Greek Anthology Book XIV: Metrodorus: 127

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Arithmetical Epigram of Metrodorus

Demochares lived for a quarter of his whole life as a boy,
for a fifth part of it a young man,
and for a third as a man,
and when he reached grey old age he lived thirteen years more on the threshold of eld.


Solution

Let $n$ be the age in years of Demochares at his death.

$\dfrac n 4$ was spent as a boy
$\dfrac n 5$ was spent as a young man
$\dfrac n 3$ was spent as a man
$13$ more years were spent as an old man.


We have:

\(\ds n\) \(=\) \(\ds \dfrac n 4 + \dfrac n 5 + \dfrac n 3 + 13\)
\(\ds \leadsto \ \ \) \(\ds 60 n\) \(=\) \(\ds 15 n + 12 n + 20 n + 60 \times 13\) multiplying through by $60 = \lcm \set {4, 5, 3}$ and simplifying
\(\ds \leadsto \ \ \) \(\ds \paren {60 - 15 - 12 - 20} n\) \(=\) \(\ds 60 \times 13\)
\(\ds \leadsto \ \ \) \(\ds 13 n\) \(=\) \(\ds 60 \times 13\)
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds 60\)


So Demochares lived for $60$ years, of which:

$15$ years were spent as a boy
$12$ years were spent as a young man
$20$ years were spent as a man

and as we know:

$13$ more years were spent as an old man.

$\blacksquare$


Also see


Source of Name

This entry was named for Metrodorus.


Sources

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