2000 AMC 8 第 23 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
23.
有一个包含七个数的列表。前四个数的平均数是 ，后四个数的平均数是 。如果全部七个数的平均数是 ，那么两组四个数中共同的那个数是
There is a list of seven numbers. The average of the first four numbers is and the average of the last four numbers is If the average of all seven numbers is then the number common to both sets of four numbers is
小提示：
把每个平均数转化为总和。
Convert each average into a sum
大提示：
共同的数在两个四数总和中被计算了两次。
The shared number is counted twice in the two four-number sums
解答：
前四个数的和是 。后四个数的和是 。
全部七个数的和是 。共同的数包含在前两个和中。
也就是说，前两个和相加会把每个数计算一次，但共同的数计算两次。
而全部七个数的和只把每个数计算一次。因此用前两个和的总和减去七个数的总和，就得到共同的数。
所以共同的数为
所以正确答案是 B。
The sum of the first four numbers is The sum of the last four numbers is
The sum of all seven numbers is We know that the number common to both sets is included in both of the first two sums.
This means that the sum of the first two sums includes every number once, except for the common number which is included twice.
The third sum, however, only includes every number once. This means that the sum of the first two sums minus the third sum yields our desired number.
Therefore, the common number is
Thus, B is the correct answer.
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