# The age of husband and wife

My husband's age can be written by reversing the figures of my own age. He is older than me and the difference between our ages is equivalent to the eleventh of the sum of both.

What is my age and that of my husband?

#### Solution

Suppose we can rejuvenate the man until he is the same age as the woman. What would we have taken at his age? Well, the difference between the two, of course. And if we add them, how much will we have decreased the sum? We will have removed that eleventh part, so that now that sum will be divided into ten equal piles. And since they are the same age now, five will be from the wife and five will be from the husband.

With all of the above, we can see that the difference between both ages is actually one fifth of the woman's age. In other words, the woman's age is divisible by 5, so it ends in 5.

The husband's age will then start at 5 and will be older than the wife's. We can prove which numbers ending in 5 increase a fifth by turning them, up to 45 (because then they decrease, of course). We will verify that it only happens to 45.

That is the woman is 45 years old and the man 54 (9 years, the fifth part of the woman's age or the eleventh part of 99, the sum of both ages).