2012 AMC 8 第 11 题

Question

正整数

3

、

4

、

5

、

6

、

6

、

7

和

x

的平均数、中位数和唯一众数都相等。

x

的值是多少？

The mean, median, and unique mode of the positive integers

3,

4,

5,

6,

6,

7,

and

x

are all equal. What is the value of

x?

5

6

7

11

12

答案：D

知识点：平均数中位数（数据）众数

难度评级：1240

小提示：

唯一众数必须是列表中已经重复的数。

The unique mode must be the repeated listed number.

大提示：

如果平均数也是那个数，那么总和就被确定。

If the mean is that number, the total sum is fixed.

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文字解答：

已列出的数中，除

6

外每个数只出现一次，而

6

出现两次。若

x

等于其他已列出的数，就会有两个众数，不再有唯一众数。因此唯一众数必须是

6

，平均数也必须是

6

。共有

7

个数，所以总和为

6\cdot 7 = 42

。另一方面，总和还等于

3+4+5+6+6+7+x

= 31+x

= 42\text{，}

所以

x = 11

。

所以正确答案是 D。

Every listed value except

6

appears once while

6

appears twice. If

x

equals any of the other listed values, then we have two modes, which means we do not have a unique mode. Otherwise, the only value that shows up more than once is

6

making that the unique mode. This also means

6

is the mean. Since there are

7

elements, the sum of the elements is

6\cdot 7 = 42.

The sum is also

3+4+5+6+6+7+x

= 31+x

= 42,

so

x = 11.

Thus, the answer is D .