A passenger train runs at the rate of $80$ kmph. It starts from the station, $6$ hours after a goods train leaves the station. The passenger train overtakes the goods train after $4$ hours. The speed of goods train is

### Concept & Logic When one vehicle overtakes another after starting from the exact same point, they have traveled the exact same total distance. We use the fundamental distance formula: $\text{Distance} = \text{Speed} \times \text{Time}$. ### Step-by-Step Solution * **Given:** Passenger train speed = $80$ kmph. It travels for $4$ hours before overtaking. The goods train traveled for those $4$ hours plus its $6$-hour head start. * **Calculation:** Distance traveled by the passenger train = $80 \text{ km/hr} \times 4 \text{ hr} = 320 \text{ km}$. * Since the passenger train overtakes the goods train, the goods train must have also traveled exactly $320 \text{ km}$ from the station. * Total time taken by the goods train = $6 \text{ hours (head start)} + 4 \text{ hours (time until overtaken)} = 10 \text{ hours}$. * Speed of the goods train = $\frac{\text{Distance}}{\text{Time}} = \frac{320}{10} = 32$ km/hr. ### Exam Strategy & Shortcut Since the total distance is constant for both trains, you can set up an inverse proportionality: $S_1 \times T_1 = S_2 \times T_2$. Therefore, $80 \times 4 = S_2 \times 10 \implies 320 = 10S_2 \implies S_2 = 32$. ### Common Pitfall A common error is forgetting to add the $4$ hours of mutual travel time to the goods train's total time. Students often mistakenly use just the $6$-hour head start. ### Final Answer Therefore, the correct answer is **$32$ km/hr**.