The mean age of Amanda's $4$ cousins is $8$, and their median age is $5$. What is the sum of the ages of Amanda's youngest and oldest cousins?
$\textbf{(A)}\ 13\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 25$
The sum of the ages of the cousins is $4$ times the mean, or $32$.
There are an even number of cousins, so there is no single median, so $5$ must be the mean of the two in the middle.
Therefore the sum of the ages of the two in the middle is $10$. Subtracting $10$ from $32$ produces $\textbf{(D)}\ \boxed{22}$.