The ratio of vapour densities of two gases at the same temperature is $\frac{4}{25}$,  then the ratio of r.m.s. velocities will be :

The root mean square (r.m.s.) velocity ($v_{rms}$) of a gas is given by the equation:

$$v_{rms} = \sqrt{\frac{3RT}{M}}$$

Since vapour density (VD) is directly proportional to molar mass ($M$), we can write:

$$M_1 \propto VD_1, \quad M_2 \propto VD_2$$

Thus, the given ratio of vapour densities is:

$$\frac{VD_1}{VD_2} = \frac{M_1}{M_2} = \frac{4}{25}$$

The ratio of r.m.s. velocities is:

$$\frac{v_{rms,1}}{v_{rms,2}} = \sqrt{\frac{M_2}{M_1}}$$

Substituting $M_1/M_2 = 4/25$:

$$\frac{v_{rms,1}}{v_{rms,2}} = \sqrt{\frac{25}{4}}$$

 $$\frac{v_{rms,1}}{v_{rms,2}} = \frac{5}{2}$$