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New Year Deserves New Age Problem

Age problems have always been a staple in mathematics textbooks... especially in algebra word problem sections. However, this problem is not a straight-forward algebra problem and requires some additional thinking...

Problem: In 1932, my dad was as old as the last two digits of his birth year. When he mentioned this interesting coincidence to his grandfather, my dad was surpised when his grandfather said the same thing was true for him as well. Believe me, it's quite possible and I am able to prove it too. How old was my father and his grandfather in 1932?

Note: Is there anything special about the year 1932? That is, could the problem still be solvable for years other than 1932?


Source: Yakov Perelman, Mathematics Can Be Fun, Moscow: MIR Publishers, 1985

Hint: Try some cases....could my dad have been born in the 1800's? Remember that his grandfather was still alive in 1932, so be realistic in your exploration as well.


Solution Commentary: After some playing, you should discover that everything can be summarized by two "key" equations:

2(10a+b) = 32 and 2(10c+d) = 132
where my father was born in 19ab and his grandfather in 18cd. The solution quickly follows from this...

Now, as to the year 1932...what happens if you shift to 1934...1960...1998? And, why am I forcing the year to end in an even digit? Could this ever happen in the 21st century?

G.H. (Vancouver, WA) adds these comments: "Some fun problems this time! Dad was born in 1916 and was 16 in 1932. G'pa was born in 1866 and was 66 in 1932. I don't think 1932 is the only year this could be done, but I think it has to be an even year, not an odd one. (If it is 1934, then Dad was born in 1917 and is 17 in 1934. G'pa was born in 1867 and is 67 in 1934.)"