Greek Anthology Book XIV: Metrodorus: 118

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Arithmetical Epigram of Metrodorus

Myrto once picked apples and divided them among her friends;
she gave the fifth part to Chrysis,
the fourth to Hero,
the nineteenth to Psamathe,
and the tenth to Cleopatra,
but she presented the twentieth part to Parthenope
and gave only twelve to Evadne.
Of the whole number a hundred and twenty fell to herself.


Solution

Let $n$ be the total number of apples.

$\dfrac n 5$ were given to Chrysis.
$\dfrac n 4$ were given to Hero.
$\dfrac n {19}$ were given to Psamathe.
$\dfrac n {10}$ were given to Cleopatra.
$\dfrac n {20}$ were given to Parthenope.
$12$ were given to Evadne.
$120$ were kept by Myrto.


Hence:

\(\ds n\) \(=\) \(\ds \dfrac n 5 + \dfrac n 4 + \dfrac n {19} + \dfrac n {10} + \dfrac n {20} + 12 + 120\)
\(\ds \leadsto \ \ \) \(\ds 380 n\) \(=\) \(\ds 76 n + 95 n + 20 n + 38 n + 19 n + 380 \times 132\) multiplying through by $380 = \lcm \set {5, 4, 19, 10, 20}$ and simplifying
\(\ds \leadsto \ \ \) \(\ds \paren {380 - 76 - 95 - 20 - 38 - 19} n\) \(=\) \(\ds 380 \times 132\)
\(\ds \leadsto \ \ \) \(\ds 132 n\) \(=\) \(\ds 380 \times 132\)
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds 380\)


So the narrator started with $380$ apples, of which:

$76$ were given to Chrysis
$95$ were given to Hero
$20$ were given to Psamathe
$38$ were given to Cleopatra
$19$ were given to Parthenope

and as we know:

$12$ were given to Evadne

and $120$ were kept by Myrto.

$\blacksquare$


Source of Name

This entry was named for Metrodorus.


Sources

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