Census and censusability
Willy and Wally Woolley were brothers, and they were sensitive about their ages. So when they had to fill in the census form, they left their ages blank. The census official pointed out that they were obliged by law to provide the information, but still the brothers hesitated.
The official could see that Willy was older than Wally, which was, he supposed, a start. “What’s your total age?” he asked, hoping at least to get an estimate, but the brothers were having none of it.
“Twelve times the difference between our ages,” said Willy cautiously, hoping not to reveal too much.
The official thought for a moment. “What’s the difference between the squares of your ages?”
This time it was Wally who answered, after much counting on fingers. “Twenty times our average age,” he said.
“Excellent,” the official replied. “Now I know what to write on the form.”
What were Wally and Willy’s ages? (Assume they are whole numbers.)
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Wally was 55 and Willy was 65.
Let Wally’s age be a and Willy’s b. Willy said that a+b = 12(b-a), so 11b = 13a. Since the ages are whole numbers and 11 and 13 have no common factor, there must be a whole number c such that a = 11c and b = 13c. Then the average is 12c, and the difference of the squares is 48c2. So 48c2 = 240c, and c = 5. Therefore a = 55 and b = 65.
The winner was Rebecca Teasdale, Cockermouth, Cumbria