Greek Anthology Book XIV: Metrodorus: 117

Arithmetical Epigram of Metrodorus

$A$. Where are thy apples gone, my child?

$B$. Ino has two-sixths

and Semele one-eighth,

and Autonoe went off with one-fourth,

while Agave snatched from my bosom and carried away a fifth.

For thee ten apples are left,

but I, yes I swear it by dear Cypris, have only this one.

Solution

Let $n$ be the total number of apples.

$\dfrac {2 n} 6$ were taken by Ino.

$\dfrac n 8$ were taken by Semele.

$\dfrac n 4$ were taken by Autonoe.

$\dfrac n 5$ were taken by Agave.

$10$ remained for the child's parent.

$1$ remained for the child.

Hence:

\(\ds n\)

\(=\)

\(\ds \dfrac {2 n} 6 + \dfrac n 8 + \dfrac n 4 + \dfrac n 5 + 10 + 1\)

\(\ds \leadsto \ \ \)

\(\ds 120 n\)

\(=\)

\(\ds 40 n + 15 n + 30 n + 24 n + 120 \times 11\)

multiplying through by $120 = \lcm \set {6, 8, 4, 5}$ and simplifying

\(\ds \leadsto \ \ \)

\(\ds \paren {120 - 40 - 15 - 30 - 24} n\)

\(=\)

\(\ds 120 \times 11\)

\(\ds \leadsto \ \ \)

\(\ds 11 n\)

\(=\)

\(\ds 120 \times 11\)

\(\ds \leadsto \ \ \)

\(\ds n\)

\(=\)

\(\ds 120\)

So the narrator started with $120$ apples, of which:

$40$ were taken by Ino

$15$ were taken by Semele

$30$ were taken by Autonoe

$24$ were taken by Agave

and as we know:

$10$ remained for the child's parent

and $1$ remained for the child.

It is understood that $\dfrac {2 n} 6$ is not expressed in canonical form,that is: $\dfrac n 3$, but such is the nature of the epigram.

$\blacksquare$

Source of Name

This entry was named for Metrodorus.

Sources

• 1918: W.R. Paton: The Greek Anthology Book XIV ... (previous) ... (next): Metrodorus' Arithmetical Epigrams: $117$