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?

$-5$

$0$

$5$

$\dfrac{15}{4}$

$\dfrac{35}{4}$

Since there are $5$ numbers, the median is the $3$ rd largest number. That would be $6$ if $x< 6,$ $8$ if $x> 8,$ and $x$ if $6 \leq x \leq 8.$

In addition, the mean is equal to $\dfrac{4+6+8+17+x}5$ $= 7+\dfrac x5.$

If the mean is $6,$ then $7+\frac x5=6,$ so $x=-5.$

If the mean is $8,$ then $7+\frac x5=8,$ so $x=5,$ which is not in the required range.

If the mean is $x,$ then $\begin{align*}7+\frac x5&=x\\7&=\frac{4x}5\\x&=8.75,\end{align*}$ which is not in the required range.

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

Thus, the answer is A .