AMC 8 1988 Problem 21

AMC8 1988 年第 21 题

统计与数据图表★★★☆☆

A fifth number,

n

, is added to the set

\{ 3,6,9,10 \}

to make the mean of the set of five numbers equal to its median. The number of possible values of

n

将第五个数

n

加入集合

\{3,6,9,10\}

中，使得这五个数的平均数等于其中位数。

n

的可能值有多少个？

正确答案 · Correct Answer

解析 · Solution

Solution 1

The mean of the set

\{3, 6, 9, 10, n\}

\displaystyle \frac{3+6+9+10+n}{5} = \frac{28+n}{5}

If n < 6, the median is 6. Solving

\displaystyle \frac{28+n}{5} = 6

yields n = 2, which is valid since 2 < 6.

6 \leq n < 9

, the median is

n

. Solving

\displaystyle \frac{28+n}{5} = n

yields n = 7, which is valid since

6 \leq 7 < 9

n \geq 9

, the median is 9. Solving

\displaystyle \frac{28+n}{5} = 9

yields n = 17, which is valid since

17 \geq 9

Therefore, there are 3 possible values for

n

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