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

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