MATHCOUNTS

2004 School Competition

Countdown Round

1. Marie owns 103 department stores. Each store has 106 shelves for towels. What is the total number of shelves for towels?

Answer: 10,918 (shelves)

2. When 5 is subtracted from three times a number, the result is 7. What is the number?

Answer: 4

3. The Battling Hummingbirds have won 40% of their first 60 games. If they win their next 12 games in a row, what percent of all of their games will they have won?

Answer: 50 (percent)

4. A 3-foot tall flower-stand casts a 9-foot shadow. At the same time, a nearby tree casts a 51-foot shadow. How many feet tall is the tree?

Answer: 17 (feet)

5. Two boxes, A and B, contain pennies. In box A, there are nine more pennies than in box B. The total number of pennies in boxes A and B is 41. What is the number of pennies in box A?

Answer: 25 (pennies)

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

Answer: 25

7. Trae is searching for a treasure on the beach. From point A, he is told to walk two meters north, three meters east, and then two meters north to point B. In meters, what is the length of ?AB

Answer: 5 (meters)

8. Mr. Cole is driving at his favorite speed of 55 miles per hour. He is 125 miles from his destination. His goal is to arrive at his destination in two hours. How many miles per hour faster than his preferred speed must he drive to meet his goal? Express your answer as a decimal to the nearest tenth.

Answer: 7.5 (miles per hour)

9. What number is one hundred less than one thousand thirty-three?

Answer: 933

10. What is the degree measure of the angle between the hour hand and the minute hand of a clock at exactly 2:30 p.m. on a 12-hour analog clock?

Answer: 105 (degrees)

11. Katrina can bicycle 50 miles in four hours. At the same speed, how many miles does she bicycle in six hours?

Answer: 75 (miles)

12. What is the least positive integer that has exactly three distinct factors?

Answer: 4

13. The cube of what integer is closest to 800?

Answer: 9

14. In a state senate, there are 24 Democrats, 32 Republicans, and 4 Independents. If one senator is selected at random, what is the probability that the senator is a Republican? Express your answer as a common fraction.

Answer: 8

15

15. The sum of x and y is 11 and the positive difference between x and y is 3. What is the value of the product of x and y?

Answer: 28

16. The total value of 90 pennies, 20 dimes and 12 quarters is what percent of the value of a ten-dollar bill?

Answer: 59 (percent)

17. Solve for x: 18 x = 10 + x.

Answer: 4

18. An 8 cm by N cm rectangle has a diagonal of length 10 cm. What is the number of centimeters in the perimeter of this rectangle?

Answer: 28 (centimeters)

19. The histogram shows the numbers of types of the 50 pets reported by the 7th graders in a school. What percent of the pets are cats? 20

15

10

5

Dogs Cats Fish Birds

Answer: 40 (percent)

20. What is the sum of the reciprocals of the distinct positive factors of 8? Express your answer as a mixed number.

Answer: 1 78

21. What is the number of degrees in the measure of angle ABD in square ABCD?

Answer: 45 (degrees)

22. What is the units digit of 62004?

Answer: 6

23. Triangle ABC is isosceles with angle B congruent to angle C. The measure of angle C is four times the measure of angle A. What is the number of degrees in the measure of angle B?

Answer: 80 (degrees)

24. What integer is closest

to ?100

7

Answer: 14

25. Sixty percent of a number is 40 less than the number. What is the number?

Answer: 100

26. What is the least positive three-digit multiple of 24?

Answer: 120

27. The Carters had a 10-foot by 10-foot patio built at a cost of $2200. How many dollars per square foot did the patio cost to build?

Answer: 22 (dollars per square foot)

28. Jenna has $1.75 in nickels and dimes. She has twice as many dimes as nickels. How many dimes does Jenna have?

Answer: 14 (dimes)

29. What is the decimal

equivalent of ? Express

your answer to the nearest

hundredth.

11

4

Answer: 2.75

30. The ratio of the measures of two complementary angles is 7:11. What is the measure, in degrees, of the larger angle?

Answer: 55 (degrees)

31. Simplify: . Express your answer as a common fraction.

( )23

3

Answer:8

27

32. From a one meter piece of yarn, Chris cuts a 31.5 cm piece and a 35.5 cm piece. In centimeters, how much yarn is left?

Answer: 33 (centimeters)

33. Simplify:

7

Answer: 7467

34. A stair-step exercise machine indicates that Leon burns 180 calories in 12 minutes. At the same rate, how many calories would Leon burn in a 4-minute workout?

Answer: 60 (calories)

35. If f (x) = x2 1, what is the value of f 

Answer: 0

36. Four friends are in line for movie tickets. Rick is behind Tom, but ahead of Mark. James and Tom are separated by one person. Who is last in line?

Answer: Mark

37. When the expression 7 + 4(x + 5) x) is simplified, what is the value of the coefficient of the x-term?

Answer: 12

38. Steven is making a scale drawing of his living room. He is using a scale of one-quarter inch to one foot. In the drawing, how many inches will represent the length of a 72-inch long couch? Express your answer as a decimal to the nearest tenth.

Answer: 1.5 (inches)

39. The ages, in years, of the ten people in the math club are: 12, 12, 12, 12, 13, 13, 13, 14, 14, 39. What is the mode of the ages?

Answer: 12 (years)

40. What is the greatest prime factor of 10!?

Answer: 7

41. Two numbers are in the ratio of 5:7. Their sum is 108. What is the lesser number?

Answer: 45

42. Rebecca remembered the first six digits of a friend’s 7-digit phone number. If she randomly selects a digit to replace the last digit she can’t remember, what is the probability she will dial the correct number? Express your answer as a common fraction.

Answer:1

10

43. Given: . What is the

value of the ratio ? Express

your answer as a common

fraction.

x

y

8

3

y

x2

Answer:3

16

44. The categories and percentages for 536 million juvenile books are given in the table. Of these juvenile books, how many books were Nonfiction? Fiction 55%

Nonfiction 25%Educational 10%Other 10%

Answer: 134,000,000 (books)

45. Twenty-five percent of what number is 12?

Answer: 48

46. What is the number of square centimeters in the area of a circle with radius 4 cm? Express your answer is terms of

Answer: 16(square centimeters)

47. A candy company puts 21 candies in each bag. There are only red and blue candies in each bag. For every two red candies in a bag there are five blue candies. How many red candies are in each bag?

Answer: 6 (candies)

48. How many positive three-digit integers are multiples of 11?

Answer: 81 (integers)

49. Express the value of

as a common fraction.

2

2

1313

Answer:5

7

50. Joshua flips three coins. What is the probability that they are neither all heads nor all tails? Express your answer as a common fraction.

Answer:3

4

51. How many distinct prime factors does 52 32have?

Answer: 1 (factor)

52. How many triangles of any size are in this figure?

Answer: 5 (triangles)

53. What is 150% of 26?

Answer: 39

54. Each bottle of perfume at The Perfume Parlor contains ounces of perfume. How many of these bottles of perfume could be made from 96 ounces of perfume?

316

Answer: 512 (bottles)

55. In how many ways can a pair of students be selected from a group of five students?

Answer: 10 (ways)

56. In the formula d = rt, what is the value of r if d = 60 and t = 4?

Answer: 15

57. On average, each American consumes 286 eggs per year. What is the number of years it would take, at this rate, for an American to eat 2000 eggs? Express your answer to the nearest whole number.

Answer: 7 (years)

58. Of the 80 students at camp, 37.5% of them are male. What is the number of students at camp who are female?

Answer: 50 (students)

59. One fair standard six-sided die is rolled. What is the probability of rolling a number greater than 4? Express your answer as a common fraction.

Answer:1

3

60. There are 16 boys in Mr. Smith’s class of 34 students. What is the ratio of boys to girls? Express your answer as a common fraction.

Answer:8

9