jmc

We have $\tan R=\frac{PQ}{PR}$ and $\sin R=\frac{PQ}{RQ}=\frac{PQ}{20}$, so $\tan R=4\sin R$ gives us $\frac{PQ}{PR}=4\cdot \frac{PQ}{20}=\frac{PQ}{5}$. From $\frac{PQ}{PR}=\frac{PQ}{5}$, we have $PR=\boxed{5}$.