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
- c. 100: Anonymous: Chiu Chang Suann Jing
- 1965: Henrietta Midonick: The Treasury of Mathematics: Volume $\text { 1 }$
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): The Nine Chapters: $62$