Greek Anthology Book XIV: 11. - Problem

Problem

I desire my two sons to receive the thousand staters of which I am possessed,

but let the fifth part of the legitimate one's share exceed by ten the fourth part of what falls to the illegitimate one.

Let $s$ staters be the share received by the legitimate heir.

Then:

$\ds \dfrac s 5$

$=$

$\ds 10 + \dfrac {1000 - s} 4$

$\ds \leadsto \ \ $

$\ds 4 s$

$=$

$\ds 20 \times 10 + 5 \times 1000 - 5 s$

multiplying both sides by $20$ to clear fractions

$\ds \leadsto \ \ $

$\ds 9 s$

$=$

$\ds 5200$

simplifying

$\ds \leadsto \ \ $

$\ds s$

$=$

$\ds \frac {5200} 9$

$\ds $

$=$

$\ds 577 \frac 7 9$

Thus the legitimate heir inherits $577 \frac 7 9$ staters, while the illegitimate one receives $1000 - 577 \frac 7 9 = 422 \frac 2 9$.

$\blacksquare$