If $f(x,y)$ is a "good" function, then
$$\sum_{xy\equiv 1\mod p}f(x,y)=\frac{1}{p}\sum_{x,y=0}^{p-1}f(x,y)-R_p[f],$$
where $R_p[f]$ is a "small" error term (see Lemma 5 [here][1]). In your case $f(x,y)=xy$, so the main term is $p^3/4.$  Usually $R_p[f]=O(p^{1/2+\varepsilon}\|f\|)$ while the main term is like $p\|f\|$. In this case error term is $O(p^{5/2+\varepsilon}).$

This observation has a lot of applications in problems cnnected with lattices (because bases are parametrized by equation $ad-bc=n$, so $ad\equiv n\mod b$ ).




  [1]: http://iam.khv.ru/articles/Ustinov/nth13_eng.pdf