Chiu Chang Suann Jing/Examples/Example 4

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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

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