Randy drove the first third of his trip on a gravel road, the next $20$ miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip?
$\textbf{(A)}\ 30\qquad\textbf{(B)}\ \frac{400}{11}\qquad\textbf{(C)}\ \frac{75}{2}\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ \frac{300}{7}$
If the first and last legs of his trip account for $\frac{1}{3}$ and $\frac{1}{5}$ of his trip, then the middle leg accounts for $1 - \frac{1}{3} - \frac{1}{5} = \frac{7}{15}$ths of his trip.  This is equal to $20$ miles.  Letting the length of the entire trip equal $x$, we have 
\[\frac{7}{15}x = 20 \implies x=\boxed{\textbf{(E)}\ \frac{300}{7}}\]