Henry Ernest Dudeney/Modern Puzzles/17 - Name their Wives/Solution
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Modern Puzzles by Henry Ernest Dudeney: $17$
- Name their Wives
- A man left a legacy of $\pounds 1000$ to $3$ relatives and their wives.
- The wives received together $\pounds 396$.
- Jane received $\pounds 10$ more than Catherine,
- and Mary received $\pounds 10$ more than Jane.
- John Smith was given just as much as his wife,
- Henry Snooks got half as much again as his wife,
- and Tom Crowe received twice as much as his wife.
- What was the Christian name of each man's wife?
Solution
- Catherine is Mrs. John Smith
- Jane is Mrs. Henry Snooks
- Mary is Mrs. Tom Crowe.
Proof
Let $C$, $M$ and $J$ be the sizes of legacies of Catherine, Mary and Jane respectively.
We are given that:
\(\ds J\) | \(=\) | \(\ds 10 + C\) | ||||||||||||
\(\ds M\) | \(=\) | \(\ds 10 + J\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 10 + \paren {10 + C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 20 + C\) | ||||||||||||
\(\ds C + J + M\) | \(=\) | \(\ds 396\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds C + \paren {10 + C} + \paren {20 + C}\) | \(=\) | \(\ds 396\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3 C + 30\) | \(=\) | \(\ds 396\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3 C\) | \(=\) | \(\ds 366\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds C\) | \(=\) | \(\ds 122\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds J\) | \(=\) | \(\ds 132\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds M\) | \(=\) | \(\ds 142\) |
Now we note that the total inherited by the husbands is $1000 - 396 = 604$.
Using the method of false position, we multiply $C$, $J$ and $M$ by $1$, $\dfrac 3 2$ and $2$ respectively, and sum them to see what we get.
Thus we find:
- $122 + \dfrac 3 2 \times 132 + 2 \times 142 = 604$
and marvel that the first guess was spot on.
So:
- Catherine is Mrs. John Smith
- Jane is Mrs. Henry Snooks
- Mary is Mrs. Tom Crowe.
$\blacksquare$
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $17$. -- Name their Wives
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $10$. Name their Wives