Greek Anthology Book XIV: 4. - Problem

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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

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