Gauss's law states that the net flux of an electric field in a closed surface is directly proportional to the enclosed electric charge. Therefore,

${\varphi }_{\text{smaller cube }}=\frac{2\mathrm{Q}}{{\epsilon }_0}$

Since the smaller cube is inside the bigger cube, so total charge contained by the bigger cube will be $2Q+3Q=5Q$.

${\varphi }_{\text{bigger cube }}=\frac{5\mathrm{Q}}{{\epsilon }_0}$

Therefore, required ratio will be, 

$\frac{{\varphi }_{\text{smaller cube }}}{{\varphi }_{\text{bigger cube }}}=\frac{2}{5}$.