[일반칼럼]미 초등학교 6학년 수학 시험 문제 전문과 해답 입니다.. 한번 해보세요..모든 문제가 주관식영어네요..
작성자candice작성시간03.11.08조회수1,880 목록 댓글 1
미국 뉴져지 주 북부에 위치한 버겐 카운티에서 해마다 실시하는 수학 경시 대회의 수학 문제 입니다. 다음은 6학년을 대상으로한 2002년도의 문제들입니다. 수 천명이 응시하는데 참고로 1등은 50문제 가운데 41문제를 맞추었고 10등은 32문제 정도를 맞추었다고 하는 군요. 주어진 시험 시간은 75분 제한입니다.
보시는 대로 모두 주관식 문제이고 객관식 문제는 단 하나도 없습니다. 미국에선 수학 뿐만 아니라 전 과목이 거의 모두 주관식 문제들이지요. 학교에서나 과학, 수학 경시 대회에서나.
1. What is the remainder when 34 is divided by 35?
2. Carol has hour pentagons, one rectangle, and six triangles. How many total sides do Carol's shapes have?
3. If 'A' is the area of a square with side length 4, determine 5A.
4. Determine W if W=5(1/5 + 1/5 + 1/5 + 1/5).
5. The Knights won exactly 3 of their first 10 games. By winning all of their remaining 'A' games, the Knights ended with victories in exactly half the games they played. What number does 'A' represent?
6. What is the result when 32x36x40x44 is divided by 22x20x18x16?
7. On a standard 12-hour clock, the numbers 12 and 6 are opposite each other. On the strange island Mopland, the people use circular 14-hour clocks with numbers 1 through 14 equally spaced. One pair of opposite numbers on this clock has a sum of 17. If the greater of these two numbers is 'a', what is 15a?
8. An alien from a strange planet has two hands but only nine fingers. One hand has four fingers. The alien puts one hand behind his back and holds up his remaining hand. He then makes three of the fingers on the one visible hand invisible. What is the maximum number of fingers that may now be visible?
9. Determine x if x is not equal to zero and (x+x)/(xx) =1/5.
10. What is 500% of 30% of 20% of 500?
11. Call a number 'N' with at least three distinct factors other than 1 and 'N' supercomposite. How many supercomposite integers are there between 179 and 183 inclusive?
12. If 'p' is a prime and 'k' is an integer, find 'k' if 21p=6k.
13. Determine x if (2/8)(4/16) = x/16.
14. What is the number halfway between 1/6 and 1/8?
15. What is the largest odd factor of 53,328?
16. What percentage of the letters in the phrase "MATH IS FUN" are vowels? Round to the nearest percent.
17. If a/b = 3 and 'b' is not equal to zero, find 5a - 15b.
18. If the multiplication 20 x 30 x 40 x 50 x 60 x 70 is done, in how many zeros does the resulting product end?
19. Registration for a quiz bowl tournament is $35 for one team and $20 for each additional team. The Academy is sending 3 teams to the tournament. If there are 5 members of every team and each player pays the same amount, how much does each pay?
20. Find the whole number 'N' satisfying 3!N! = 6! (K! denotes the product of all the positive integers less than or equal to 'K'.)
21. How many positive 5-digit numbers are there that are each divisible by 5 and have no digit repeated?
22. Evaluate [2003/2002] + [2002/2001] + [2000/1999] if [x] denotes the greatest integers less than or equal to 'x'.
23. A huge rectangular box is 10 feet by 20 feet by 40 feet. If Carrie wants to pack the box with cubes, each 2 feet by 2 feet by 2 feet with no space left over, how many cubes will she need?
24. A fan spins at a constant rate of 5 revolutions per second. How many revolutions will the fan make in 1 minute?
25. What is the height of a triangle with a base of length 37 and an area of 37?
26. Stephen needs to mop a circular floor of radius 10/(root ㅠ) feet and a square floor with side length 5 feet. If it takes Stephen 8 minutes to mop 10 square feet, how many minutes will the job take? ('ㅠ' = pi, 3.141592...)
27. What is the least positive integer divisible by 1,2,3,4,5, and 6?
28. Evaluate: root[(10-0)(9-1)(8-2)(7-3)(6-4)(5-5)].
29. If 4 cubes of side length 8 are glued together, what is the least possible surface area of the resulting figure?
30. What is the sum of all the positive multiples of 5 less than 100?
31. Of the integers -1, -8, -2 and -4, which has the greatest reciprocal?
32. 100 students are designing a 400 page field guide. The students work in groups of 4. If each group takes 15 minutes to write 1 page, how many minutes will it take to complete the field guide if all the students work at the same time?
33. A square with perimeter 24root2 has the same area as an isosceles right triangle. If 'P' is the perimeter of the triangle, find (P - 12root2).
34. Determine 'x' if root36 + root25 + root1 = root(x).
35. A circle of radius 2 passes through the center of a second circle of radius 2. Let 'D' be the distance from the center of one circle to the center of the other. Find the length of the diagonal of a square with side length 'D'.
36. Solve for 'n' if (the 7th power of 5) + (the 7th power of 5) + (the 7th power of 5) + (the 7th power of 5) + (the 7th power of 5) = (the 'n'th power of 5).
37. How many multiples of 2 or 3 (but not both) are there between 97 and 299?
38. Nathan goes shopping for items for his new puppy. At the first dtore he visits, he spends one third of his money on a collar. In the second store, he spends one third of the money he has left on puppy food. Finally, he spends his remaining $12 on dog bowls. How much money, in dollars, did Nathan have at the start?
39. Grace has 1 dollar bill, 1 quarter, 7 dimes, and 1 penny. She wants to buy a candy bar for 77 cents, but the store has no change. Grace decides she will buy the candy bar anyway, paying extra if she cannot make 77 cents exactly. What is the least amount extra that Grace can pay? ( 1 dollar = 100 cents, 1 quarter = 25 cents, 1 dime = 10 cents, 1 penny = 1 cent.)
40. Alana wants to buy an ice cream cone, ice cream, and a topping. If the ice cream costs three times as much as the cone and the topping costs one third the price of the cone, how much must she pay in total if the ice cream costs $3.33?
41. A world famous chef knows how to make over 100 different cake layer flovors. He is making a gigantic cake with 100 layers. If each layer has exactly 1 flovor, what is the greatest number of flavors that may be used so that one flavor must be used at least three times?
42. Daniel is painting a 3 foot by 4 foot rectangular wall. He paints half a square foot per hour. If Daniel paints for 9 straight hours, how many square feet of the wall are left unpainted?
43. Two cars depart from one location at exactly the same time, heading for a picnic 100 miles away. If one car travels toward the destination at a constant rate of 60 miles per hour and the other travels toward the destination at a constant rate of 40 miles per hour, how much time, in minutes, will ellapse between the arrival of the two cars?
44. Which integer from 9995 to 9999 inclusive may be written as the product of two whole numbers whose difference is 4?
45. A movie ticket at a local movie costs $10 for an adult and $7 for a child. A child with $50 wants to take some of his friends to see a movie. However, he must bring along his two parents. If the child uses his own money to pay for himself, his parents ( who are adults), and his friends ( who are all children), what is the maximum number of friends he can bring?
46. A perfect 9th power is a number which can be represented as the product of 9 equal integers. How many of the first 1,000,000,001 positive integers are perfect '9th power's?
47. One of Steve's pet ants is placed on the coordinate plane at the point (1,1). He crawls to the point (2, 1+root3), then to (3,1), and finally back to
(1,1). Determine the area of the triangle Steve's ant has just traced out.
48. The hockey season consists of two parts: the regular season and playoffs. Each team plays 82 games in the regular season. The top teams advance to the playoffs. In the playoffs, two teams play each other until one team wins 4 games. That team moves on to the next round of playoffs. There are a total of 4 rounds in the playoffs. What is the maximum number of games a team can play in the season?
49. Steve and Eve play a game. Five regular coins are flipped, and if an even number of heads come up, Eve gives Steve $10. How much should Eve charge Steve for each game to be fair?
50. Points A, B, C, D are on a straight line in that order, and there is another point 'P' not on this line such that PD = 5/2 PB and AB = CD. Given that the area of triangle PAB is 10, compute the area of triangle PCD.
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1. 35 goes into 34 zero times with a remainder of 34. The remainder is 34.
2. A pentagon has 5 sides, a rectangle has 4 sides, and a triangle has 3 sides, so the answer is 4x5 + 1x4 + 6x3 = 20 + 4 + 18 = 42.
3. A = 4x4 = 16, so 5A = 5x16 = 80.
4. The 5 cancels out all the denominators, so the answer is W = 1 + 1 + 1 + 1 =4.
5. Since the Knights won 3 games, they must have not won 7 games. Hence 7 represents half of the final number of games, or 14. A = 14 - 10 = 4.
6. 32/16 = 2; 36/18 = 2; 40/20 = 2; 44/22 = 2 so the answer is 2x2x2x2 = 16.
7. The desired pair is 12 and 5, so a = 12 and the answer is 12 x 15 = 180.
8. The maximum is achieved when the alien puts the hand with 4 fingers behind his back and makes 3 of the fingers on the hand with 5 fingers invisible, leaving 2 visible.
9. Cross multiplication and division through by 'x' yields x = 10.
10. 20% of 500 is 100; 30% of 100 is 30; 500% of 30 is 150.
11. Note that 179 is prime, 180 = (10)(9)(2), 181 is prime, 182 = (2)(7)(13), and 183 = (3)(61). So 180 and a82 are supercomposite and the answer is 2.
12. The right hand side of the given equation will always be even. Hence the left hand side must be even as well, implying 'p' must be even. This means 'p' = 2 because 2 is the only even prime. So 42 = 6k, from which it follows that k = 7.
13. Reduce the two fractions on the left hand side so it becomes (1/4)(1/4), whisch evaluates tp 1/16. Hence x = 1.
14. We must find the arithmatic mean of the two numbers: (1/6 + 1/8)/2 = 7/48.
15. Divide by 2 ( or appropriate powers of 2) until obtaining an odd number. This gives us 3333.
16. "MATH IS FUN" has 9 letters, of which 3 are vowels. 3/9 to the nearest percent is 33%.
17. Multiplying both sides of the given equation by 'b' gives a = 3b. Multiplying through by 5 yields 5a = 15b so 5a - 15b = 0.
18. Each of the numbers in the multiplication contributes 1 zero except for 50 which, when combined with a 2 from another number, contributes 2 terminal zeros. Hence the answer is 7.
19. The Academy pays $35 + ($20)(2) = $75. There are (3)(5) = 15 players. So each player pays $75/15 = $5.
20. Note that 3! = 6. Dividing both sides through by 6 gives N! = 5! so N = 5.
21. The number either ends in 5 or 0. If it ends in 5, there are 8 choices for the first digit (it can't be 0), 8 choices for the second digit, 7 for the third, and 6 for the fourth, for a total of 8x8x7x6 = 2688. If it ends in 0, there are 9 choices for the first , 8 for the second, 7 for the third, and 6 for the fourth, for a total of 9x8x7x6 = 3024. The total count of numbers is therefore 2688 + 3024 = 5712.
22. [2003/2002] + [2002/2001] + [2000/1999] = 1 + 1 + 1 = 3.
23. The total volume of the box is 8000 cubic feet. Each cube has a volume of 8 cubic heet. Thus Carrie needs 1000 cubes.
24. There are 60 seconds in one minute, so the fan makes (5)(60) = 300 revolutions.
25. Area = 1/2 (base)(height), so 37 =1/2 37(height). Hence the height must be 2.
26. Area = ㅠ(r)(r) + (s)(s) = ㅠ(100/ㅠ) + 5x5 = 100 + 25 = 125 square feet. He can mop 10 square feet in 8 minutes, so the job takes 125(8/10) = 100 minutes.
27. The greatest power of 2 that occurs is 2x2, the greatest power of 3 is 3 to the power of 1, and of 5 is 5 to the power of 1. Hence the answer is (4)(3)(5) = 60.
28. The term (5-5) evaluates to 0, so the entire product is 0, as is the answer.
29. An 8x16x16 rectangular prism will have at least surface area. The answer is (2)(16)(16) + (4)(8)(16) = 1024.
30. 5 + 10 + 15 + ... + 95 = (5(1 + 2 + 3 +... + 19) = 5(19x20/2) = 950.
31. The reciprocals are -1, -1/8, -1/2, -1/4, respectively, and the largest of these is -1/8, so the answer is -8.
32. There are 100/4 = 25 groups. Each group will complete 400/25 = 16 pages since they all work at the same rate. It will take a total of (16)(15) = 240 minutes to complete.
33. Each side of the square has length 24(root2)/4 = 6(root2), which makes the area (6root2)(6root2) = (36)(2) = 72. Let x be the length of the legs of the right triangle. Then (1/2)(x)(x) = 72, so x = 12. By the Pythagorean Theorem, the hypotenuse of the triangle is root(2)(12)(12) = 12(root2). Thus P = 12 + 12 + 12(root2) and P - 12(root2) = 24.
34. root36 + root25 + root1 = 6 + 5 + 1 = 12 so x= 12x12 = 144.
35. Since the center of each circle must pass through the center of the other, D is equal to the radius of the circles, 2. The length of the diagonal of a square with side length 2 is equal to the length of the hypotenuse of a right triangle with two legs of length 2, which is root(2x2 + 2x2) = 2(root2).
36. (the 7th power of 5) + (the 7th power of 5) + (the 7th power of 5) + (the 7th power of 5) + (the 7th power of 5) = 5(the 7th power of 5) = the 8th power of 5, so n = 8.
37. There are 101 multiples of 2 and 67 multiples of 3 between 97 and 299. However, simply adding the two will count the numbers that are multiples of 6, of which there are 33. So the answer is 101 + 67 - 2(33) = 102.
38. $12 represents two thirds of the money he had going into the second store, so he must have had $18 after making his first purchase. $18 represents two thirds of the money Nathan had to start, so he started with $27.
39. Paying with the quarter and 6 dimes, Grace will spend 85 cents, which is 85 - 77 = 8 cents more than would be necessary if she had correct change.
40. Let I, C, and T be the price of the ice cream, the cone, and the topping, respectively. Then I = 3C so $3.33 = 3C and $1.11 = C. Also, T = (1/3)C so T is one third of $1.11, or $0.37. The total is $3.33 + $1.11 + $0.37 = $4.81.
41. Using 50 layer flavors, it is possible that each is used for exactly two layers, which does not satisfy the requirement. With 49 flavors, however, at least one must be used at least 3 times. Hence 49 is the answer.
42. The first car takes 100/60 hours to arrive while the second car takes 100/40 hours. The desired difference is 10/4 - 10/6 = 5/6 hour, or 50 minutes.
44. The number can be written in the form n(n+4). Because n(n+4) = (n)(n) + 4n = (nn + 4n + 4) - 4 = (n+2)square -4, it can be concluded that the number must be 4 less than a perfect square. 9996 is the correct answer because it is 4 less than 10,000, a perfect square, and 9996 = 10000 - 4 = 100 square - 2 square = (100+2)(100-2) = (102)(98).
45. The child must spend 2($10) = $20 on his parents' tickets, which leave $30 to spend on child's tickets. Since each costs $7, he can afford to buy 4, one of which must be for himself. Thus he can bring at most 3 friends.
46. 1,000,000,000 = the 9th power of 10, so there is one perfect 9th power for each of the integers 1 through 10. (The 9th power of 11) > 1,000,000,001, so the answer is 10.
47. Translating the triangle down and to the left 1 unit brings the vertices of the triangle to (0,0), (1, root3), and (2,0). It is clear that this triangle has base 2 and height root3 so the desired area is (1/2)(2)(root3) = root3.
48. Each round of the playoffs can last no longer than 7 games. Since there are four rounds, the playoffs are at most 28 games long. So the longest a season can be is 82 + 28 = 110 games long.
49. Using brute force, you could calculate that probability of an even number of heads is 1/2, or you could reason that the probability of an even number of heads is equal to the probability of an even number of tails, so it must be 1/2. Either way, Steve should win half the time and lose half the time, so Eve should charge $5 to balance out between wins and losses.
50. Let E be the point on the line containing points A, B, C, and D, such that PE is perpendicular to this line. Because the two triangles PAB and PCD have the same altitude, namely PE, and it was given that AB = CD, implying that they have the same base, their areas must be equal. Therefore, 10 is the correct answer.
http://cafe.daum.net/eacie 출처는 요기의 윤택님께서 올리신 글입니다..
보시는 대로 모두 주관식 문제이고 객관식 문제는 단 하나도 없습니다. 미국에선 수학 뿐만 아니라 전 과목이 거의 모두 주관식 문제들이지요. 학교에서나 과학, 수학 경시 대회에서나.
1. What is the remainder when 34 is divided by 35?
2. Carol has hour pentagons, one rectangle, and six triangles. How many total sides do Carol's shapes have?
3. If 'A' is the area of a square with side length 4, determine 5A.
4. Determine W if W=5(1/5 + 1/5 + 1/5 + 1/5).
5. The Knights won exactly 3 of their first 10 games. By winning all of their remaining 'A' games, the Knights ended with victories in exactly half the games they played. What number does 'A' represent?
6. What is the result when 32x36x40x44 is divided by 22x20x18x16?
7. On a standard 12-hour clock, the numbers 12 and 6 are opposite each other. On the strange island Mopland, the people use circular 14-hour clocks with numbers 1 through 14 equally spaced. One pair of opposite numbers on this clock has a sum of 17. If the greater of these two numbers is 'a', what is 15a?
8. An alien from a strange planet has two hands but only nine fingers. One hand has four fingers. The alien puts one hand behind his back and holds up his remaining hand. He then makes three of the fingers on the one visible hand invisible. What is the maximum number of fingers that may now be visible?
9. Determine x if x is not equal to zero and (x+x)/(xx) =1/5.
10. What is 500% of 30% of 20% of 500?
11. Call a number 'N' with at least three distinct factors other than 1 and 'N' supercomposite. How many supercomposite integers are there between 179 and 183 inclusive?
12. If 'p' is a prime and 'k' is an integer, find 'k' if 21p=6k.
13. Determine x if (2/8)(4/16) = x/16.
14. What is the number halfway between 1/6 and 1/8?
15. What is the largest odd factor of 53,328?
16. What percentage of the letters in the phrase "MATH IS FUN" are vowels? Round to the nearest percent.
17. If a/b = 3 and 'b' is not equal to zero, find 5a - 15b.
18. If the multiplication 20 x 30 x 40 x 50 x 60 x 70 is done, in how many zeros does the resulting product end?
19. Registration for a quiz bowl tournament is $35 for one team and $20 for each additional team. The Academy is sending 3 teams to the tournament. If there are 5 members of every team and each player pays the same amount, how much does each pay?
20. Find the whole number 'N' satisfying 3!N! = 6! (K! denotes the product of all the positive integers less than or equal to 'K'.)
21. How many positive 5-digit numbers are there that are each divisible by 5 and have no digit repeated?
22. Evaluate [2003/2002] + [2002/2001] + [2000/1999] if [x] denotes the greatest integers less than or equal to 'x'.
23. A huge rectangular box is 10 feet by 20 feet by 40 feet. If Carrie wants to pack the box with cubes, each 2 feet by 2 feet by 2 feet with no space left over, how many cubes will she need?
24. A fan spins at a constant rate of 5 revolutions per second. How many revolutions will the fan make in 1 minute?
25. What is the height of a triangle with a base of length 37 and an area of 37?
26. Stephen needs to mop a circular floor of radius 10/(root ㅠ) feet and a square floor with side length 5 feet. If it takes Stephen 8 minutes to mop 10 square feet, how many minutes will the job take? ('ㅠ' = pi, 3.141592...)
27. What is the least positive integer divisible by 1,2,3,4,5, and 6?
28. Evaluate: root[(10-0)(9-1)(8-2)(7-3)(6-4)(5-5)].
29. If 4 cubes of side length 8 are glued together, what is the least possible surface area of the resulting figure?
30. What is the sum of all the positive multiples of 5 less than 100?
31. Of the integers -1, -8, -2 and -4, which has the greatest reciprocal?
32. 100 students are designing a 400 page field guide. The students work in groups of 4. If each group takes 15 minutes to write 1 page, how many minutes will it take to complete the field guide if all the students work at the same time?
33. A square with perimeter 24root2 has the same area as an isosceles right triangle. If 'P' is the perimeter of the triangle, find (P - 12root2).
34. Determine 'x' if root36 + root25 + root1 = root(x).
35. A circle of radius 2 passes through the center of a second circle of radius 2. Let 'D' be the distance from the center of one circle to the center of the other. Find the length of the diagonal of a square with side length 'D'.
36. Solve for 'n' if (the 7th power of 5) + (the 7th power of 5) + (the 7th power of 5) + (the 7th power of 5) + (the 7th power of 5) = (the 'n'th power of 5).
37. How many multiples of 2 or 3 (but not both) are there between 97 and 299?
38. Nathan goes shopping for items for his new puppy. At the first dtore he visits, he spends one third of his money on a collar. In the second store, he spends one third of the money he has left on puppy food. Finally, he spends his remaining $12 on dog bowls. How much money, in dollars, did Nathan have at the start?
39. Grace has 1 dollar bill, 1 quarter, 7 dimes, and 1 penny. She wants to buy a candy bar for 77 cents, but the store has no change. Grace decides she will buy the candy bar anyway, paying extra if she cannot make 77 cents exactly. What is the least amount extra that Grace can pay? ( 1 dollar = 100 cents, 1 quarter = 25 cents, 1 dime = 10 cents, 1 penny = 1 cent.)
40. Alana wants to buy an ice cream cone, ice cream, and a topping. If the ice cream costs three times as much as the cone and the topping costs one third the price of the cone, how much must she pay in total if the ice cream costs $3.33?
41. A world famous chef knows how to make over 100 different cake layer flovors. He is making a gigantic cake with 100 layers. If each layer has exactly 1 flovor, what is the greatest number of flavors that may be used so that one flavor must be used at least three times?
42. Daniel is painting a 3 foot by 4 foot rectangular wall. He paints half a square foot per hour. If Daniel paints for 9 straight hours, how many square feet of the wall are left unpainted?
43. Two cars depart from one location at exactly the same time, heading for a picnic 100 miles away. If one car travels toward the destination at a constant rate of 60 miles per hour and the other travels toward the destination at a constant rate of 40 miles per hour, how much time, in minutes, will ellapse between the arrival of the two cars?
44. Which integer from 9995 to 9999 inclusive may be written as the product of two whole numbers whose difference is 4?
45. A movie ticket at a local movie costs $10 for an adult and $7 for a child. A child with $50 wants to take some of his friends to see a movie. However, he must bring along his two parents. If the child uses his own money to pay for himself, his parents ( who are adults), and his friends ( who are all children), what is the maximum number of friends he can bring?
46. A perfect 9th power is a number which can be represented as the product of 9 equal integers. How many of the first 1,000,000,001 positive integers are perfect '9th power's?
47. One of Steve's pet ants is placed on the coordinate plane at the point (1,1). He crawls to the point (2, 1+root3), then to (3,1), and finally back to
(1,1). Determine the area of the triangle Steve's ant has just traced out.
48. The hockey season consists of two parts: the regular season and playoffs. Each team plays 82 games in the regular season. The top teams advance to the playoffs. In the playoffs, two teams play each other until one team wins 4 games. That team moves on to the next round of playoffs. There are a total of 4 rounds in the playoffs. What is the maximum number of games a team can play in the season?
49. Steve and Eve play a game. Five regular coins are flipped, and if an even number of heads come up, Eve gives Steve $10. How much should Eve charge Steve for each game to be fair?
50. Points A, B, C, D are on a straight line in that order, and there is another point 'P' not on this line such that PD = 5/2 PB and AB = CD. Given that the area of triangle PAB is 10, compute the area of triangle PCD.
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1. 35 goes into 34 zero times with a remainder of 34. The remainder is 34.
2. A pentagon has 5 sides, a rectangle has 4 sides, and a triangle has 3 sides, so the answer is 4x5 + 1x4 + 6x3 = 20 + 4 + 18 = 42.
3. A = 4x4 = 16, so 5A = 5x16 = 80.
4. The 5 cancels out all the denominators, so the answer is W = 1 + 1 + 1 + 1 =4.
5. Since the Knights won 3 games, they must have not won 7 games. Hence 7 represents half of the final number of games, or 14. A = 14 - 10 = 4.
6. 32/16 = 2; 36/18 = 2; 40/20 = 2; 44/22 = 2 so the answer is 2x2x2x2 = 16.
7. The desired pair is 12 and 5, so a = 12 and the answer is 12 x 15 = 180.
8. The maximum is achieved when the alien puts the hand with 4 fingers behind his back and makes 3 of the fingers on the hand with 5 fingers invisible, leaving 2 visible.
9. Cross multiplication and division through by 'x' yields x = 10.
10. 20% of 500 is 100; 30% of 100 is 30; 500% of 30 is 150.
11. Note that 179 is prime, 180 = (10)(9)(2), 181 is prime, 182 = (2)(7)(13), and 183 = (3)(61). So 180 and a82 are supercomposite and the answer is 2.
12. The right hand side of the given equation will always be even. Hence the left hand side must be even as well, implying 'p' must be even. This means 'p' = 2 because 2 is the only even prime. So 42 = 6k, from which it follows that k = 7.
13. Reduce the two fractions on the left hand side so it becomes (1/4)(1/4), whisch evaluates tp 1/16. Hence x = 1.
14. We must find the arithmatic mean of the two numbers: (1/6 + 1/8)/2 = 7/48.
15. Divide by 2 ( or appropriate powers of 2) until obtaining an odd number. This gives us 3333.
16. "MATH IS FUN" has 9 letters, of which 3 are vowels. 3/9 to the nearest percent is 33%.
17. Multiplying both sides of the given equation by 'b' gives a = 3b. Multiplying through by 5 yields 5a = 15b so 5a - 15b = 0.
18. Each of the numbers in the multiplication contributes 1 zero except for 50 which, when combined with a 2 from another number, contributes 2 terminal zeros. Hence the answer is 7.
19. The Academy pays $35 + ($20)(2) = $75. There are (3)(5) = 15 players. So each player pays $75/15 = $5.
20. Note that 3! = 6. Dividing both sides through by 6 gives N! = 5! so N = 5.
21. The number either ends in 5 or 0. If it ends in 5, there are 8 choices for the first digit (it can't be 0), 8 choices for the second digit, 7 for the third, and 6 for the fourth, for a total of 8x8x7x6 = 2688. If it ends in 0, there are 9 choices for the first , 8 for the second, 7 for the third, and 6 for the fourth, for a total of 9x8x7x6 = 3024. The total count of numbers is therefore 2688 + 3024 = 5712.
22. [2003/2002] + [2002/2001] + [2000/1999] = 1 + 1 + 1 = 3.
23. The total volume of the box is 8000 cubic feet. Each cube has a volume of 8 cubic heet. Thus Carrie needs 1000 cubes.
24. There are 60 seconds in one minute, so the fan makes (5)(60) = 300 revolutions.
25. Area = 1/2 (base)(height), so 37 =1/2 37(height). Hence the height must be 2.
26. Area = ㅠ(r)(r) + (s)(s) = ㅠ(100/ㅠ) + 5x5 = 100 + 25 = 125 square feet. He can mop 10 square feet in 8 minutes, so the job takes 125(8/10) = 100 minutes.
27. The greatest power of 2 that occurs is 2x2, the greatest power of 3 is 3 to the power of 1, and of 5 is 5 to the power of 1. Hence the answer is (4)(3)(5) = 60.
28. The term (5-5) evaluates to 0, so the entire product is 0, as is the answer.
29. An 8x16x16 rectangular prism will have at least surface area. The answer is (2)(16)(16) + (4)(8)(16) = 1024.
30. 5 + 10 + 15 + ... + 95 = (5(1 + 2 + 3 +... + 19) = 5(19x20/2) = 950.
31. The reciprocals are -1, -1/8, -1/2, -1/4, respectively, and the largest of these is -1/8, so the answer is -8.
32. There are 100/4 = 25 groups. Each group will complete 400/25 = 16 pages since they all work at the same rate. It will take a total of (16)(15) = 240 minutes to complete.
33. Each side of the square has length 24(root2)/4 = 6(root2), which makes the area (6root2)(6root2) = (36)(2) = 72. Let x be the length of the legs of the right triangle. Then (1/2)(x)(x) = 72, so x = 12. By the Pythagorean Theorem, the hypotenuse of the triangle is root(2)(12)(12) = 12(root2). Thus P = 12 + 12 + 12(root2) and P - 12(root2) = 24.
34. root36 + root25 + root1 = 6 + 5 + 1 = 12 so x= 12x12 = 144.
35. Since the center of each circle must pass through the center of the other, D is equal to the radius of the circles, 2. The length of the diagonal of a square with side length 2 is equal to the length of the hypotenuse of a right triangle with two legs of length 2, which is root(2x2 + 2x2) = 2(root2).
36. (the 7th power of 5) + (the 7th power of 5) + (the 7th power of 5) + (the 7th power of 5) + (the 7th power of 5) = 5(the 7th power of 5) = the 8th power of 5, so n = 8.
37. There are 101 multiples of 2 and 67 multiples of 3 between 97 and 299. However, simply adding the two will count the numbers that are multiples of 6, of which there are 33. So the answer is 101 + 67 - 2(33) = 102.
38. $12 represents two thirds of the money he had going into the second store, so he must have had $18 after making his first purchase. $18 represents two thirds of the money Nathan had to start, so he started with $27.
39. Paying with the quarter and 6 dimes, Grace will spend 85 cents, which is 85 - 77 = 8 cents more than would be necessary if she had correct change.
40. Let I, C, and T be the price of the ice cream, the cone, and the topping, respectively. Then I = 3C so $3.33 = 3C and $1.11 = C. Also, T = (1/3)C so T is one third of $1.11, or $0.37. The total is $3.33 + $1.11 + $0.37 = $4.81.
41. Using 50 layer flavors, it is possible that each is used for exactly two layers, which does not satisfy the requirement. With 49 flavors, however, at least one must be used at least 3 times. Hence 49 is the answer.
42. The first car takes 100/60 hours to arrive while the second car takes 100/40 hours. The desired difference is 10/4 - 10/6 = 5/6 hour, or 50 minutes.
44. The number can be written in the form n(n+4). Because n(n+4) = (n)(n) + 4n = (nn + 4n + 4) - 4 = (n+2)square -4, it can be concluded that the number must be 4 less than a perfect square. 9996 is the correct answer because it is 4 less than 10,000, a perfect square, and 9996 = 10000 - 4 = 100 square - 2 square = (100+2)(100-2) = (102)(98).
45. The child must spend 2($10) = $20 on his parents' tickets, which leave $30 to spend on child's tickets. Since each costs $7, he can afford to buy 4, one of which must be for himself. Thus he can bring at most 3 friends.
46. 1,000,000,000 = the 9th power of 10, so there is one perfect 9th power for each of the integers 1 through 10. (The 9th power of 11) > 1,000,000,001, so the answer is 10.
47. Translating the triangle down and to the left 1 unit brings the vertices of the triangle to (0,0), (1, root3), and (2,0). It is clear that this triangle has base 2 and height root3 so the desired area is (1/2)(2)(root3) = root3.
48. Each round of the playoffs can last no longer than 7 games. Since there are four rounds, the playoffs are at most 28 games long. So the longest a season can be is 82 + 28 = 110 games long.
49. Using brute force, you could calculate that probability of an even number of heads is 1/2, or you could reason that the probability of an even number of heads is equal to the probability of an even number of tails, so it must be 1/2. Either way, Steve should win half the time and lose half the time, so Eve should charge $5 to balance out between wins and losses.
50. Let E be the point on the line containing points A, B, C, and D, such that PE is perpendicular to this line. Because the two triangles PAB and PCD have the same altitude, namely PE, and it was given that AB = CD, implying that they have the same base, their areas must be equal. Therefore, 10 is the correct answer.
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