A and B walk around a circular park. They start at 8 a.m. from the same point in the opposite directions. A and B walk at a speed of 2 rounds per hour and 3 rounds per hour respectively. How many times shall they cross each other after 8.00 a.m. and before 9.30 a.m. ?

The correct option is 7.

Let's break down the solution step-by-step to understand why A and B cross each other exactly 7 times between 8:00 a.m. and 9:30 a.m.

Step 1: Understand the speeds and relative speed
A walks at a speed of 2 rounds per hour.
B walks at a speed of 3 rounds per hour.
Since they start from the same point and walk in opposite directions around the circular park, their relative speed (the speed at which they cover the circular track relative to each other) is the sum of their individual speeds:
Relative speed = Speed of A + Speed of B
Relative speed = 2 rounds/hour + 3 rounds/hour = 5 rounds/hour.

This means that, relative to each other, they cover 5 complete rounds of the circular park in one hour. Since they are moving in opposite directions, every time they cover a combined distance equal to 1 complete round of the circle, they cross each other once. Therefore, they cross each other 5 times every hour.

Step 2: Calculate the time duration
The time interval is from 8:00 a.m. to 9:30 a.m.
Duration = 1 hour and 30 minutes = 1.5 hours.

Step 3: Calculate the number of crossings
To find the total number of crossings, we multiply the duration by the rate of crossings per hour:
Number of crossings = Relative speed in rounds per hour × Time duration in hours

$$\text{Number of crossings}=5\times 1.5=7.5$$

Since crossings can only occur as discrete, completed events, we look at the integer part of this value. They will cross each other exactly 7 times. The 8th crossing would occur after 9:30 a.m. (specifically at 9:36 a.m.). Thus, the number of crossings after 8:00 a.m. and before 9:30 a.m. is 7.