Greek Anthology Book XIV: Metrodorus: 132

Arithmetical Epigram of Metrodorus

 * This is Polyphemus the brazen Cyclops, and as if on him someone made an eye, a mouth, and a hand, connecting them with pipes.
 * He looks quite as if he were dripping water and seems also to be spouting it from his mouth.


 * None of the spouts are irregular;
 * that from his hand when running will fill the cistern in three days only,
 * that from his eye in one day,
 * and his mouth in two-fifths of a day.


 * Who will tell me the time it takes when all three are running?

Solution
Let $t$ be the number of hours it takes to fill the cistern.

Let $a, b, c$ be the flow rate in numbers of cisterns per hour of (respectively) the eye, the mouth and the hand.

In $t$ hours, the various contributions of each of the spouts is $a t$, $b t$ and $c t$ respectively.

So for the total contribution to be $1$ cistern, we have:

We have:

and so:

So the cistern will be filled in $\dfrac 6 {23}$ of a day.