Greek Anthology Book XIV: 4. - Problem

Problem

 * On the Dung of Augeas

Heracles the mighty was questioning Augeas, seeking to learn the number of his herds, and Augeas replied: "About the streams of Alpheius, my friend, are the half of them; the eighth part pasture around the hill of Cronos, the twelfth part far away by the precinct of Taraxippus; the twentieth part feed in holy Elis, and I left the thirtieth part in Arcadia; but here you see the remaining fifty herds."

Solution
Let $n$ be the number of herds owned by Augeas.

Then:
 * $\dfrac n 2$ are about the streams of Alpheius


 * $\dfrac n 8$ pasture around the hill of Cronos


 * $\dfrac n {12}$ pasture far away by the precinct of Taraxippus


 * $\dfrac n {20}$ feed in holy Elis


 * $\dfrac n {30}$ were left in Arcadia


 * $50$ remain to be seen by Heracles.

Thus:

Thus:


 * $120$ are about the streams of Alpheius


 * $30$ pasture around the hill of Cronos


 * $20$ pasture far away by the precinct of Taraxippus


 * $12$ feed in holy Elis


 * $8$ were left in Arcadia

in addition to:


 * the $50$ remain to be seen by Heracles

and it is seen that:


 * $120 + 30 + 20 + 12 + 8 + 50 = 140$