[ Word problems ]

The king said to the queen: "I am now twice as old as you were when I was as old as you are now. When you are as old as I am now, together we will be sixty-three years old." How old is each of them?
Try to denote the king's age "then" by $x$, the queen's age "then" by $y$, and form a system of equations. Think about how to solve this problem without forming a system of equations.

## Solution

Let $t$ be the age difference between the king and the queen "now," "then," and "always." Since the queen is now as old as the king was "then," the time that has passed from "then" to "now" is also $t$ years. The difference between the king's age "now" and the queen's age "then" is the sum of two numbers - the age difference between them "always" and the period from "then" to "now." This sum is $-2t$. Therefore, the queen's age "then" was $-2t$ years, and the king's age "now" is $4t$ years. "Now," the queen is $3t$ years old, and the king "then" was $3t$ years old. When the queen is $4t$ years old, the king will be $5t$ years old. Together, these $9t$ years make up 63 years. Thus, $t=7$. Therefore, "now" the king is 28 years old, and the queen is 21 years old.