Linear equations with rational fractions of bounded height and stochastic matrices

We obtain a tight, up to a logarithmic factor, upper bound on the number of solutions to the equation $$\mathop {\mathop \sum}_{j = 1}^n {a_j}{{{s_j}} \over {{r_j}}} = {a_0},$$ with variables ${r_1}, \ldots ,{r_n}$ in an arbitrary box at the origin and variables ${s_1}, \ldots ,{s_n}$ in an essentially arbitrary translation of this box. We apply this result to get an upper bound on the number of stochastic matrices with rational entries of bounded height.