[ Motion problems]

The road from home to school takes Petya 20 minutes. One day, on his way to school, he remembered that he had forgotten his pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell, but if he returns home for the pen, he will, walking at the same speed, be 7 minutes late for the start of the lesson. What part of the way had he walked before he remembered the pen?
On how much more time will Petya spend on the entire journey if he returns home for a pen, compared to how much time he would have spent if he did not return?

## Solution

If Petya returns home for a pen, he will spend $3+7=10$ minutes more on the entire journey than he would have if he did not return. This means that the round trip from the place where he remembered the pen to his home and back takes 10 minutes. Therefore, Petya remembered the pen 5 minutes into his walk from home, i.e., he had walked $\frac{1}{4}$ of the journey.

## Answer

$\frac{1}{4}$ of the journey.