Sun Tzu Suan Ching/Examples/Example 1

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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.


\(\ds \dfrac x 2 + \dfrac x 3 + \dfrac x 4\) \(=\) \(\ds 65\)
\(\ds 13 x\) \(=\) \(\ds 780\) multiplying through by $12$ and simplifying
\(\ds x\) \(=\) \(\ds 60\)