1. Speed of sound in an ideal gas

   For an ideal gas, the speed of sound is $v = \sqrt{\gamma \frac{P}{\rho}}$ where $\gamma = \dfrac{C_p}{C_v}$ is the adiabatic index.

2. Given condition

   All gases have the same value of $\dfrac{P}{\rho}$.

   Therefore, $v \propto \sqrt{\gamma}$

   So we only need the values of $\gamma$ for $\mathrm{He}, \mathrm{CH_4}, \mathrm{CO_2}$.

3. Find $\gamma$ for each gas

   • Helium ($\mathrm{He}$) is a monoatomic gas: $\gamma_{\mathrm{He}} = \frac{5}{3}$

   • Methane ($\mathrm{CH_4}$) is a non-linear polyatomic gas. For such gases (ignoring vibration modes), $C_v = 3R, \quad C_p = 4R$ hence $\gamma_{\mathrm{CH_4}} = \frac{4}{3}$

   • Carbon dioxide ($\mathrm{CO_2}$) is a linear polyatomic gas. For such gases (ignoring vibration modes), $C_v = \frac{5R}{2}, \quad C_p = \frac{7R}{2}$ hence $\gamma_{\mathrm{CO_2}} = \frac{7}{5}$

4. Write the ratio of speeds

   Since $v \propto \sqrt{\gamma}$, $V_{\mathrm{He}} : V_{\mathrm{CH_4}} : V_{\mathrm{CO_2}} = \sqrt{\frac{5}{3}} : \sqrt{\frac{4}{3}} : \sqrt{\frac{7}{5}}$

5. Match with the options

   This matches Option C.

Final Answer: $\boxed{\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{7}{5}}}$