  We show that, for sufficiently large integers $m$ and $X$, for almost all $a
=1, ..., m$ the ratios $a/x$ and the products $ax$, where $|x|\le X$, are very
uniformly distributed in the residue ring modulo $m$. This extends some recent
results of Garaev and Karatsuba. We apply this result to show that on average
over $r$ and $s$, ranging over relatively short intervals, the distribution of
Kloosterman sums $$ K_{r,s}(p) = \sum_{x=1}^{p-1} \exp(2 \pi i (rn +
sn^{-1})/p), $$ for primes $p\le T$ is in accordance with the Sato--Tate
conjecture.
