Greek Anthology Book XIV: 50. - Problem

Problem

Throw me in, silversmith, besides the bowl itself,

the third of its weight,

and the fourth,

and the twelfth;

and casting them into the furnace stir them, and mixing them all up take out, please, the mass, and let it weigh one mina.

Let $w$ be the weight in minae of the bowl.

We have:

$\ds w + \frac w 3 + \frac w 4 + \frac w {12}$

$=$

$\ds 1$

$\ds \leadsto \ \ $

$\ds 12 w + 4 w + 3 w + w$

$=$

$\ds 12$

multiplying through by $12$

$\ds \leadsto \ \ $

$\ds 20 w$

$=$

$\ds 12$

$\ds \leadsto \ \ $

$\ds w$

$=$

$\ds \dfrac {12} {20}$

$\ds $

$=$

$\ds \dfrac 3 5$

Thus the bowl weighs $\dfrac 3 5$ of a mina.

Assuming $100$ drachmae to the mina, that is, $60$ drachmae.

$\blacksquare$