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Recently I met a women which said: Do you know something funny? If you reverse my own age, the figures represent my husbands age. He is, of course, senior to me, and the difference between our age is $\frac{1}{11}$ of their sum.

Can you find out the women's age as well as her husband's age?

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Their ages are:

Wife: $45$, Husband: $54$

Reason:

Let wife's age be represented as: $10x+y$ where $x$ and $y$ are single digit natural numbers. Now, husband's age is $10y+x$. So, now, difference of their ages: $9y-9x = \frac{1}{11}(11x+11y) = x+y = k$. So, $9x+9y=9k$ and $9y-9x = k$. Thus, $y=\frac{5k}{9}, x=\frac{4k}{9}$. So, since $x, y$ are digits and are strictly integers, the only possible answer is $k=9$. Thus leading us to the answer.

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  • $\begingroup$ In fact, it is not mention in the problem that the ages are two-digit numbers. Other answers include $495, 4995, 49995, \dots$ and there are even more, such as $4545$ and $495495$. It is easy to show that any answer must begin with $4$ and end with $5$. $\endgroup$ – WhatsUp Aug 18 '20 at 2:06
  • $\begingroup$ And actually it is this OEIS sequence. $\endgroup$ – WhatsUp Aug 18 '20 at 2:22

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