Let's solve this problem step by step.

We are given:

• M runs 5 rounds per hour.
• N runs 2 rounds per hour.
• They run in opposite directions, starting from the same point.
• We need to find how many times they will meet in 1.5 hours.

Step 1: Calculate relative speed

Since M and N run in opposite directions, their speeds will add up. The relative speed is: $\text{Relative Speed} = 5 \text{ rounds/hour} + 2 \text{ rounds/hour} = 7 \text{ rounds/hour}.$

Step 2: Calculate total rounds they cover together in 1.5 hours

In 1.5 hours, they will cover: $\text{Total rounds} = 7 \text{ rounds/hour} \times 1.5 \text{ hours} = 10.5 \text{ rounds}.$

Step 3: Determine how many times they meet

Each time they meet corresponds to one complete round they cover together. Since they cover 10.5 rounds in total, they will meet 10 times (since the last half-round means they haven't fully completed the 11th meeting).

So, M and N will meet 10 times in 1.5 hours.

Would you like any further details or clarifications?

Here are 5 related questions to explore:

1. What if M and N were running in the same direction? How would their meeting frequency change?
2. If the track length was given, how would we incorporate that into the solution?
3. What if N increased their speed? How would the number of meetings change?
4. How long would it take for them to meet 15 times if they continued running?
5. Can this problem be modeled using relative motion in physics?

Tip: When solving problems involving relative motion, remember to consider whether the two objects are moving in the same or opposite directions, as this will affect their relative speed.