Greek Anthology Book XIV: Metrodorus: 143/Historical Note

Historical Note on Metrodorus' Arithmetical Epigram no. $143$
In 's $1918$ translation of, he gives the answer as:


 * the elder brother had $1 \frac 5 7$


 * the narrator had $2 \frac 2 7$


 * the mother had $1$.

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.

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

Let $b$ be the narrator's share.

Let $c$ be the mother's share.

We have:

So:
 * the elder brother takes $1 \dfrac 5 7$


 * the narrator takes $2 \dfrac 2 7$


 * the mother takes $1$.