Someone says this to their friend: I am seven times as old as you were when your brother was half as old as you are now. All three of our ages are whole numbers and the sum of them is 74 years.

Which of us is how old?
Let the person addressed be $x$ years old at the earlier time, and his brother $y$ years older, so "then" he was $x+y$ years old. Thus, the speaker is now $7 x$ years old, the current age of the person addressed is twice his brother's age at that time, $2 x+2 y$ years, and his brother is now $2 x+3 y$ years old. Their combined age is

$$
7 x+(2 x+2 y)+(2 x+3 y)=11 x+5 y=74
$$

Since $x$ and $y$ are positive integers, $11 x$ must end in 4 or 9, so $x$ can only be 4, and then $y$ is 6. Therefore, the speaker is 28 years old, the person addressed is 20, his brother is 26, and it is easy to see that all conditions of the problem are satisfied.