Chiu Chang Suann Jing/Examples/Example 5
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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: Curious and Interesting Puzzles ... (previous) ... (next): The Nine Chapters: $63$