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

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