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 $$\sum _{j=1}^n{a}_j\frac{s_j}{r_j}=a_0,$$ with variables $r_1,\mathrm{.}\mathrm{.}\mathrm{.},r_n$ in an arbitrary box at the origin and variables $s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_n$ 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.