. (D) 0.0000025 and 0.000025

5. (E) 0.00000025 and 0.0000025

81. Ada and Paul received their scores on three tests. On the first test, Ada”s score was 10 points higher than Paul”s score. On the second test, Ada”s score was 4 points higher than Paul”s score. If Paul”s average (arithmetic mean) score on the three tests was 3 points higher than Ada”s average score on the three tests, then Paul”s score on the third test was how many points higher than Ada”s score?

1. (A) 9

2. (B) 14

3. (C) 17

4. (D) 23

5. (E) 25

82. The price of a certain stock increased by 0.25 of 1 percent on a certain day. By what fraction did the price of the stock increase that day?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

83. Three business partners, Q, R, and S, agree to divide their total profit for a certain year in the ratios 2:5:8, respectively. If Q”s share was \$4,000, what was the total profit of the business partners for the year?

1. (A) \$26,000

2. (B) \$30,000

3. (C) \$52,000

4. (D) \$60,000

5. (E) \$300,000

84. In the rectangular coordinate system above, the area of is

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

85. What is the largest integer n such that ?

1. (A) 5

2. (B) 6

3. (C) 7

4. (D) 10

5. (E) 51

86. The average (arithmetic mean) length per film for a group of 21 films is t minutes. If a film that runs for 66 minutes is removed from the group and replaced by one that runs for 52 minutes, what is the average length per film, in minutes, for the new group of films, in terms of t?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

87. If x = − |w|, which of the following must be true?

1. (A) x = − w

2. (B) x = w

3. (C) x2 = w

4. (D) x2 = w2

5. (E) x3 = w3

88. Which of the following lines in the xy-plane does not contain any point with integers as both coordinates?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

89. One inlet pipe fills an empty tank in 5 hours. A second inlet pipe fills the same tank in 3 hours. If both pipes are used together, how long will it take to fill of the tank?

1. (A) hr

2. (B) hr

3. (C) hr

4. (D) hr

5. (E) hr

90.

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

91. If the length and width of a rectangular garden plot were each increased by 20 percent, what would be the percent increase in the area of the plot?

1. (A) 20%

2. (B) 24%

3. (C) 36%

4. (D) 40%

5. (E) 44%

92. The population of a bacteria culture doubles every 2 minutes. Approximately how many minutes will it take for the population to grow from 1,000 to 500,000 bacteria?

1. (A) 10

2. (B) 12

3. (C) 14

4. (D) 16

5. (E) 18

93. For a light that has an intensity of 60 candles at its source, the intensity in candles, S, of the light at a point d feet from the source is given by the formula , where k is a constant. If the intensity of the light is 30 candles at a distance of 2 feet from the source, what is the intensity of the light at a distance of 20 feet from the source?

1. (A) candle

2. (B) candle

3. (C) 1 candle

4. (D) 2 candles

5. (E) 3 candles

94. In the correctly worked addition problem shown, where the sum of the two-digit positive integers AB and BA is the three-digit integer AAC, and A, B, and C are different digits, what is the units digit of the integer AAC?

1. (A) 9

2. (B) 6

3. (C) 3

4. (D) 2

5. (E) 0

95. Given the inequalities above, which of the following CANNOT be the value of r?

1. (A) −20

2. (B) −5

3. (C) 0

4. (D) 5

5. (E) 20

96. A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. If n is a positive integer, for which of the following values of k is divisible by 9?

1. (A) 9

2. (B) 16

3. (C) 23

4. (D) 35

5. (E) 47

97. On the number line, the shaded interval is the graph of which of the following inequalities?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

98. Of all the students in a certain dormitory, are first-year students and the rest are second-year students. If of the first-year students have not declared a major and if the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major, what fraction of all the students in the dormitory are second-year students who have not declared a major?

1. A.

2. B.

3. C.

4. D.

5. E.

99. If the average (arithmetic mean) of x, y, and z is 7x and x ≠ 0, what is the ratio of x to the sum of y and z?

1. (A) 1:21

2. (B) 1:20

3. (C) 1:6

4. (D) 6:1

5. (E) 20:1

100.

1. (A) − 1.2

2. (B) −0.12

3. (C) 0

4. (D) 0.12

5. (E) 1.2

101. René earns \$8.50 per hour on days other than Sundays and twice that rate on Sundays. Last week she worked a total of 40 hours, including 8 hours on Sunday. What were her earnings for the week?

1. (A) \$272

2. (B) \$340

3. (C) \$398

4. (D) \$408

5. (E) \$476

102. In a shipment of 120 machine parts, 5 percent were defective. In a shipment of 80 machine parts, 10 percent were defective. For the two shipments combined, what percent of the machine parts were defective?

1. (A) 6.5%

2. (B) 7.0%

3. (C) 7.5%

4. (D) 8.0%

5. (E) 8.5%

103. If , then x =

1. (A) − 3

2. (B) − 1

3. (C) 0

4. (D) 1

5. (E) 3

104. Of the following, the closest approximation to is

1. (A) 5

2. (B) 15

3. (C) 20

4. (D) 25

5. (E) 225

105. Which of the following CANNOT be the greatest common divisor of two positive integers x and y?

1. (A) 1

2. (B) x

3. (C) y

4. (D)

5. (E)

106. Last year Carlos saved 10 percent of his annual earnings. This year he earned 5 percent more than last year and he saved 12 percent of his annual earnings. The amount saved this year was what percent of the amount saved last year?

1. (A) 122%

2. (B) 124%

3. (C) 126%

4. (D) 128%

5. (E) 130%

107. A corporation that had \$115.19 billion in profits for the year paid out \$230.10 million in employee benefits. Approximately what percent of the profits were the employee benefits? (Note: )

1. (A) 50%

2. (B) 20%

3. (C) 5%

4. (D) 2%

5. (E) 0.2%

108. In the coordinate plane, line k passes through the origin and has slope 2. If points (3,y) and (x,4) are on line k, then

1. (A) 3.5

2. (B) 7

3. (C) 8

4. (D) 10

5. (E) 14

109. If a, b, and c are constants, , and for all numbers x, what is the value of b?

1. (A) −3

2. (B) −1

3. (C) 0

4. (D) 1

5. (E) 3

110. On the number line, if , if p is halfway between r and s, and if t is halfway between p and r, then

1. (A)

2. (B)

3. (C)

4. (D) 3

5. (E) 4

111. Company K”s earnings were \$12 million last year. If this year”s earnings are projected to be 150 percent greater than last year”s earnings, what are Company K”s projected earnings this year?

1. (A) \$13.5 million

2. (B) \$15 million

3. (C) \$18 million

4. (D) \$27 million

5. (E) \$30 million

112. If and , then

1. (A)

2. (B)

3. (C)

4. (D)

5. (E) 5

113.

1. (A) 177

2. (B) 173(18)

3. (C) 176(18)

4. (D)

5. (E)

114. A certain clock marks every hour by striking a number of times equal to the hour, and the time required for a stroke is exactly equal to the time interval between strokes. At 6:00 the time lapse between the beginning of the first stroke and the end of the last stroke is 22 seconds. At 12:00, how many seconds elapse between the beginning of the first stroke and the end of the last stroke?

1. (A) 72

2. (B) 50

3. (C) 48

4. (D) 46

5. (E) 44

115. What is the greatest number of identical bouquets that can be made out of 21 white and 91 red tulips if no flowers are to be left out? (Two bouquets are identical whenever the number of red tulips in the two bouquets is equal and the number of white tulips in the two bouquets is equal.)

1. (A) 3

2. (B) 4

3. (C) 5

4. (D) 6

5. (E) 7

116. In the xy-plane, the points (c,d), (c,-d), and (-c,-d) are three vertices of a certain square. If c < 0 and d > 0, which of the following points is in the same quadrant as the fourth vertex of the square?

1. (A) (−5,−3)

2. (B) (−5,3)

3. (C) (5,−3)

4. (D) (3,−5)

5. (E) (3,5)

117. For all numbers s and t, the operation is defined by . If , then

1. (A) 2

2. (B) 3

3. (C) 5

4. (D) 6

5. (E) 11

118. Salesperson A”s compensation for any week is \$360 plus 6 percent of the portion of A”s total sales above \$1,000 for that week. Salesperson B”s compensation for any week is 8 percent of B”s total sales for that week. For what amount of total weekly sales would both salespeople earn the same compensation?

1. (A) \$21,000

2. (B) \$18,000

3. (C) \$15,000

4. (D) \$4,500

5. (E) \$4,000

119. If = x%, then x =

1. (A) 0.3

2. (B) 0.03

3. (C) 0.003

4. (D) 0.0003

5. (E) 0.00003

120. If a basketball team scores an average (arithmetic mean) of x points per game for n games and then scores y points in its next game, what is the team”s average score for the games?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

121. If and , which of the following must be negative?

1. (A) xyz

2. (B) xyz2

3. (C) xy2z

4. (D) xy2z2

5. (E) x2y2z2

122. At a certain pizzeria, of the pizzas sold in one week were mushroom and of the remaining pizzas sold were pepperoni. If n of the pizzas sold were pepperoni, how many were mushroom?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E) 3n

123. Two trains, X and Y, started simultaneously from opposite ends of a 100-mile route and traveled toward each other on parallel tracks. Train X, traveling at a constant rate, completed the 100-mile trip in 5 hours; Train Y, traveling at a constant rate, completed the 100-mile trip in 3 hours. How many miles had Train X traveled when it met Train Y?

1. (A) 37.5

2. (B) 40.0

3. (C) 60.0

4. (D) 62.5

5. (E) 77.5

124. What is the value of for ?

1. (A) − 0.72

2. (B) − 1.42

3. (C) − 1.98

4. (D) − 2.40

5. (E) − 2.89

125. What is the remainder when 324 is divided by 5?

1. (A) 0

2. (B) 1

3. (C) 2

4. (D) 3

5. (E) 4

126. If the volume of a ball is 32,490 cubic millimeters, what is the volume of the ball in cubic centimeters? (1 millimeter = 0.1 centimeter)

1. (A) 0.3249

2. (B) 3.249

3. (C) 32.49

4. (D) 324.9

5. (E) 3,249

127. David used part of \$100,000 to purchase a house. Of the remaining portion, he invested of it at 4 percent simple annual interest and of it at 6 percent simple annual interest. If after a year the income from the two investments totaled \$320, what was the purchase price of the house?

1. (A) \$96,000

2. (B) \$94,000

3. (C) \$88,000

4. (D) \$75,000

5. (E) \$40,000

128. The cost to rent a small bus for a trip is x dollars, which is to be shared equally among the people taking the trip. If 10 people take the trip rather than 16, how many more dollars, in terms of x, will it cost per person?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

129. If x is an integer and , which of the following CANNOT be a divisor of y?

1. (A) 4

2. (B) 5

3. (C) 6

4. (D) 7

5. (E) 8

130. A certain electronic component is sold in boxes of 54 for \$16.20 and in boxes of 27 for \$13.20. A customer who needed only 54 components for a project had to buy 2 boxes of 27 because boxes of 54 were unavailable. Approximately how much more did the customer pay for each component due to the unavailability of the larger boxes?

1. (A) \$0.33

2. (B) \$0.19

3. (C) \$0.11

4. (D) \$0.06

5. (E) \$0.03

131. As a salesperson, Phyllis can choose one of two methods of annual payment: either an annual salary of \$35,000 with no commission or an annual salary of \$10,000 plus a 20 percent commission on her total annual sales. What must her total annual sales be to give her the same annual pay with either method?

1. (A) \$100,000

2. (B) \$120,000

3. (C) \$125,000

4. (D) \$130,000

5. (E) \$132,000

132. Last year Department Store X had a sales total for December that was 4 times the average (arithmetic mean) of the monthly sales totals for January through November. The sales total for December was what fraction of the sales total for the year?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

133. Working alone, Printers X, Y, and Z can do a certain printing job, consisting of a large number of pages, in 12, 15, and 18 hours, respectively. What is the ratio of the time it takes Printer X to do the job, working alone at its rate, to the time it takes Printers Y and Z to do the job, working together at their individual rates?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

134. In the sequence x0, x1, x2, . . . , xn, each term from x1 to xk is 3 greater than the previous term, and each term from to xn is 3 less than the previous term, where n and k are positive integers and . If and if , what is the value of n?

1. (A) 5

2. (B) 6

3. (C) 9

4. (D) 10

5. (E) 15

135. If , then

1. (A)

2. (B)

3. (C) 3x2

4. (D)

5. (E)

136. In the figure shown above, line segment QR has length 12, and rectangle MPQT is a square. If the area of rectangular region MPRS is 540, what is the area of rectangular region TQRS?

1. (A) 144

2. (B) 216

3. (C) 324

4. (D) 360

5. (E) 396

137. A train travels from New York City to Chicago, a distance of approximately 840 miles, at an average rate of 60 miles per hour and arrives in Chicago at 6:00 in the evening, Chicago time. At what hour in the morning, New York City time, did the train depart for Chicago? (Note: Chicago time is one hour earlier than New York City time.)

1. (A) 4:00

2. (B) 5:00

3. (C) 6:00

4. (D) 7:00

5. (E) 8:00

138. Last year Manfred received 26 paychecks. Each of his first 6 paychecks was \$750; each of his remaining paychecks was \$30 more than each of his first 6 paychecks. To the nearest dollar, what was the average (arithmetic mean) amount of his paychecks for the year?

1. (A) \$752

2. (B) \$755

3. (C) \$765

4. (D) \$773

5. (E) \$775

139. Machines A and B always operate independently and at their respective constant rates. When working alone, Machine A can fill a production lot in 5 hours, and Machine B can fill the same lot in x hours. When the two machines operate simultaneously to fill the production lot, it takes them 2 hours to complete the job. What is the value of x?

1. (A)

2. (B) 3

3. (C)

4. (D)

5. (E)

140. A driver completed the first 20 miles of a 40-mile trip at an average speed of 50 miles per hour. At what average speed must the driver complete the remaining 20 miles to achieve an average speed of 60 miles per hour for the entire 40-mile trip? (Assume that the driver did not make any stops during the 40-mile trip.)

1. (A) 65 mph

2. (B) 68 mph

3. (C) 70 mph

4. (D) 75 mph

5. (E) 80 mph

141. The figure shown above consists of three identical circles that are tangent to each other. If the area of the shaded region is , what is the radius of each circle?

1. (A) 4

2. (B) 8

3. (C) 16

4. (D) 24

5. (E) 32

142. A positive integer n is a perfect number provided that the sum of all the positive factors of n, including 1 and n, is equal to 2n. What is the sum of the reciprocals of all the positive factors of the perfect number 28?

1. (A)

2. (B)

3. (C) 2

4. (D) 3

5. (E) 4

143. The infinite sequence a1, a2, . . . , an, . . . is such that a1 = 2, a2 = −3, a3 = 5, a4 = −1, and an = an − 4 for n > 4. What is the sum of the first 97 terms of the sequence?

1. (A) 72

2. (B) 74

3. (C) 75

4. (D) 78

5. (E) 80

144. The sequence a1, a2, . . . , an, . . . is such that an = 2an − 1x for all positive integers n ≥ 2 and for a certain number x. If a5 = 99 and a3 = 27, what is the value of x?

1. (A) 3

2. (B) 9

3. (C) 18

4. (D) 36

5. (E) 45

145. A window is in the shape of a regular hexagon with each side of length 80 centimeters. If a diagonal through the center of the hexagon is w centimeters long, then w =

1. (A) 80

2. (B) 120

3. (C) 150

4. (D) 160

5. (E) 240

146. On a certain transatlantic crossing, 20 percent of a ship”s passengers held round-trip tickets and also took their cars aboard the ship. If 60 percent of the passengers with round-trip tickets did not take their cars aboard the ship, what percent of the ship”s passengers held round-trip tickets?

1. (A)

2. (B) 40%

3. (C) 50%

4. (D) 60%

5. (E)

147. If x and k are integers and , what is the value of k?

1. (A) 5

2. (B) 7

3. (C) 10

4. (D) 12

5. (E) 14

148. For every even positive integer m, f(m) represents the product of all even integers from 2 to m, inclusive. For example, . What is the greatest prime factor of f(24)?

1. (A) 23

2. (B) 19

3. (C) 17

4. (D) 13

5. (E) 11

149. In pentagon PQRST, , , , and . Which of the lengths 5, 10, and 15 could be the value of PT?

1. (A) 5 only

2. (B) 15 only

3. (C) 5 and 10 only

4. (D) 10 and 15 only

5. (E) 5, 10, and 15

3, k, 2, 8, m, 3

150. The arithmetic mean of the list of numbers above is 4. If k and m are integers and , what is the median of the list?

1. (A) 2

2. (B) 2.5

3. (C) 3

4. (D) 3.5

5. (E) 4

151. If the variables, X, Y, and Z take on only the values 10, 20, 30, 40, 50, 60, or 70 with frequencies indicated by the shaded regions above, for which of the frequency distributions is the mean equal to the median?

1. (A) X only

2. (B) Y only

3. (C) Z only

4. (D) X and Y

5. (E) X and Z

152. When the figure above is cut along the solid lines, folded along the dashed lines, and taped along the solid lines, the result is a model of a geometric solid. This geometric solid consists of 2 pyramids, each with a square base that they share. What is the sum of the number of edges and the number of faces of this geometric solid?

1. (A) 10

2. (B) 18

3. (C) 20

4. (D) 24

5. (E) 25

153. For how many ordered pairs (x,y) that are solutions of the system above are x and y both integers?

1. (A) 7

2. (B) 10

3. (C) 12

4. (D) 13

5. (E) 14

154. The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is , what is the length of line segment RU?

1. (A)

2. (B)

3. (C) 3

4. (D) 4

5. (E) 6

155. A certain university will select 1 of 7 candidates eligible to fill a position in the mathematics department and 2 of 10 candidates eligible to fill 2 identical positions in the computer science department. If none of the candidates is eligible for a position in both departments, how many different sets of 3 candidates are there to fill the 3 positions?

1. (A) 42

2. (B) 70

3. (C) 140

4. (D) 165

5. (E) 315

156. A survey of employers found that during 1993 employment costs rose 3.5 percent, where employment costs consist of salary costs and fringe-benefit costs. If salary costs rose 3 percent and fringe-benefit costs rose 5.5 percent during 1993, then fringe-benefit costs represented what percent of employment costs at the beginning of 1993?

1. (A) 16.5%

2. (B) 20%

3. (C) 35%

4. (D) 55%

5. (E) 65%

157. The subsets of the set {w, x, y} are {w}, {x}, {y}, {w, x}, {w, y}, {x, y}, {w, x, y}, and { } (the empty subset). How many subsets of the set {w, x, y, z} contain w?

1. (A) Four

2. (B) Five

3. (C) Seven

4. (D) Eight

5. (E) Sixteen

158. There are 10 books on a shelf, of which 4 are paperbacks and 6 are hardbacks. How many possible selections of 5 books from the shelf contain at least one paperback and at least one hardback?

1. (A) 75

2. (B) 120

3. (C) 210

4. (D) 246

5. (E) 252

159. If x is to be chosen at random from the set {1, 2, 3, 4} and y is to be chosen at random from the set {5, 6, 7}, what is the probability that xy will be even?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

160. The function f is defined for each positive three-digit integer n by f(n) = 2x 3y 5z, where x, y, and z are the hundreds, tens, and units digits of n, respectively. If m and v are three-digit positive integers such that f(m) = 9f(v), then mv =

1. (A) 8

2. (B) 9

3. (C) 18

4. (D) 20

5. (E) 80

161. If 1050 − 74 is written as an integer in base 10 notation, what is the sum of the digits in that integer?

1. (A) 424

2. (B) 433

3. (C) 440

4. (D) 449

5. (E) 467

162. A certain company that sells only cars and trucks reported that revenues from car sales in 1997 were down 11 percent from 1996 and revenues from truck sales in 1997 were up 7 percent from 1996. If total revenues from car sales and truck sales in 1997 were up 1 percent from 1996, what is the ratio of revenue from car sales in 1996 to revenue from truck sales in 1996?

1. (A) 1:2

2. (B) 4:5

3. (C) 1:1

4. (D) 3:2

5. (E) 5:3

163. , which of the following must be true?

I.

II.

III. is positive.

1. (A) II only

2. (B) III only

3. (C) I and II only

4. (D) II and III only

5. (E) I, II, and III

164. A certain right triangle has sides of length x, y, and z, where . If the area of this triangular region is 1, which of the following indicates all of the possible values of y?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

165. A set of numbers has the property that for any number t in the set, is in the set. If −1 is in the set, which of the following must also be in the set?

I. −3

II. 1

III. 5

1. (A) I only

2. (B) II only

3. (C) I and II only

4. (D) II and III only

5. (E) I, II, and III

166. A couple decides to have 4 children. If they succeed in having 4 children and each child is equally likely to be a boy or a girl, what is the probability that they will have exactly 2 girls and 2 boys?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

167. In the figure above, point O is the center of the circle and . What is the value of x?

1. (A) 40

2. (B) 36

3. (C) 34

4. (D) 32

5. (E) 30

168. When 10 is divided by the positive integer n, the remainder is . Which of the following could be the value of n?

1. (A) 3

2. (B) 4

3. (C) 7

4. (D) 8

5. (E) 12

169. An airline passenger is planning a trip that involves three connecting flights that leave from Airports A, B, and C, respectively. The first flight leaves Airport A every hour, beginning at 8:00 a.m., and arrives at Airport B hours later. The second flight leaves Airport B every 20 minutes, beginning at 8:00 a.m., and arrives at Airport C hours later. The third flight leaves Airport C every hour, beginning at 8:45 a.m. What is the least total amount of time the passenger must spend between flights if all flights keep to their schedules?

1. (A) 25 min

2. (B) 1 hr 5 min

3. (C) 1 hr 15 min

4. (D) 2 hr 20 min

5. (E) 3 hr 40 min

170. If n is a positive integer and n2 is divisible by 72, then the largest positive integer that must divide n is

1. (A) 6

2. (B) 12

3. (C) 24

4. (D) 36

5. (E) 48

171. If n is a positive integer and , which of the following could NOT be a value of k?

1. (A) 1

2. (B) 4

3. (C) 7

4. (D) 25

5. (E) 79

172. A certain grocery purchased x pounds of produce for p dollars per pound. If y pounds of the produce had to be discarded due to spoilage and the grocery sold the rest for s dollars per pound, which of the following represents the gross profit on the sale of the produce?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

173. If x, y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

1. (A)

2. (B)

3. (C)

4. (D)

5. (E)

174. Running at their respective constant rates, Machine X takes 2 days longer to produce w widgets than Machine Y. At these rates, if the two machines together produce widgets in 3 days, how many days would it take Machine X alone to produce 2w widgets?

1. (A) 4

2. (B) 6

3. (C) 8

4. (D) 10

5. (E) 12

175. A square wooden plaque has a square brass inlay in the center, leaving a wooden strip of uniform width around the brass square. If the ratio of the brass area to the wooden area is 25 to 39, which of the following could be the width, in inches, of the wooden strip?

1. I. 1

2. II. 3

3. III. 4

4. (A) I only

5. (B) II only

6. (C) I and II only

7. (D) I and III only

8. (E) I, II, and III

176.

1. (A) 16

2. (B) 14

3. (C) 3

4. (D) 1

5. (E) −1

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