Henry Ernest Dudeney/Puzzles and Curious Problems/27 - Buying Tobacco/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $27$
- Buying Tobacco
- A box of $50$ cigarettes cost the same in shillings and pence as some tobacco bought in pence and shillings.
- The change out of a $10 \shillings$ note was the same as the cost of the cigarettes.
- What did the cigarettes cost?
Solution
The cigarettes cost $2 \shillings 5 \oldpence$, while the tobacco cost $5 \shillings 2 \oldpence$
Proof
Recall that $1 \shillings$ is worth $12 \oldpence$
Suppose the cigarettes cost $x \shillings y \oldpence$
Then the tobacco would cost $y \shillings x \oldpence$
In pence, the cigarettes cost $\paren {12 x + y} \oldpence$ while the tobacco cost $\paren {12 y + x} \oldpence$
We are given that:
- $10 \times 12 - \paren {12 x + y} - \paren {12 y + x} = \paren {12 x + y}$
which can be rewritten as:
- $25 x + 14 y = 120$
Taking $\bmod 5$, we have:
- $14 y \equiv 0 \pmod 5$
which by Euclid's Lemma implies that $y$ is divisible by $5$.
$y \ne 0$ as $120$ is not divisible by $25$.
$y < 10$ as $14 y > 120$ otherwise.
Therefore $y = 5$, and $x = 2$ follows.
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $27$. -- Buying Tobacco