Speed of sound in a gas mixture

[Rick4]

In a mixture of argon (atomic mass = 39.9 u) and neon (atomic mass = 20.2 u), the speed of sound is 363 m/s at 300K. Assume that both monatomic gases behave as ideal gases. Find the percentage of the atoms that are argon and the percentage that are neon.

I found the solution (in the attachments) but what I can't figure out is where the equation that was used in step 4 of the solution came from.

This is the equation that was used to solve the problem.
Where did it come from?

v = (v₁v₂(V₁+ V₂)^1/2)/(V₂v₁² + V₁v₂²)^1/2

 

[jbsteele]

The equation is derived from the fact that the speed of sound in a mixture of gases is given by the following relation: v = (∑xivi)1/2, where xi is the mole fraction of gas i and vi is the speed of sound in gas i.

 

[baba89]

The equation used in step 4 of the solution is known as the root-mean-square (rms) velocity formula for gases. It is derived from the kinetic theory of gases and is used to calculate the average speed of gas molecules in a mixture. The equation takes into account the individual velocities and masses of the gas molecules in the mixture. In this case, the equation is used to calculate the speed of sound in the gas mixture, which is related to the average speed of the gas molecules.