Question

What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers?

$-5$

$0$

$5$

$\dfrac{15}{4}$

$\dfrac{35}{4}$

Answer: A

Concepts:mean median (data)casework

Difficulty rating: 1490

Solution:

The mean is $7+\frac{x}{5}$ . If $x<6$ , the median is $6$ , so $7+\frac{x}{5}=6$ gives the valid value $x=-5$ .

If $6\le x\le8$ , the median is $x$ , but $7+\frac{x}{5}=x$ gives $x=\frac{35}{4}$ , outside this interval. If $x>8$ , the median is $8$ , but $7+\frac{x}{5}=8$ gives $x=5$ , again outside the required range.

Therefore, the only possible $x$ is $-5,$ making the sum $-5.$

Thus, the answer is A .

2019 AMC 10B Problem 13

Answer: A

Difficulty rating: 1490

Solution:

Problem 13 in Other Years