Chiu Chang Suann Jing/Examples/Example 3

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: Curious and Interesting Puzzles ... (previous) ... (next): The Nine Chapters: $61$