Greek Anthology Book XIV: 6. - Problem

Problem

"Best of clocks, how much of the day is past?"

There remain twice two-thirds of what is gone.

Let $h$ be the number of hours that have passed.

A day of $12$ hours is assumed.

Then:

$\ds 12 - h$

$=$

$\ds 2 \times \dfrac 2 3 h$

$\ds \leadsto \ \ $

$\ds 36 - 3 h$

$=$

$\ds 4 h$

clearing fractions

$\ds \leadsto \ \ $

$\ds 36$

$=$

$\ds 7 h$

$\ds \leadsto \ \ $

$\ds h$

$=$

$\ds \dfrac {36} 7$

$\ds $

$=$

$\ds 5 \frac 1 7$

So $5 \frac 1 7$ hours have passed, and $6 \frac 6 7$ remain.

$\blacksquare$