Greek Anthology Book XIV: 7. - Problem

Problem
I am a brazen lion; my spouts are my two eyes, my mouth, and the flat of my right foot. My right eye fills a jar in two days, my left eye in three, and my foot in four. My mouth is capable of filling it in six hours; tell me how long all four together will take to fill it.

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

Let $r, l, m, f$ be the flow rate in numbers of jars per day of (respectively) right eye, left eye, mouth and foot.

In $t$ hours, the various contributions of each of the spouts is:


 * Right eye: $r t$


 * Left eye: $l t$


 * Right foot: $f t$


 * Mouth: $m t$

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

There are $24$ hours in a day.

Hence, we have:

and so:

So the jar will be filled in $4 \frac {44} {61}$ hours.