A flask contains argon and chlorine in the ratio of 2:1 by mass....

The root mean square speed of a gas molecule is given by $v_{rms} = \sqrt{\dfrac{3RT}{M}}$, where $T$ is the absolute temperature and $M$ is the molar mass.

For argon and chlorine in the same flask, the temperature $T$ is the same.

Therefore, the ratio of their rms speeds is inversely proportional to the square root of their molar masses:

$\dfrac{v_{rms}^{Ar}}{v_{rms}^{Cl}} = \sqrt{\dfrac{M_{Cl}}{M_{Ar}}}$

Given the molecular mass of chlorine $M_{Cl} = 70.0$ u and atomic mass of argon $M_{Ar} = 40.0$ u:

$\dfrac{v_{rms}^{Ar}}{v_{rms}^{Cl}} = \sqrt{\dfrac{70.0}{40.0}} = \sqrt{\dfrac{7}{4}} = \dfrac{\sqrt{7}}{2}$

The mass ratio of the gases in the mixture is extra information and not required for this calculation.

Answer: $\dfrac{\sqrt{7}}{2}$