# A problem of Age

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?

Wife: $$45$$, Husband: $$54$$
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.
• 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 $5$. – WhatsUp Aug 18 '20 at 2:06