OIM 1988 Problem 2

Let $a,b,c,d,p$ and $q$ be positive integers satisfying $ad-bc=1$ and $\frac{a}{b}>\frac{p}{q}>\frac{c}{d}$ .

Prove that: $(a)$ $q\ge b+d$ $(b)$ If $q=b+d$ , then $p=a+c$ .