Greek Anthology Book XIV: Metrodorus: 143

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Arithmetical Epigram of Metrodorus

The father perished in the shoals of the Syrtis, and this, the eldest of the brothers, came back from that voyage with five talents.
To me he gave twice two-thirds of his share,
on our mother he bestowed two-eighths of my share,
nor did he sin against divine justice.


Solution

Let $a$ talents be the elder brother's share.

Let $b$ talents be the narrator's share.

Let $c$ talents be the mother's share.


We have:

\(\ds a + b + c\) \(=\) \(\ds 5\)
\(\ds b\) \(=\) \(\ds 2 \times \dfrac 2 3 a\)
\(\ds \) \(=\) \(\ds \dfrac {4 a} 3\)
\(\ds c\) \(=\) \(\ds \dfrac 2 8 b\)
\(\ds \leadsto \ \ \) \(\ds c\) \(=\) \(\ds \dfrac 2 8 \times \dfrac {4 a} 3\)
\(\ds \) \(=\) \(\ds \dfrac a 3\)
\(\ds \leadsto \ \ \) \(\ds a + b + \frac a 3\) \(=\) \(\ds 5\)
\(\ds \leadsto \ \ \) \(\ds a + \dfrac {4 a} 3 + \frac a 3\) \(=\) \(\ds 5\)
\(\ds \leadsto \ \ \) \(\ds \paren {3 + 4 + 1} a\) \(=\) \(\ds 15\)
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds \dfrac {15} 8\)
\(\ds \) \(=\) \(\ds 1 \dfrac 7 8\)
\(\ds \leadsto \ \ \) \(\ds b\) \(=\) \(\ds \dfrac 4 3 \times \dfrac {15} 8\)
\(\ds \) \(=\) \(\ds \dfrac {20} 8\)
\(\ds \) \(=\) \(\ds 2 \dfrac 1 2\)
\(\ds \leadsto \ \ \) \(\ds c\) \(=\) \(\ds \dfrac 1 4 \times \dfrac {20} 8\)
\(\ds \) \(=\) \(\ds \dfrac 5 8\)


So:

the elder brother takes $1 \frac 7 8$ talents
the narrator takes $2 \frac 1 2$ talents
the mother takes $\dfrac 5 8$ of a talents.

$\blacksquare$


Source of Name

This entry was named for Metrodorus.


Historical Note

In W.R. Paton's $1918$ translation of The Greek Anthology Book XIV, he gives the answer as:

the elder brother had $1 \frac 5 7$ talents
the narrator had $2 \frac 2 7$ talents
the mother had $1$ talent.


The discrepancy between this and the shares reported in the calculation appears to be due to a misinterpretation of the wording.


If we replace:

on our mother he bestowed two-eighths of my share

with:

on our mother he bestowed two-eighths of our combined share

then we arrive at the solution given by The Greek Anthology Book XIV.


Sources

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