Partial pressures are really just fractions of the total pressure. You can add any fraction together to achieve a new total, in accordance with Dalton's Law of Partial Pressures. So the math is valid; it's really in the measured pressures that you can go wrong.
Suppose a total pressure
Then we could have a situation where the
By summing each contributed pressure, you get the total contribution to the pressure, i.e. you get the total pressure.
REMARKS ABOUT REAL GASES
This works fairly well so long as the gas itself can be assumed ideal without losing accuracy in terms of what its volume per
But, there are characteristics that real gases have, and ideal gases don't:
The above two points also can blend together at extreme pressure amounts, as some gases can deviate drastically from ideality depending on the pressure and temperature of the system.
CONCLUSIONS
As long as every gas in a system is able to be assumed ideal, and as long as they don't react, we can treat contributions to the partial pressure with approximately equal weight.