# Chiu Chang Suann Jing/Examples/Example 3

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## Example of Problem from

## Example of Problem from *Chiu Chang Suann Jing*

*A number of men bought a number of articles, neither of which are known;**it is only known that if each man paid $8$ cash, there would be a surplus of $3$ cash,**and if each man paid $7$ cash, there would be a deficiency of $4$ cash.*

*Required the respective numbers?*

## Solution

There are $7$ men and $53$ objects.

## Proof

Let $m$ be the number of men.

Let $p$ be the number of objects.

It is assumed that the objects each cost $1$ cash each.

Then:

\(\text {(1)}: \quad\) | \(\ds 8 m\) | \(=\) | \(\ds p + 3\) | |||||||||||

\(\text {(2)}: \quad\) | \(\ds 7 m\) | \(=\) | \(\ds p - 4\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds m\) | \(=\) | \(\ds 7\) | $(1) - (2)$ | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds p - 4\) | \(=\) | \(\ds 49\) | substituting back into $2$ | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds p\) | \(=\) | \(\ds 53\) |

$\blacksquare$

## Sources

- c. 100: Anonymous:
*Chiu Chang Suann Jing* - 1913: Yoshio Mikami:
*The Development of Mathematics in China and Japan* - 1965: Henrietta Midonick:
*The Treasury of Mathematics: Volume $\text { 1 }$* - 1992: David Wells:
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