## 39. How old am I?

Try to answer this question if it is known that Alfred's age is obtained by rearranging the digits of my age, that the quotient of the number equal to my age divided by the sum of these two digits differs from the analogous quotient for Alfred by a number equal to the difference between these two digits, and that, finally, the product of these two quotients is exactly equal to my age.
39. Let $x$ be my age, $d$ be the number of tens in it, and $u$ be the number of units. Then $x=10 d+u$, and Alfred's age is $10 u+d$. The first condition of the problem can be written as:

$$
\left|\frac{10 d+u}{d+u}-\frac{10 u+d}{d+u}\right|=|d-u| \cdot
$$

Simplifying this equality, we get

$$
9|d-u|=(d+u) \cdot|d-u|
$$

from which

$$
d+u=9 \text {, i.e., } d=9-u \text {. }
$$

192

Now let's write the second condition of the problem

$$
\frac{10 d+u}{d+u} \cdot \frac{10 u+d}{d+u}=10 d+u
$$

or

$$
10 u+d=(d+u)^{2}
$$

Replacing $d$ with $9-u$, we get

$$
9 u+9=81
$$

Therefore,

$$
u=8 \text { and } d=1 \text {. }
$$

So, I am 18 years old, and Alfred is 81 years old.