2. Two friends walked out of School No. 15 after the Olympiad and went in different directions, pondering a difficult problem. Misha walked 8 km per hour, while Vasya walked half the distance. After 45 minutes, they came up with a solution to the problem and started running towards each other to share the solution. Each of them increased their speed by one and a half times. How long did it take for them to meet? What distance did they cover before meeting?
2. The first method of solving.
1) $8: 2=4$ km $/ h$ - Vasya's speed;
2) $8+4=12 \text{km} / h=12000: 60=200 \text{m} /$ min - the speed of moving away from each other;
3) $200 \times 45=9000 \text{m}=9$ km - the distance the boys have traveled;
4) $8+4=12$ km $/ h$ - Misha's new speed;
5) $4+2=6$ km/h - Vasya's new speed;
6) $12+6=18$ km $/ h=18000: 60=300$ m/min - the speed of approaching each other;
7) $9000: 300=30$ min - the time of meeting;
8) $9+9=18$ km - the distance traveled until the meeting.

The second method of solving. Since they have traveled and run the same distance, when the speed increases by $3 / 2$ times, the time should decrease by the same factor - $45: 3 \times 2=30$.

Answer: They met after 30 minutes, having traveled 18 km until the meeting.