Quick Prime

If $a = b$ then $\dfrac{ap}{b} = p$ will be prime, but if, say, $a = 2b$ , then $\dfrac{ap}{b} = 2p$ will not be prime, so the answer is $\boxed{Sometimes}$ .

(Note that we could have $a = 6, b = 10, p = 5$ and thus $\dfrac{ap}{b} = 3$ , so even if the condition that $a \ne b$ were applied the answer would still be "Sometimes".)