jmc

Multiplying both sides of the given equation by $p$, we see that $1+\frac{p}{q}=p\Rightarrow \frac{p}{q}=p-1\Rightarrow p=q(p-1)$. But then $(p-1)(q-1)=q(p-1)-(p-1)=p-p+1=\boxed{1}$.