Greek Anthology Book XIV: Metrodorus: 138

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Arithmetical Epigram of Metrodorus

Nicarete, playing with five companions of her own age,
gave a third of the nuts she had to Cleis,
the quarter to Sappho,
and the fifth to Aristodice,
the twentieth and again the twelfth to Theano,
and the twenty-fourth to Philinnis.
Fifty nuts were left for herself.


Solution

Let $n$ be the number of nuts in total.

$\dfrac n 3$ went to Cleis
$\dfrac n 4$ went to Sappho
$\dfrac n 5$ went to Aristodice
$\dfrac n {20} + \dfrac n {12}$ went to Theano
$\dfrac n {24}$ went to Philinnis
$50$ stayed with Nicarete.


We have:

\(\ds n\) \(=\) \(\ds \dfrac n 3 + \dfrac n 4 + \dfrac n 5 + \dfrac n {20} + \dfrac n {12} + \dfrac n {24} + 50\)
\(\ds \leadsto \ \ \) \(\ds 120 n\) \(=\) \(\ds 40 n + 30 n + 24 n + 6 n + 10 n + 5 n + 120 \times 50\) multiplying through by $120 = \lcm \set {3, 4, 5, 20, 12, 24}$
\(\ds \leadsto \ \ \) \(\ds \paren {120 - 40 - 30 - 24 - 6 - 10 - 5} n\) \(=\) \(\ds 120 \times 50\)
\(\ds \leadsto \ \ \) \(\ds 5 n\) \(=\) \(\ds 120 \times 50\)
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds \frac {120 \times 50} {10}\)
\(\ds \) \(=\) \(\ds 1200\)


So there were $1200$ nuts.

$400$ went to Cleis
$300$ went to Sappho
$240$ went to Aristodice
$60 + 100 = 160$ went to Theano
$50$ went to Philinnis

and we know that Nicarete kept $50$.

$\blacksquare$


Source of Name

This entry was named for Metrodorus.


Sources