$A$ and $B$ walk around a circular track. 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 before 9.30 a.m.?

### Concept & Relative Speed on Circular Tracks When two entities move in opposite directions on a circular track, the number of times they cross each other is proportional to the sum of their relative speeds measured in rounds per unit of time. ### Step-by-Step Solution * **Given:** Speed of $A = 2$ rounds/hr. Speed of $B = 3$ rounds/hr. Total time from 8 a.m. to 9:30 a.m. = $1.5$ hours. * **Calculation:** Since they travel in opposite directions, their relative speed is the sum of their individual speeds. * Relative speed $= 2 + 3 = 5$ rounds per hour. * This means they cross each other $5$ times every full hour. * In $1.5$ hours, the total calculated number of crossings $= 5 \times 1.5 = 7.5$. * Since they can only cross in whole discrete events, they cross exactly $7$ times. ### Exam Strategy & Shortcut Simply add the speeds in rounds/hr to get the crossing frequency ($5$ times/hr). Multiply by total hours ($1.5$) to get $7.5$. Drop the decimal to find the exact number of full meetings. ### Common Pitfall Rounding up $7.5$ to $8$ assuming they somehow complete the final meeting at exactly 9:30 a.m. The 8th meeting would mathematically occur slightly later at 9:36 a.m. ### Final Answer Therefore, the correct answer is **$7$**.