Harmonic Mean - Problem 130 in I.M. Gelfand's Algebra book

I'm having trouble understanding how to approach Problem 130 in I.M. Gelfand's book. Preceding this problem he shows us the definition of the harmonic mean:

$$\frac{1}{\quad\frac{\frac{1}{a} + \frac{1}{b}}{2}\quad} = \frac{2}{\frac{1}{a}+\frac{1}{b}}$$

He then states problem 130, which is:

A swimming pool is divided into two equal sections. Each section has its own water supply pipe. To fill one section (using its pipe) you need $a$ hours. To fill the other section you need $b$ hours. How many hours would you need if you turn on both pipes and remove the wall dividing the pool into sections.

So I'm having trouble understanding how to apply the harmonic mean to this problem. Any hints or guidance would really be appreciated.

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1 Answer

Say each section of the swimming pool is $1$ unit in volume (to simplify our lives, we are going to measure volume in half-swimming-pools, so each half is exactly one half-swimming-pool of water). The fact that the first section is filled in $a$ hours means that you are pouring water through that pipe at a rate of $\frac{1}{a}$ units per hour. Similarly, the water in the second pipe must be flowing at a rate of $\frac{1}{b}$ units per hour.

So, the water flowing into the pool through both pipes is flowing at a rate of $$\frac{1}{a}+\frac{1}{b}\ \frac{\text{units}}{\text{hour}},$$ by adding both rates.

Now, that's the rate at which water is flowing. How much volume do you need to fill? How long will it take? How is it related to the harmonic mean?

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