# Chiu Chang Suann Jing/Examples/Example 4

## 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 \ \$ $\ds x$ $=$ $\ds 4 - \tfrac {50} {21}$ $\ds$ $=$ $\ds 1 \tfrac {11} {21}$

$\blacksquare$