AMC8 2012

AMC8 2012 · Q11

AMC8 2012 · Q11. It mainly tests Averages (mean), Logic puzzles.

The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, $x$ are all equal. What is the value of $x$?

正整数 3, 4, 5, 6, 6, 7, $x$ 的均值、中位数和唯一众数都相等。$x$ 的值为多少？

(A) 5 5

(B) 6 6

(D) 11 11

(E) 12 12

Answer

Correct choice: (D)

正确答案：（D）

Solution

We can eliminate answer choices ${\textbf{(A)}\ 5}$ and ${\textbf{(C)}\ 7}$, because of the above statement. Now we need to test the remaining answer choices. Case 1: $x = 6$ Mode: $6$ Median: $6$ Mean: $\frac{37}{7}$ Since the mean does not equal the median or mode, ${\textbf{(B)}\ 6}$ can also be eliminated. Case 2: $x = 11$ Mode: $6$ Median: $6$ Mean: $6$ We are done with this problem, because we have found when $x = 11$, the condition is satisfied. Therefore, the answer is $\boxed{{\textbf{(D)}\ 11}}$.

我们可以排除选项 ${\textbf{(A)}\ 5}$ 和 ${\textbf{(C)}\ 7}$，因为上述陈述。接下来测试剩余选项。情况 1: $x = 6$ 众数: $6$ 中位数: $6$ 均值: $\frac{37}{7}$ 由于均值不等于中位数或众数，排除 ${\textbf{(B)}\ 6}$。情况 2: $x = 11$ 众数: $6$ 中位数: $6$ 均值: $6$ 当 $x = 11$ 时条件满足，因此答案是 $\boxed{{\textbf{(D)}\ 11}}$。

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