In triangle $PQR$, we have $\angle P = 90^\circ$, $QR = 20$, and $\tan R = 4\sin R$.  What is $PR$?
I know that $\tan R = \frac{\text{opposite}}{\text{adjacent}}$, so $\tan R = \frac{PQ}{PR}$. I also know that $\sin R = \frac{\text{opposite}}{\text{hypotenuse}}$, so $\sin R = \frac{PQ}{QR}$. Since $\tan R = 4\sin R$, I can substitute these expressions and get $\frac{PQ}{PR} = 4\frac{PQ}{QR}$. I can cross-multiply and simplify to get $PQ \cdot QR = 4PQ \cdot PR$. I can divide both sides by $PQ$ and get $QR = 4PR$. I can plug in $QR = 20$ and get $20 = 4PR$. I can divide both sides by $4$ and get $PR = 5$. 