2004 AMC amc8 真题 答案 详解

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中英双语真题

2004 AMC8
Problem 1
On a map, a 12-centimeter length represents 72 kilometers. How many kilometers does a 17-centimeter length represent?
在一张地图上，12厘米长代表72公里。则17厘米长代表多少公里？
(A) 6 (B) 102 (C) 204 (D) 864 (E) 1224

Problem 2
How many different four-digit numbers can be formed by rearranging the four digits in 2004?
将2004的四个数字重新排列可以形成多少个不同的四位数？
(A) 4 (B) 6 (C) 16 (D) 24 (E) 81

Problem 3
Twelve friends met for dinner at Oscar's Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for 18 people. If they shared, how many meals should they have ordered to have just enough food for the 12 of them?
12位朋友在奥斯卡的“牡蛎屋”共进晚餐，每人点了1份晚餐。食物份量很大，足够18人吃。如果他们一起分享晚餐，他们本应该只需点多少份就恰好够12人吃？
(A) 8 (B) 9 (C) 10 (D) 15 (E) 18

Problem 4
Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. Lance, Sally, Joy, and Fred are chosen for the team. In how many ways can the three starters be chosen?
Hamilton女士的八年级班级想参加一年一度的三人篮球团体赛。Lance、Sally、Joy和Fred被选为队员。要从中选择三名作为先发球员，一共有多少种方法？
(A) 2 (B) 4 (C) 6 (D) 8 (E) 10

英文真题

Compiled on: January 1, 2021 AMC8 2004 Problems Page 77 of 109 Q1. On a map, a 12-centimeter length represents 72 kilometers. How many kilometers does a 17-centimeter length represent? A) 6 B) 102 C) 204 D) 864 E) 1224 Q2. How many different four-digit numbers can be formed by rearranging the four digits in 2004? A) 4 B) 6 C) 16 D) 24 E) 81 Q3. Twelve friends met for dinner at Oscar's Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for 18 people. If they shared, how many meals should they have ordered to have just enough food for the 12 of them? A) 8 B) 9 C) 10 D) 15 E) 18 Q4. Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. Lance, Sally, Joy, and Fred are chosen for the team. In how many ways can the three starters be chosen? A) 2 B) 4 C) 6 D) 8 E) 10 Q5. The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner? A) 4 B) 7 C) 8 D) 15 E) 16 Q6. After Sally takes 20 shots, she has made 55% of her shots. After she takes 5 more shots, she raises her percentage to 56%. How many of the last 5 shots did she make? A) 1 B) 2 C) 3 D) 4 E) 5 Q7. An athlete's target heart rate, in beats per minute, is 80% of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from 220. To the nearest whole number, what is the target heart rate of an athlete who is 26 years old? A) 134 B) 155 C) 176 D) 194 E) 243 Q8. Find the number of two-digit positive integers whose digits total 7. A) 6 B) 7 C) 8 D) 9 E) 10 Q9. The average of the five numbers in a list is 54. The average of the first two numbers is 48. What is the average of the last three numbers? A) 55 B) 56 C) 57 D) 58 E) 59 Q10. Handy Aaron helped a neighbor (1\frac{1}{4}) hours on Monday, 50 minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid $3 per hour. How much did he earn for the week? A) $8 B) $9 C) $10 D) $12 E) $15 Q11. The numbers -2, 4, 6, 9 and 12 are rearranged according to these rules:

真题文字详解(英文)

Problem 1 On a map, a 12-centimeter length represents 72 kilometers. How many kilometers does a 17-centimeter length represent?
(A) 6 (B) 102 (C) 204 (D) 864 (E) 1224

Solution
We set up the proportion ( \frac{12cm}{72km} = \frac{17cm}{x km} ). Thus ( x = 102 \Rightarrow \boxed{\text{(B)}\ 102} ).

Solution 2
Since we know 12 centimeters on the map is equal to 72 kilometers, 1 centimeter on the map is simply ( \frac{72}{12} ), which is 6. The problem is asking how many kilometers a 17-centimeter length represents, so we multiply 6 by 17 which is simply 102, your answer.

Problem 2 How many different four-digit numbers can be formed by rearranging the four digits in 2004?
(A) 4 (B) 6 (C) 16 (D) 24 (E) 81

Solution 1
We can solve this problem easily, just by calculating how many choices there are for each of the four digits. First off, we know there are only 2 choices for the first digit, because 0 isn't a valid choice, or the number would be a 3-digit number, which is not what we want. We have 3 choices for the second digit, since we already used up one of the digits, and 2 choices for the third, and finally just 1 choices for the fourth and final one. ( 2 * 3 * 2 * 1 = 12 ), but there are 2 zeros that have been counted as different numbers, so divide by 2 to get ( \boxed{\text{(B)}\ 6} ).

Solution 2
Note that the four-digit number must start with either a 2 or a 4. The four-digit numbers that start with 2 are 2400, 2040, and 2004. The four-digit numbers that start with 4 are 4200, 4020, and 4002 which gives us a total of ( \boxed{\text{(B)}\ 6} ).

Solution 3
In order for the resulting numbers to have four digits, they cannot start with 0. Therefore, both zeroes must be in the last three places. There are ( \binom{3}{2} = 3 ) ways to choose which two of the last three places are zeroes. Then there are ( 2 * 1 = 2 ) ways to arrange the 2 and the 4 in the remaining two places, giving us a total of ( 3 * 2 = \boxed{\text{(B)}\ 6} ).

Problem 3 Twelve friends met for dinner at Oscar's Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for 18 people. If they shared, how many meals should they have ordered to have just enough food for the 12 of them?
(A) 8 (B) 9 (C) 10 (D) 15 (E) 18

Solution
Set up the proportion ( \frac{12 meals}{18 people} = \frac{x meals}{12 people} ). Solving for ( x ) gives us ( x = \boxed{\text{(A)}\ 8} ).

Problem 4

真题文字详解(中英双语)

Problem 1 On a map, a 12-centimeter length represents 72 kilometers. How many kilometers does a 17-centimeter length represent?
在地图上，12厘米长度代表72千米。那么17厘米长度代表多少千米？
(A) 6 (B) 102 (C) 204 (D) 864 (E) 1224

Solution We set up the proportion ( \frac{12cm}{72km} = \frac{17cm}{x km} ). Thus ( x = 102 \Rightarrow ) (B) 102
我们设立比例 ( \frac{12cm}{72km} = \frac{17cm}{x km} )。因此 ( x = 102 \Rightarrow ) (B) 102

Solution 2 Since we know 12 centimeters on the map is equal to 72 kilometers, 1 centimeter on the map is simply ( \frac{72}{12} ), which is 6. The problem is asking how many kilometers a 17-centimeter length represents, so we multiply 6 by 17 which is simply 102, your answer.
由于我们知道12厘米在地图上等于72千米，1厘米在地图上就是 ( \frac{72}{12} )，即6。问题是问17厘米长度代表多少千米，所以我们乘以6得到17，即102，这就是你的答案。

Problem 2 How many different four-digit numbers can be formed by rearranging the four digits in 2004?
通过重新排列2004中的四个数字，可以组成多少个不同的四位数？
(A) 4 (B) 6 (C) 16 (D) 24 (E) 81

Solution 1 We can solve this problem easily, just by calculating how many choices there are for each of the four digits. First off, we know there are only 2 choices for the first digit, because 0 isn't a valid choice, or the number would be a 3-digit number, which is not what we want. We have 3 choices for the second digit, since we already used up one of the digits, and 2 choices for the third, and finally just 1 choice for the fourth and final one. ( 2 * 3 * 2 * 1 ) is 12, but there are 2 zeros that have been counted as different numbers, so divide by 2 to get (B) 6.
我们可以很容易地解决这个问题，只需计算每个四位数的每个数字的选择有多少。首先，我们知道第一个数字只有2种选择，因为0不是一个有效的选择，否则这个数将是一个三位数，这不是我们想要的。我们有3种选择用于第二个数字，因为我们已经用掉了一个数字，并且有2种选择用于第三个数字，最后只有1种选择用于第四个也是最后一个数字。\ ( 2 * 3 * 2 * 1 \ ) 是12，但是有两个零被计为不同的数字，所以除以2得到 (B) 6。

Solution 2 Note that the four-digit number must start with either a 2 or a 4. The four-digit numbers that start with 2 are 2400, 2040, and 2004. The four-digit numbers that start with 4 are 4200, 4020, and 4002 which gives us a total of (B) 6.
注意，这个四位数必须以2或4开头。以2开头的四位数是2400、2040和2004。以4开头的四位数是4200、4020和4002，这给我们总共(B) 6个。