1、2013 AMC8 Problems 1. Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way? 2. A sign at the fish market says, “50% off, today only:
2、half-pound packages for just $3 per package.“ What is the regular price for a full pound of fish, in dollars? What is the value of ? 3. 4. Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her
3、 portion of the total bill. What was the total bill? 5. Hammie is in the grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds? 6. The number
4、 in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, . What is the missing number in the top row? 7. Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds.
5、 It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train? 8. A fair coin is tossed 3 times. What is the probability of at least two consecutive heads? 9. The Incredible Hulk can double the distance h
6、e jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer? 10. What is the ratio of the least common multiple of 180 and 594 to the greatest common factor
7、 of 180 and 594? 11. Teds grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour,
8、he would have spent less time on the treadmill. How many minutes less? 12. At the 2013 Winnebago County Fair a vendor is offering a “fair special“ on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular pri
9、ce. Javier took advantage of the “fair special“ to buy three pairs of sandals. What percentage of the $150 regular price did he save? 13. When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have
10、differed from the correct one? 14. Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match? 15. If , , and , what is the product of , , and ?
11、16. A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of -graders to -graders is , and the the ratio of -graders to -graders is . What is the smallest number of students that could be participating in the project? 17. The sum of six consecuti
12、ve positive integers is 2013. What is the largest of these six integers? 18. Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain? 19. Bridget,
13、 Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie dont show theirs to anyone. Cassie says, I didnt get the lowest score in our class, and Bridget adds, I didnt get the highest score. What is the ranking of the thr
14、ee girls from highest to lowest? 20. A rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of the semicircle? 21. Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast
15、 corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take? 22. Toothpicks are u
16、sed to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether? 23. Angle of is a right angle. The sides of are the diameters of semicircles as shown. The area of the semicircle on equals , and the arc of the semicircle on has length . What is the radiu
17、s of the semicircle on ? 24. Squares , , and are equal in area. Points and are the midpoints of sides and , respectively. What is the ratio of the area of the shaded pentagon to the sum of the areas of the three squares? 25. A ball with diameter 4 inches starts at point A to roll along the track sho
18、wn. The track is comprised of 3 semicircular arcs whose radii are inches, inches, and inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B? 2013 AMC8 Problems/Solutions 1. Problem Da
19、nica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way? Solution: In order to have her model cars in perfect, complete rows of 6, Danica
20、must have a number of cars that is a multiple of 6. The smallest multiple of 6 which is larger than 23 is 24, so shell need to buy more model car. 2. A sign at the fish market says, “50% off, today only: half-pound packages for just $3 per package.“ What is the regular price for a full pound of fish
21、, in dollars? Problem Solution: The 50% off price of half a pound of fish is $3, so the 100%, or the regular price, of a half pound of fish is $6. Consequently, if half a pound of fish costs $6, then a whole pound of fish is dollars. What is the value of ? 3. Problem Notice that we can pair up every
22、 two numbers to make a sum of 1: Solution Therefore, the answer is . 4. Problem Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill? Each o
23、f her seven friends paid to cover Judis portion. Therefore, Judis portion must be . Since Judi was supposed to pay of the total bill, the total bill must be . Solution 5. Hammie is in the grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is gr
24、eater, the average (mean) weight of these five children or the median weight, and by how many pounds? Problem Lining up the numbers (5, 5, 6, 8, 106), we see that the median weight is 6 pounds. Solution The average weight of the five kids is . Therefore, the average weight is bigger, by pounds, maki
25、ng the answer . 6. The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, . What is the missing number in the top row? Problem Solution Let the value in the empty box in the middle row be , and the value in the empty box in the top ro
26、w be . is the answer were looking for. Solution 1: Working Backwards We see that , making . It follows that , so . Another way to do this problem is to realize what makes up the bottommost number. This method doesnt work quite as well for this problem, but in a larger tree, it might be faster. (In t
27、his case, Solution 1 would be faster since theres only two missing numbers.) Solution 2: Jumping Back to the Start Again, let the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer were looking for. We can write some equations: Now we can s
28、ubstitute into the first equation using the two others: 7. Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the
29、following was the most likely number of cars in the train? Problem If Trey saw , then he saw . Solution 1 2 minutes and 45 seconds can also be expressed as seconds. Treys rate of seeing cars, , can be multiplied by on the top and bottom (and preserve the same rate): . It follows that the most likely
30、 number of cars is . 2 minutes and 45 seconds is equal to . Solution 2 Since Trey probably counts around 6 cars every 10 seconds, there are groups of 6 cars that Trey most likely counts. Since , the closest answer choice is . 8. A fair coin is tossed 3 times. What is the probability of at least two
31、consecutive heads? Problem First, there are ways to flip the coins, in order. Solution The ways to get two consecutive heads are HHT and THH. The way to get three consecutive heads is HHH. Therefore, the probability of flipping at least two consecutive heads is . 9. The Incredible Hulk can double th
32、e distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer? Problem This is a geometric sequence in which the common ratio is 2. To find the ju
33、mp that would be over a 1000 meters, we note that . Solution However, because the first term is and not , the solution to the problem is 10. What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594? Problem To find either the LCM or the GCF of two nu
34、mbers, always prime factorize first. Solution 1 The prime factorization of . The prime factorization of . Then, find the greatest power of all the numbers there are; if one number is one but not the other, use it (this is ). Multiply all of these to get 5940. For the GCF of 180 and 594, use the leas
35、t power of all of the numbers that are in both factorizations and multiply. = 18. Thus the answer = = . We start off with a similar approach as the original solution. From the prime factorizations, the GCF is 18. Similar Solution It is a well known fact that . So we have, . Dividing by 18 yields . T
36、herefore, . 11. Teds grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he wo
37、uld have spent less time on the treadmill. How many minutes less? Problem We use that fact that . Let d= distance, r= rate or speed, and t=time. In this case, let represent the time. Solution On Monday, he was at a rate of . So, . For Wednesday, he walked at a rate of . Therefore, . On Friday, he wa
38、lked at a rate of . So, . Adding up the hours yields + + = . We now find the amount of time Grandfather would have taken if he walked at per day. Set up the equation, . To find the amount of time saved, subtract the two amounts: - = . To convert this to minutes, we multiply by 60. Thus, the solution
39、 to this problem is 12. At the 2013 Winnebago County Fair a vendor is offering a “fair special“ on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the “fair special“ to
40、buy three pairs of sandals. What percentage of the $150 regular price did he save? Problem First, find the amount of money one will pay for three sandals without the discount. We have . Solution Then, find the amount of money using the discount: . Finding the percentage yields . To find the percent
41、saved, we have 13. Problem When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one? Let the two digits be and . Solution The correct score was . Clara misinterprete
42、d it as . The difference between the two is which factors into . Therefore, since the difference is a multiple of 9, the only answer choice that is a multiple of 9 is . 14. Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each random
43、ly picks a jelly bean to show the other. What is the probability that the colors match? Problem The probability that both show a green bean is . The probability that both show a red bean is . Therefore the probability is Solution 15. If , , and , what is the product of , , and ? Problem Solution The
44、refore, . Therefore, . To most people, it would not be immediately evident that , so we can multiply 6s until we get the desired number: , so . Therefore the answer is 16. A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of -graders to -grad
45、ers is , and the the ratio of -graders to -graders is . What is the smallest number of students that could be participating in the project? Problem Solution We multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8th graders, in order that we can put the
46、two ratios together: Solution 1: Algebra Therefore, the ratio of 8th graders to 7th graders to 6th graders is . Since the ratio is in lowest terms, the smallest number of students participating in the project is . The number of 8th graders has to be a multiple of 8 and 5, so assume it is 40 (the sma
47、llest possibility). Then there are 6th graders and 7th graders. The numbers of students is Solution 2: Fakesolving 17. The sum of six consecutive positive integers is 2013. What is the largest of these six integers? Problem The mean of these numbers is . Therefore the numbers are , so the answer is
48、Solution 1 Let the number be . Then our desired number is . Solution 2 Our integers are , so we have that . Let the first term be . Our integers are . We have, Solution 3 18. Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The fl
49、oor and the four walls are all one foot thick. How many blocks does the fort contain? Problem There are cubes on the base of the box. Then, for each of the 4 layers above the bottom (as since each cube is 1 foot by 1 foot by 1 foot and the box is 5 feet tall, there are 4 feet left), there are cubes. Hence, the answer is . Solution 1 We can just calculate the volume of the prism that was cut out of the original box. Each interior side of the fort will be 2 feet shorter than each side of the outside. Since the floor is 1 foot, the height will be 4 feet. So the volume of the int
