Sun Tzu Suan Ching/Examples/Example 1

Example of Problem from Sun Tzu Suan Ching

A woman was washing dishes in a river, when an official whose business was overseeing the waters demanded of her:

"Why are there so many dishes here?"

"Because a feasting was entertained in the house," the woman replied.

Thereuon the official enquired the number of guests.

"I don't know," the woman said, "how many guests there had been;

but every $2$ used a dish for rice between them;

every $3$ a dish for broth;

every $4$ a dish for meat;

and there were $65$ dishes in all."

$60$ guests.

Let $x$ be the number of guests.

Then:

$\ds \dfrac x 2 + \dfrac x 3 + \dfrac x 4$

$=$

$\ds 65$

$\ds 13 x$

$=$

$\ds 780$

multiplying through by $12$ and simplifying

$\ds x$

$=$

$\ds 60$

$\blacksquare$