$PV = nRT$

$\therefore n \propto P \text { at constant } T \& V.$

$\because \quad \text { mole }=\frac{\text { Mass }}{\text { Molar mass }}$

$\text { For gas } X : \frac{10}{ M _{ X }} \propto 2 atm$      $. . . . (1)$

For gas $X \& Y : \frac{10}{ M _{ X }}+\frac{80}{ M _{ Y }} \propto 6 atm$    $. . . . (2)$

From $(2)$ - $(1)$

$\frac{80}{ M _y} \propto 4$          $. . . . (3)$

On dividing $(1)$ by $(3)$

$\frac{M_{ Y }}{8 M _{ X }}=\frac{1}{2}$

$\therefore \quad  \frac{ M _{ Y }}{ M _{ X }}=4$    $. . . . (4)$

$\because \quad  V _{ rms }=\sqrt{\frac{3 R T}{M}} \Rightarrow V _{ rms } \propto \frac{1}{\sqrt{ M }}$

$\therefore  \frac{\left( V _{ rms }\right)_{ X }}{\left( V _{ rms }\right)_{ Y }}=\sqrt{\frac{ M _{ Y }}{ M _{ X }}}=\sqrt{\frac{4}{1}}=\frac{2}{1}$