5. (7 points) Mom is walking around a lake with a stroller and completes a full lap around the lake in 12 minutes. Vanya rides a scooter on the same path in the same direction and meets (overtakes) mom every 12 minutes. At what intervals will Vanya meet mom if he rides at the same speed but in the opposite direction?
Answer: Every 4 minutes.

Solution. Since Mom completes a full lap around the lake in 12 minutes and meets Vanya every 12 minutes, in 12 minutes Vanya rides around the lake exactly 2 times, while Mom completes one lap. Therefore, Vanya's speed is twice that of Mom's. From this, it follows that when Vanya was riding in the same direction as Mom, their closing speed was equal to Mom's speed. If Vanya rides in the opposite direction, their closing speed will be three times Mom's speed, i.e., three times faster. Therefore, he will meet Mom three times more frequently, i.e., every 4 minutes.

This reasoning can be conducted by introducing a notation for the length of the path.

Let \( l \) be the length of the path around the lake (in meters), then Mom's speed is \( \frac{l}{12} \) (m/min), and Vanya's speed is \( -\frac{l}{6} \) (m/min). The closing speed when Mom and Vanya are riding towards each other is \( \frac{3l}{12} = \frac{l}{4} \) (m/min). Therefore, with this speed, they will cover \( l \) meters together in 4 minutes, i.e., they will meet every 4 minutes.

## Grading Criteria.

- Any complete and correct solution - 7 points.
- Correctly found that Vanya's speed is twice that of Mom's, correctly found the sum of speeds, but the final conclusion is incorrect - 2 points.
- Correctly and justifiedly found that Vanya's speed is twice that of Mom's, but further reasoning is either unjustified or not completed - 1 point.
- A solution that provides specific distances and speeds and arrives at the correct answer - 1 point.
- Only the correct answer - 0 points.

Maximum score for all completed tasks - 35.