Density of mixture $= \rho_{\text{mix}} = \frac{V_{\mathrm{O}_2} \rho_{\mathrm{O}_2} + V_{\mathrm{H}_2} \rho_{\mathrm{H}_2}}{V_{\mathrm{O}_2} + V_{\mathrm{H}_2}} = \frac{V(\rho_{\mathrm{O}_2} + \rho_{\mathrm{H}_2})}{2V} = \frac{\rho_{\mathrm{O}_2} + \rho_{\mathrm{H}_2}}{2} \quad (\text{since } V_{\mathrm{O}_2} = V_{\mathrm{H}_2} = V) = \frac{\rho_{\mathrm{H}_2} + 16 \rho_{\mathrm{H}_2}}{2} = 8.5 \rho_{\mathrm{H}_2} \Rightarrow v \propto \frac{1}{\sqrt{\rho}} \Rightarrow \frac{V_{\text{mix}}}{V_{\mathrm{H}_2}} = \sqrt{\frac{\rho_{\mathrm{H}_2}}{\rho_{\text{mix}}}} = \sqrt{\frac{\rho_{\mathrm{H}_2}}{8.5 \rho_{\mathrm{H}_2}}} \approx \sqrt{\frac{1}{8}}$