Greek Anthology Book XIV: Metrodorus: 131/Historical Note

Historical Note on Metrodorus' Arithmetical Epigram no. $131$
In 's $1918$ translation of, he gives the answer as $2 \frac 2 {11}$ hours.

The discrepancy between this and the $2 \frac 2 7$ reported in the calculation is explained by the imprecision of the wording of the epigram.

If we take:


 * the third twice as much

to mean:


 * the third spout takes twice as much time to fill a cistern as the second spout

then we deduce that the third spout takes $16$ hours (that is, twice four plus four hours) to fill the cistern.

Hence we arrive at the solution $2 \frac 2 7$ hours.

However, if we interpret:


 * the third twice as much

to mean:


 * the third spout takes twice as much longer (that is, eight hours) than the first spout than the second one does to fill a cistern

that would mean the third spout takes $4$ hours plus twice $4$ hours, that is $12$ hours, to fill a cistern.

If this assumption is made, then $2 \frac 2 {11}$ is correct.

Proof
Thus we have:

and so:

So the cistern will be filled in $2 \frac 2 {11}$ hours.