Henry Ernest Dudeney/Puzzles and Curious Problems/136 - Juvenile Highwaymen/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $136$

Juvenile Highwaymen
Three juvenile highwaymen called upon an apple-woman to "stand and deliver."
Tom seized half of the apples, but returned $10$ to the basket;
Ben took one-third of what were left, but returned $2$ that he did not fancy;
Jim took half of the remainder, but threw back one that was worm-eaten.
The woman was then left with only $12$ in her basket.
How many had she before the raid was made?


Solution

The apple-woman had $40$ apples of which Tom had $10$, Ben had $8$ and Jim had $10$.


Proof

Let $n$ be the number of apples the apple-woman started with.

Let $T$, $B$ and $J$ be the number of apples stolen by Tom, Ben and Jim respectively.

We have:

\(\ds T\) \(=\) \(\ds \dfrac n 2 - 10\) Tom seized half of the apples, but returned $10$ to the basket;
\(\ds B\) \(=\) \(\ds \dfrac {n - T} 3 - 2\) Ben took one-third of what were left, but returned $2$ that he did not fancy;
\(\ds J\) \(=\) \(\ds \dfrac {n - T - B} 2 - 1\) Jim took half of the remainder, but threw back one that was worm-eaten.
\(\ds n - T - B - J\) \(=\) \(\ds 12\) The woman then left with only $12$ in her basket.

These can be more conveniently written as:

\(\text {(1)}: \quad\) \(\ds n - 2 T\) \(=\) \(\ds 20\)
\(\text {(2)}: \quad\) \(\ds n - T - 3 B\) \(=\) \(\ds 6\)
\(\text {(3)}: \quad\) \(\ds n - T - B - 2 J\) \(=\) \(\ds 2\)
\(\text {(4)}: \quad\) \(\ds n - T - B - J\) \(=\) \(\ds 12\)
\(\ds \leadsto \ \ \) \(\ds J\) \(=\) \(\ds 10\) $(4) - (3)$
\(\text {(5)}: \quad\) \(\ds \leadsto \ \ \) \(\ds n - T - B\) \(=\) \(\ds 22\) susbtituting for $J$ in $(4)$ and simplifying
\(\ds \leadsto \ \ \) \(\ds 2 B\) \(=\) \(\ds 16\) $(5) - (2)$
\(\ds \leadsto \ \ \) \(\ds B\) \(=\) \(\ds 8\)
\(\text {(6)}: \quad\) \(\ds \leadsto \ \ \) \(\ds n - T\) \(=\) \(\ds 30\) substituting for $B$ in $(2)$ or $(5)$ (either will do) and simplifying
\(\ds \leadsto \ \ \) \(\ds T\) \(=\) \(\ds 10\) $(6) - (1)$
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds 40\) substituting for $B$ in $(1)$ or $(6)$ (either will do) and simplifying

$\blacksquare$


Sources

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