Henry Ernest Dudeney/Puzzles and Curious Problems/5 - Dividing the Legacy/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $5$

Dividing the Legacy
A man left $\pounds 100$ to be divided between his two sons Alfred and Benjamin.
If one-third of Alfred's legacy be taken from one-fourth of Benjamin's, the remainder would be $\pounds 11$.
What was the amount of each legacy?


Solution

Alfred gets $\pounds 24$ and Benjamin gets $\pounds 76$.


Proof

Let $A$ and $B$ be the amount of Alfred's and Benjamin's legacies respectively.

We have:

\(\text {(1)}: \quad\) \(\ds A + B\) \(=\) \(\ds 100\) $\pounds 100$ to be divided between his two sons
\(\text {(2)}: \quad\) \(\ds \dfrac B 4 - \dfrac A 3\) \(=\) \(\ds 11\) If one-third of Alfred's legacy be taken from one-fourth of Benjamin's, the remainder would be $\pounds 11$.
\(\text {(3)}: \quad\) \(\ds \leadsto \ \ \) \(\ds 3 B - 4 A\) \(=\) \(\ds 132\) $(2) \times 12$
\(\text {(4)}: \quad\) \(\ds 4 B + 4 A\) \(=\) \(\ds 400\) $(1) \times 4$
\(\ds \leadsto \ \ \) \(\ds 7 B\) \(=\) \(\ds 532\) $(3) + (4)$
\(\ds \leadsto \ \ \) \(\ds B\) \(=\) \(\ds 76\)
\(\ds \leadsto \ \ \) \(\ds A\) \(=\) \(\ds 24\)

$\blacksquare$


Sources

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