Henry Ernest Dudeney/Puzzles and Curious Problems/5 - Dividing the Legacy/Solution
Jump to navigation
Jump to search
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
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $5$. -- Dividing the Legacy
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $15$. Dividing the Legacy