A walks around a circular field at the rate of one round per hour while B runs around it at the rate of six rounds per hour. They start in the same direction from the same point at 7.30 a.m. They shall first cross each other at :

### Concept & Relative Speed When two bodies move in the same direction along a circular path, their relative speed is the difference of their individual speeds. They will cross each other for the first time when the faster body gains exactly one complete round over the slower body. $$Relative Speed = Speed_B - Speed_A$$ ### Step-by-Step Solution 1. **Identify individual speeds:** Speed of A = 1 round/hour. Speed of B = 6 rounds/hour. 2. **Calculate relative speed:** Since they move in the same direction, relative speed = 6 - 1 = 5 rounds/hour. 3. **Determine time to meet:** They will cross each other when B gains exactly 1 round on A. Time taken = (Distance to gain) / (Relative Speed) = 1 / 5 hour. 4. **Convert time to minutes:** 1/5 hour = (1/5) * 60 minutes = 12 minutes. 5. **Calculate meeting time:** They start at 7.30 a.m. Adding 12 minutes gives 7.42 a.m. ### Exam Strategy & Shortcut For circular tracks with same direction, meetings occur at intervals of $T = \frac{Distance}{Relative Speed}$. Here distance is 1 lap, relative speed is 5 laps/hr. Time is simply 1/5 hr or 12 mins. Add directly to start time. ### Common Pitfall A common mistake is adding the speeds instead of subtracting them. Remember, speeds are subtracted when objects move in the same direction. ### Final Answer Therefore, the correct answer is **7.42 a.m.**