Greek Anthology Book XIV: Metrodorus: 128

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Arithmetical Epigram of Metrodorus

What violence my brother has done me, dividing our father's fortune of five talents unjustly!
Poor tearful I have this fifth part of the seven-elevenths of my brother's share.
Zeus, thou sleepest sound.


Solution

Let $n$ talents be the narrator's share of the fortune.

We have that the brother's share is therefore $5 - n$ talents.

Hence:

\(\ds n\) \(=\) \(\ds \frac 1 5 \paren {\frac 7 {11} \paren {5 - n} }\)
\(\ds \leadsto \ \ \) \(\ds 55 n\) \(=\) \(\ds 35 - 7 n\) multiplying through by $55$ and simplifying
\(\ds \leadsto \ \ \) \(\ds 62 n\) \(=\) \(\ds 35\)
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds \dfrac {35} {62}\)


So the narrator has $\dfrac {35} {62}$ of a talent.

The brother, meanwhile, has $4 \frac {27} {62}$ talents.

$\blacksquare$


Source of Name

This entry was named for Metrodorus.


Historical Note

The translator plaintively remarks on the solution:

The one offered is that the one brother had $4 \frac 4 {11}$ of a talent, the other $\frac 7 {11}$, but I cannot work it out.

Doubtless, as is apparent in the calculation given here on $\mathsf{Pr} \infty \mathsf{fWiki}$.


This would have been correct had the question been worded:

... I have this fifth part of the seven-elevenths of the total inheritance

but it isn't so it's not.


Sources

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