Two different ideal gases, X and Y, are kept in separate identical...

For an ideal gas, the number density $n$ is given by $n = \dfrac{P}{k_B T}$.

Since both gases are at the same pressure and temperature, their number densities are equal ($n_X = n_Y$).

The mean free path is given by $\lambda = \dfrac{1}{\sqrt{2}\pi d^2 n}$.

Since $n$ is constant across both flasks, $\lambda \propto \dfrac{1}{d^2}$.

Therefore, $\dfrac{\lambda_X}{\lambda_Y} = \left(\dfrac{d_Y}{d_X}\right)^2$

Given $\lambda_X = 9\lambda_Y$, we have:

$9 = \left(\dfrac{d_Y}{d_X}\right)^2$

$\dfrac{d_Y}{d_X} = 3$

$\dfrac{d_X}{d_Y} = \dfrac{1}{3}$

The ratio of their molecular diameters is $1:3$.

Answer: $1:3$