# Chiu Chang Suann Jing/Examples/Example 5

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## Example of Problem from

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

*There are $3$ classes of corn, of which**$3$ bundles of the first class,**$2$ of the second class, and**$1$ of the third class*

*make $39$ measures.*

*$2$ of the first,**$3$ of the second, and**$1$ of the third*

*make $34$ measures.*

*And:**$1$ of the first,**$2$ of the second, and**$3$ of the third*

*make $26$ measures.*

*How many measures of grain are contained in $1$ bundle of each class?*

## Solution

The first class bundle contains $9 \frac 1 4$ measures.

The second class bundle contains $4 \frac 1 4$ measures.

The third class bundle contains $2 \frac 3 4$ measures.

## Proof

Let $x$, $y$ and $z$ denote the measures of grain contained in one bundle of each of the $1$st, $2$nd and $3$rd class respectively.

We have:

\(\text {(1)}: \quad\) | \(\ds 3 x + 2 y + z\) | \(=\) | \(\ds 39\) | |||||||||||

\(\text {(2)}: \quad\) | \(\ds 2 x + 3 y + z\) | \(=\) | \(\ds 34\) | |||||||||||

\(\text {(3)}: \quad\) | \(\ds x + 2 y + 3 z\) | \(=\) | \(\ds 26\) | |||||||||||

\(\text {(4)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds x - y\) | \(=\) | \(\ds 5\) | $(1) - (2)$ | |||||||||

\(\ds \leadsto \ \ \) | \(\ds 3 x + 2 \paren {x - 5} + z\) | \(=\) | \(\ds 39\) | substituting for $y$ in $(1)$ | ||||||||||

\(\text {(5)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds 5 x + z\) | \(=\) | \(\ds 49\) | simplifying | |||||||||

\(\ds \leadsto \ \ \) | \(\ds 2 x - 2 z\) | \(=\) | \(\ds 13\) | $(1) - (3)$ | ||||||||||

\(\text {(6)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds x - z\) | \(=\) | \(\ds 6 \tfrac 1 2\) | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds 6 x\) | \(=\) | \(\ds 55 \tfrac 1 2\) | $(5) - (6)$ | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds 9 \tfrac 1 4\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds 4 \tfrac 1 4\) | substituting for $x$ in $(4)$ | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds 9 \tfrac 1 4 - 6 \tfrac 1 2\) | substituting for $x$ in $(6)$ | ||||||||||

\(\ds \) | \(=\) | \(\ds 2 \tfrac 3 4\) |

$\blacksquare$

## Sources

- c. 100: Anonymous:
*Chiu Chang Suann Jing* - 1913: Yoshio Mikami:
*The Development of Mathematics in China and Japan* - 1992: David Wells:
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