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:

$\ds n$

$=$

$\ds \dfrac n 2 + \dfrac n 8 + \dfrac n {12} + \dfrac n {20} + \dfrac n {30} + 50$

$\ds \leadsto \ \ $

$\ds 120 n$

$=$

$\ds 60 n + 15 n + 10 n + 6 n + 4 n + 120 \times 50$

multiplying through by $120$

$\ds \leadsto \ \ $

$\ds 25 n$

$=$

$\ds 120 \times 50$

gathering terms

$\ds \leadsto \ \ $

$\ds n$

$=$

$\ds 120 \times 2$

dividing both sides by $25$

$\ds $

$=$

$\ds 240$

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:

$120 + 30 + 20 + 12 + 8 + 50 = 240$

$\blacksquare$

Sources

1918: W.R. Paton: The Greek Anthology Book XIV ... (previous) ... (next): $4$. -- Problem