If $a, b$, and $c$ are positive integers, the radicals $\sqrt{a+\frac{b}{c}}$ and $a\sqrt{\frac{b}{c}}$ are equal when and only when: 
$\textbf{(A)}\ a=b=c=1\qquad\textbf{(B)}\ a=b\text{ and }c=a=1\qquad\textbf{(C)}\ c=\frac{b(a^2-1)}{a}\\ \textbf{(D)}\ a=b\text{ and }c\text{ is any value}\qquad\textbf{(E)}\ a=b\text{ and }c=a-1$
Here, we are told that the two quantities are equal. 
Squaring both sides, we get: $a+\frac{b}{c}=a^2*\frac{b}{c}$.
Multiply both sides by $c$: $ac+b=ba^2$.
Looking at all answer choices, we can see that $ac=ba^2-b=b(a^2-1)$. 
This means that $c=\frac{b(a^2-1)}{a}$, and this is option C. Therefore chose $\boxed{C}$ as your answer.
~hastapasta