How long will it take to fill the pool?

The pool can be filled by a big pump in 5 hours and by a small pump in 20 hours.

How long will both pumps take to fill the pool?

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Solution

This is a question about rates.
In this case the volume of water pumped per hour.
For the big pump this is $\frac{V}{5}$ and $\frac{V}{20}$ for the little pump, where $V$ is the volume of the pool.
If both pumps are used together, then these rates are added to give the rate at which the pool is filled, i.e.,

$$\frac{V}{\mathrm{t}}=\frac{V}{5}+\frac{V}{20}$$

where $t$ is the time to fill the pool with both pumps.

The $V$'s cancel,

$$\frac{1}{\mathrm{t}}=\frac{1}{5}+\frac{1}{20}$$.

Rearranging,

$$t=\frac{5\times 20}{5+20}$$

Giving,

$$t=4\mathrm{hours}$$

Is this answer reasonable?

Well, it took 5 hours to fill with the big pump, so if we add another no matter how slow it is it'll help fill the pool quicker. Our calculated time is quicker, so we should be happy with that.

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