P and Q walk along a circular track. They start at 5:00 a.m. from the same point in opposite directions. P walks at an average speed of 5 rounds per hour and Q walks at an average speed of 3 rounds per hour. How many times will they cross each other between 5:20 a.m. and 7:00 a.m.?

To solve this, we use the concept of relative speed in a circular track.

1. Calculate Relative Speed: Since P and Q move in opposite directions, their relative speed is the sum of their individual speeds. Relative Speed = 5 rounds per hour + 3 rounds per hour = 8 rounds per hour. This means they cross each other 8 times every hour.

2. Determine the Total Time: The total duration from 5:00 a.m. to 7:00 a.m. is 2 hours. In 2 hours, they cross a total of 8 times 2 = 16 times.

3. Find the Crossing Times: Since they start at 5:00 a.m., crossings occur at regular intervals of 60 minutes divided by 8, which is every 7.5 minutes. The crossings happen at: 5:07.5, 5:15, 5:22.5, 5:30, 5:37.5, 5:45, 5:52.5, 6:00, 6:07.5, 6:15, 6:22.5, 6:30, 6:37.5, 6:45, 6:52.5, and 7:00.

4. Filter for the Specific Interval: We need to count crossings between 5:20 a.m. and 7:00 a.m. The first two crossings (5:07.5 and 5:15) happen before 5:20. The total crossings by 7:00 a.m. are 16. Subtracting the 2 early crossings: 16 minus 2 = 14. However, the question asks for crossings between the times. If we exclude the start and end of the interval, or strictly follow the 7.5 minute gap logic starting from 5:22.5 up to 7:00, there are exactly 13 instances.

Starting from 5:22.5 as the first crossing after 5:20, and counting up to 6:52.5 (the last crossing before 7:00), there are 13 crossings.

The correct answer is B.