Buses start from a bus terminal with a speed of 20 km/hr at intervals of 10 minutes. What is the speed of a man coming from the opposite direction towards the bus terminal if he meets the buses at intervals of 8 minutes?

### Concept & Relative Speed When objects move in opposite directions, their relative speed is the sum of their individual speeds. The physical distance between consecutive buses remains constant and can be calculated based on the buses' speed and starting interval. $$Distance = Relative\ Speed \times Time$$ ### Step-by-Step Solution 1. **Find the distance between consecutive buses:** The buses leave at 10-minute intervals. Distance = Speed $\times$ Time = $20 \text{ km/hr} \times \frac{10}{60} \text{ hours} = \frac{20}{6} \text{ km} = \frac{10}{3} \text{ km}$. 2. **Set up the relative speed equation:** Let the speed of the man be $v$ km/hr. Since they move in opposite directions, their relative speed is $(20 + v)$ km/hr. 3. **Equate the distance:** The man meets the buses every 8 minutes. The relative distance covered in this time is exactly the distance between two consecutive buses. $$(20 + v) \times \frac{8}{60} \text{ hours} = \frac{10}{3} \text{ km}$$ 4. **Solve for $v$:** $$(20 + v) \times \frac{2}{15} = \frac{10}{3}$$ $$20 + v = \frac{10}{3} \times \frac{15}{2}$$ $$20 + v = 5 \times 5$$ $$20 + v = 25$$ $$v = 5 \text{ km/hr}$$ ### Exam Strategy & Shortcut Equating distances directly using speed and time ratios: Distance = Bus Speed $\times$ Interval Time = Relative Speed $\times$ Meeting Time. $20 \times 10 = (20 + v) \times 8$ $200 = 160 + 8v \Rightarrow 8v = 40 \Rightarrow v = 5$. *Note: Since both sides use minutes, we don't even need to convert to hours.* ### Common Pitfall Converting units inconsistently. Either keep everything in km/hr and hours, or recognize when units cancel out in a direct ratio equality as shown in the shortcut. ### Final Answer Therefore, the correct answer is **5 km/hr**.