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
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $19$. -- A Poultry Poser
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $25$. A Poultry Poser