Henry Ernest Dudeney/Puzzles and Curious Problems/19 - A Poultry Poser/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $19$

A Poultry Poser
Three chickens and one duck sold for as much as two geese;
one chicken, two ducks, and three geese were sold together for $25 \shillings$
What was the price of each bird in an exact number of shillings?


Solution

A chicken costs $2 \shillings$, a duck costs $4 \shillings$ and a goose costs $5 \shillings$


Proof

Let $c$, $d$ and $g$ be the price in shillings for one chicken, one duck and one goose respectively.

We have:

\(\text {(1)}: \quad\) \(\ds 3 c + d\) \(=\) \(\ds 2 g\) Three chickens and one duck sold for as much as two geese;
\(\text {(2)}: \quad\) \(\ds c + 2 d + 3 g\) \(=\) \(\ds 25\) one chicken, two ducks, and three geese were sold together for $25 \shillings$.
\(\ds \leadsto \ \ \) \(\ds d\) \(=\) \(\ds 2 g - 3 c\) rearranging $(1)$
\(\ds \leadsto \ \ \) \(\ds c + 2 \paren {2 g - 3 c} + 3 g\) \(=\) \(\ds 25\) substituting for $d$ in $(2)$
\(\ds \leadsto \ \ \) \(\ds 7 g\) \(=\) \(\ds 5 \paren {c + 5}\) simplifying

It follows that $g$ is a multiple of $5$.

If $g \ge 10$ we have from $(2)$ that $c + 2 d < 0$ which cannot happen.

Hence $g = 5$ and so:

\(\text {(3)}: \quad\) \(\ds 3 c + d\) \(=\) \(\ds 10\) substituting $g = 5$ in $(1)$
\(\text {(4)}: \quad\) \(\ds c + 2 d\) \(=\) \(\ds 10\) substituting $g = 5$ in $(2)$ and simplifying
\(\text {(5)}: \quad\) \(\ds \leadsto \ \ \) \(\ds 6 c + 2 d\) \(=\) \(\ds 20\) $(3) \times 2$
\(\ds \leadsto \ \ \) \(\ds 5 c\) \(=\) \(\ds 10\) $(1) - (2)$
\(\ds \leadsto \ \ \) \(\ds c\) \(=\) \(\ds 2\)
\(\ds \leadsto \ \ \) \(\ds 3 \times 2 + d\) \(=\) \(\ds 10\)
\(\ds \leadsto \ \ \) \(\ds d\) \(=\) \(\ds 4\)

$\blacksquare$


Sources

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