P and Q walk along a circular track. They start at 5:00 am. 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.?

Correct Option

13

To determine the number of times P and Q cross each other, the relative speed must first be calculated. Since they are moving in opposite directions on a circular track, their speeds add up.

• P's speed = 5 rounds per hour
• Q's speed = 3 rounds per hour
• Relative speed = 5 + 3 = 8 rounds per hour.

This relative speed indicates that they complete 8 crossings in one hour. Therefore, the time taken for one crossing is:

• Time per crossing = 1 hour / 8 crossings = 60 minutes / 8 = 7.5 minutes.

Next, calculate the total duration of the specified interval:

• Start time = 5:20 a.m.
• End time = 7:00 a.m.
• Duration = 1 hour 40 minutes = 60 minutes + 40 minutes = 100 minutes.

Finally, divide the total duration by the time taken for one crossing to find the number of crossings:

• Number of crossings = Total duration / Time per crossing = 100 minutes / 7.5 minutes/crossing = 13.33 crossings.

Since the question asks for the number of times they cross each other, only complete crossings are counted. Thus, they will cross each other 13 times between 5:20 a.m. and 7:00 a.m.

Incorrect Options

Options 1 (12), 3 (14), and 4 (15) are incorrect as they do not align with the calculated number of complete crossings based on the relative speeds and the specified time interval. The precise calculation shows 13 complete crossings.