Problem 6.3. Vitya and his mother left home at the same time and walked in opposite directions at the same speed: Vitya - to school, and his mother - to work. After 10 minutes, Vitya realized he didn't have the keys to the house, and he would return from school earlier than his mother, so he started to catch up with her, increasing his speed fivefold. How many minutes after he realized he needed to get the keys will Vitya catch up with his mother?
Answer: 5 minutes.

Solution. Let Vitya and his mother initially walk at a speed of $s$ meters per minute. After 10 minutes, when Vitya realized he had forgotten his keys, the distance between him and his mother became $10 s + 10 s = 20 s$. When Vitya started to catch up with his mother, he walked at a speed of $5 s$ meters per minute, so the distance between him and his mother decreased by $4 s$ meters per minute. Therefore, it would take $20 s : 4 s = 5$ minutes for the distance to reduce to zero.