Henry Ernest Dudeney/Puzzles and Curious Problems/44 - Their Ages/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $44$
- Their Ages
- A man, on being asked the ages of his two sons, stated that
- eighteen more than the sum of their ages is double the age of the elder,
- and six less than the difference of their ages is the age of the younger.
- What are their ages?
Solution
The sons are aged $30$ and $12$.
Proof
Let $a$ and $b$ be the ages of the older and younger son respectively.
We have:
| \(\ds a + b + 18\) | \(=\) | \(\ds 2 a\) | eighteen more than the sum of their ages is double the age of the elder | |||||||||||
| \(\ds a - b - 6\) | \(=\) | \(\ds b\) | six less than the difference of their ages is the age of the younger | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds b + 18\) | \(=\) | \(\ds a\) | |||||||||||
| \(\ds a\) | \(=\) | \(\ds 2 b + 6\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds b + 18\) | \(=\) | \(\ds 2 b + 6\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds b\) | \(=\) | \(\ds 12\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds 30\) |
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $44$. -- Their Ages
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $39$. Their Ages