Calculate the ratio of the mean free path of the molecule of two gases if the ratio of the numbers density per $c{m}^3$ of the gases is 5:3 and the ratio of the diameters of the molecules of the gases is 4:5

Given ratio of the number density $\frac{n_1}{n_2}=\frac{5}{3}$
Ratio of the diameters  $\frac{d_1}{d_2}=\frac{4}{5}$
Mean free path of the molecules of first gas ${\lambda }_1=\frac{1}{\sqrt{2}\pi n_1{d}_1^2}$
Similarly mean free path of the second gas
${\lambda }_2=\frac{1}{\sqrt{2}\pi n_2{d}_2^2}$
∴Ratio of  ${\lambda }_1$ and ${\lambda }_2$
$\frac{{\lambda }_1}{{\lambda }_2}=\frac{n_2{d}_2^2}{n_1{d}_1^2}=\frac{3}{5}\times {\left(\frac{5}{4}\right)}^2$

$\frac{{\lambda }_1}{{\lambda }_2}=\frac{3}{5}\times \frac{5}{4}\times \frac{5}{4}=\frac{15}{16}$