/sci/ - Science & Math

I'm sorry for the stupid question, but would you mind telling me where I did a mistake in my thinking?
So one machine takes 2a to fill up the whole pool, the other one takes 2b to do the same. Then if they're turned on at the same time we take the harmonic mean, and [math] \frac{4ab}{b+a} [/math] remains. But I tried googling and everyone gets 2ab instead of 4ab on top.