jmc

Let our numbers be $a$ and $b$ with $a>b.$ Then $ab+a+b=95$. With Simon's Favorite Factoring Trick in mind, we add $1$ to both sides and get $ab+a+b+1=96$, which factors as $(a+1)(b+1)=96$. We consider pairs $(a+1,b+1)$ of factors of $96$: $(96,1),(48,2),(32,3),(24,4),(16,6),\text{and}(12,8)$. Since $a<20$, we can rule out the first 4 pairs. The pair $(16,6)$ gives us $a=15,b=5$ which doesn't work because $a$ and $b$ are relatively prime, so we are left with the last pair, which gives $a=11$ and $b=7$, so $a+b=\boxed{18}$.