If $78$ is divided into three parts which are proportional to $1, \frac13, \frac16,$ the middle part is:
$\textbf{(A)}\ 9\frac13 \qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 17\frac13 \qquad\textbf{(D)}\ 18\frac13\qquad\textbf{(E)}\ 26$
Let the part proportional to $1$ equal $x$. Then, the parts are $x, \frac{1}{3}x$, and $\frac{1}{6}x$. The sum of these parts should be $78$, so $\frac{9}{6}x=1\frac{1}{2}x=78$. Solving for $x$, $x=52$. The middle part is $\frac{1}{3}x$, so the answer is $17\frac{1}{3}$, or $\boxed{\textbf{C}}$