MATHCOUNTS

2002 Chapter Competition

Countdown Round

1. What is the sum of the number of faces, vertices and edges in a triangular prism?

Answer: 20

2. One-third of a 30-student class is absent today. One-half of those were also absent yesterday. What percent of the class has been absent for two straight days? Express your answer to the nearest whole number.

Answer: 17 (percent)

3. Compute:

( ) ( ) .17 10 17 102 2

Answer: 680

4. Two of Mr. Bernard’s classes took the same test. His class of 20 students had an average score of 80. His other class of 30 students had an average of 70. What was the average score for all 50 students?

Answer: 74

5. The radius of a circle is increased by 100%. By what percent is the area of the circle increased?

Answer: 300 (percent)

6. Compute:

( ) .5 5 32 2 2

Answer: 1

7. What is the least common multiple of 12, 18 and 30?

Answer: 180

8. In any given year, the dates (represented as month/day) 4/4, 6/6, 8/8, 10/10 and 12/12 all fall on the same day of the week. June 3, 2020 is a Wednesday. What day of the week is December 15, 2020?

Answer: Tuesday

9. Taylor wants to buy cases to hold her 86 compact discs. Each case holds 9 discs. How many cases does she need to buy?

Answer: 10 (cases)

10. A board whose length is 84 inches is cut into three pieces in the ratio 1:2:3. What is the number of inches in the length of the shortest piece?

Answer: 14 (inches)

11. How many integers can be represented as a difference of two distinct members of the set {1, 2, 3}?

Answer: 4 (integers)

12. At what time is the sum of the digits which represent the hours and minutes on a 12-hour digital watch the greatest?

Answer: 9:59

13. What is the sum of the coordinates of the midpoint of the segment with endpoints (6, 12) and (0, -6)?

Answer: 6

14. A refrigerator was originally priced at $250. It was then put on sale for 20% off. What is the number of dollars in the final price of the refrigerator if an additional 15% is taken off of the sale price?

Answer: 170 (dollars)

15. What is the sum of all the prime numbers less than 10?

Answer: 17

16. Sixteen is 64% of what number?

Answer: 25

17. Molly has seven U.S. coins with a total value of 88 cents. She does not have any half-dollars. How many dimes does Molly have?

Answer: 1 (dime)

18. What is the number of square units in the area of a triangle whose sides are 3, 4 and 5 units?

Answer: 6 (square units)

19. The point A(-7, 4) is reflected across the x-axis onto point B. Point B is reflected over the y-axis onto point C. What is the sum of the coordinates of point C?

Answer: 3

20. If x = 3 and y = 2, then what

is the value of ?2 3

6

3 2x y

Answer: 7

21. What is the value of

?

12

18

132

1128

1512

4 16 64

256 1024

Answer: 32

22. Set A has 16 elements and set B has 37 elements. The union of sets A and B has 43 elements. How many elements are in the intersection of sets A and B?

Answer: 10 (elements)

23. TV screens are described by the lengths of their diagonals. A 19" TV has a rectangular screen with a diagonal length of 19 inches. The screen of a 20" TV is 12 inches tall. How many inches wide is the screen?

Answer: 16 (inches)

24. Jonathan drove at an average rate of 48 miles per hour. How many miles did he travel in 40 minutes?

Answer: 32 (miles)

25. For what value

of n does ?

Express your answer as a common fraction.

2 2

22

23

2

n

Answer: 5

6

26. What part of 15 hours is 15 seconds? Express your answer as a common fraction.

Answer:1

3600

27. What is the ratio of 1 pound, 4 ounces to 3 pounds, 10 ounces? Express your answer as a common fraction.

Answer:10

29

28. The cost of the daily school lunch increased from $1.50 to $1.95. What was the percent increase?

Answer: 30 (percent)

29. For class president, Tom received 50% of the votes, John received 30% of the votes and Alana received the remaining 88 votes. How many votes did Tom receive?

Answer: 220 (votes)

30. Compute: (2 + 12 + 22 + 32) + (8 + 18 + 28 + 38).

Answer: 160

31. The average of nine consecutive integers is 13. What is the sum of the least and greatest of these integers?

Answer: 26

32. If the sides of a triangle are tripled, then the new area is what percent of the original area?

Answer: 900 (percent)

33. Ervin made 37.5% of the shots he took during his basketball game. If he took exactly 40 shots during the game, how many shots did he make?

Answer: 15 (shots)

34. The numbers 1 through 999, inclusive, are printed on a piece of paper. How many digits are printed on the paper?

Answer: 2889 (digits)

35. How many pairs of prime numbers have a sum of 40?

Answer: 3 (pairs)

36. What is the sum of the first 6 positive odd integers?

Answer: 36

37. What is the greatest real number that is at least as large as its square?

Answer: 1

38. The Catch The Spirit group is conducting a raffle. Each ticket costs $2, and the total expenses are $500. What is the minimum number of tickets that must be sold to yield a profit of $2000?

Answer: 1250 (tickets)

39. What is the sum of the reciprocals of all the positive divisors of 8? Express your answer as a mixed number.

Answer: 1 78

40. If , what is the value of x ?

2 2 220 19 x

Answer: 19

41. Solve for n: .( ) ,2 5 10 0002 2n

Answer: 2

42. Of the following numbers, what is the sum of the two smallest, to the nearest thousandth:

0.15 0.42 0.063 0.1657 ?

Answer: 0.213

43. How many solutions does the

equation have?3 3x x

Answer: 2 (solutions)

44. If n!5! = 6!, then what is value of n ?

Answer: 3

45. If a * b = ab + ba, for all positive integer values of a and b, then what is the value of 4 * 3?

Answer: 145

46. Each bounce of a ball goes as high as the previous bounce. The second bounce was 24 inches high. What was the height, in inches, of the first bounce?

34

Answer: 32 (inches)

47. The perimeter of an isosceles triangle is 36 cm, and the altitude to its base is 12 cm. What is the number of square centimeters in the area of the triangle?

Answer: 60 (square centimeters)

48. There are 30 equally-weighted questions on Mr. Daven’s math final. If a student must score 68% or greater to pass, what is the minimum number of questions that must be answered correctly to pass?

Answer: 21 (questions)

49. A pizza parlor offers six toppings. What is the greatest number of four-topping pizzas that can be made such that no two pizzas have the same topping combination?

Answer: 15 (pizzas)

50. Jacob bought a CD for $15 and sold it for $20. He then bought it back for $25 and sold it again for $28. How many dollars profit did he make?

Answer: 8 (dollars)

51. What is 150% of 0.84, to the nearest hundredth?

Answer: 1.26

52. Given and ,

what is the value of ?

xy 2

3yz 3

2

xz

Answer: 1

53. What is the greatest odd integer that is a factor of 5! ?

Answer: 15

54. What is the number of centimeters in the diameter of a circle whose area is 100 cm2?

Answer: 20 (centimeters)

55. Compute:

55 1212 15 1212 .

Answer: 48,480

56. One leg of a right triangle is increased by 10%, and the other leg is decreased by 10%. By what percent does the area of the triangle decrease?

Answer: 1 (percent)

57. Data can be entered at the rate of 150 pieces of information in 15 minutes. At this rate, how many pieces of information can be entered in hours?1 1

2

Answer: 900 (pieces)

58. A suitcase lock has 3 dials with the digits 0, 1, 2,..., 9 on each. How many different settings are possible if all three digits have to be different?

Answer: 720 (settings)

59. Ralph can do one-third of a job in two-thirds of an hour. At this rate, how many hours will it take him to finish the entire job?

Answer: 2 (hours)

60. Compute:

6 6 6 6 6 .

Answer: 31

61. It is known that

and n is a

positive integer. What is the largest possible value for n?

2006

n

Answer: 33

62. Fifty cards, numbered 1- 50, are placed in a box. One card is randomly selected. What is the probability that the number on the card is prime and is a multiple of 7? Express your answer as a common fraction.

Answer:1

50

63. Tyler’s quiz scores in Math Investigations were 5, 7, 9, 10, 13, 19 and 21. Determine the ratio of the median of his scores to the arithmetic mean of his scores. Express your answer as a common fraction.

Answer:5

6

64. A discount card offers $5 off for any purchase from $50 to $99.99, and $15 off any purchase of $100 or more. What is the maximum percent discount that can be obtained using this card?

Answer: 15 (percent)

65. A rectangle has perimeter 26 inches and integer length sides, in inches. What is the number of square inches in the greatest possible area?

Answer: 42 (square inches)

66. What is 1/2 of 1/3 of 1/5 of 60?

Answer: 2

67. Compute and express as a

common fraction: 34

15

45

12

.

Answer:19

26

68. During a 24-hour period, 1440 cars pass through a toll booth. What is the mean number of cars that pass through per minute?

Answer: 1 (car)

69. What is the greatest common factor of 68 and 92?

Answer: 4

70. The angle measures of the three angles of a triangle are in the ratio 1:3:6. What is the number of degrees in the measure of the largest angle?

Answer: 108 (degrees)

71. What is the least common multiple of 1, 2, 3, 4, 5, 6, 7 and 8?

Answer: 840

72. Express as a common

fraction: 5 49 .

Answer:7

3

73. The number 115 can be written as 12q + r where q and r are integers and What is the value of

0 12 r .q r ?

Answer: 2

74. How many seconds are in hours?1 1

4

Answer: 4500 (seconds)

75. If Mike drinks eight 8-ounce glasses of water each day during 2002, how many gallons of water will he consume? (A gallon is 128 ounces.) Express your answer as a decimal to the nearest tenth.

Answer: 182.5 (gallons)

76. Two opposite sides of a square are each increased by 40%, while the other two sides are each decreased by 30%. The perimeter of the original square is increased by what percent?

Answer: 5 (percent)

77. A book of tickets for 30 games of a local university’s baseball team sold for $120, and a book of tickets for 7 games of the same university’s football team sold for $161. By how many dollars did the average cost of one football game ticket exceed that of one baseball game ticket?

Answer: 19 (dollars)

78. A portion of a number line is divided into 4 equal parts, as shown. What is the value of p, to the nearest ten-thousandth?

0.2304 p 0.4304

Answer: 0.2804

79. Ham- Cheese- Fries Sodas burgers burgersMr. Jones

Needs: 15 12 25 30 1 Meal Deal

includes: 2 2 3 4

What is the least number of Meal Deals that Mr. Jones must purchase to get all of the food and beverage that he needs?

Answer: 9 (Meal Deals)

80. A rectangle has an area of 36 m2 and a width of meters. What is the number of meters in the length of the rectangle?

23

Answer: 54 (meters)