# Chiu Chang Suann Jing/Examples/Example 5

## 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 $(6)$ $\ds$ $=$ $\ds 2 \tfrac 3 4$

$\blacksquare$