# Chiu Chang Suann Jing/Examples/Example 4

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## Example of Problem from

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

*If $5$ oxen and $2$ sheep together cost $10$ taels of gold,**and $2$ oxen and $5$ sheep together cost $8$ taels,**what are the prices of oxen and sheep respectively?*

## Solution

An ox costs $1 \frac {11} {21}$ tael.

A sheep costs $\frac {20} {21}$ tael.

It would appear that the numbers for the problem were chosen with no thought to the convenience or plausibility of the answer.

## Proof

Let $x$ be the price per ox.

Let $y$ be the price per sheep.

Then:

\(\text {(1)}: \quad\) | \(\ds 5 x + 2 y\) | \(=\) | \(\ds 10\) | |||||||||||

\(\text {(2)}: \quad\) | \(\ds 2 x + 5 y\) | \(=\) | \(\ds 8\) | |||||||||||

\(\text {(3)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds 10 x + 4 y\) | \(=\) | \(\ds 20\) | $(1) \times 2$ | |||||||||

\(\text {(4)}: \quad\) | \(\ds 10 x + 25 y\) | \(=\) | \(\ds 40\) | $(2) \times 5$ | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds 21 y\) | \(=\) | \(\ds 20\) | $(4) - (3)$ | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \tfrac {20} {21}\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds 2 x + 5 \tfrac {20} {21}\) | \(=\) | \(\ds 8\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds 4 - \tfrac {50} {21}\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 1 \tfrac {11} {21}\) |

$\blacksquare$

## Sources

- c. 100: Anonymous:
*Chiu Chang Suann Jing* - 1965: Henrietta Midonick:
*The Treasury of Mathematics: Volume $\text { 1 }$* - 1992: David Wells:
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