Paschim Express left Delhi for Mumbai at 14.30 hrs travelling at a speed of 60 kmph and August Kranti Express left Delhi for Mumbai on the same day at 16.30 hrs travelling at a speed of 80 kmph. How far away from Delhi will the two trains meet (excluding stoppages)?

### Concept & Trains in Same Direction When two trains travel in the same direction, the faster train starting later will eventually overtake the slower train. The time taken to overtake depends on the initial head start distance and their relative speed. $$Time\ to\ Overtake = \frac{Head\ Start\ Distance}{Relative\ Speed}$$ ### Step-by-Step Solution 1. **Determine the time gap:** Paschim Express leaves at 14:30. August Kranti leaves at 16:30. Time difference = 2 hours. 2. **Calculate initial head start:** In these 2 hours, Paschim Express covers a certain distance. Head start distance = Speed of Paschim $\times$ Time = $60 \text{ km/h} \times 2 \text{ h} = 120 \text{ km}$. 3. **Find relative speed:** Since both travel from Delhi to Mumbai (same direction), relative speed = Speed of August Kranti - Speed of Paschim. Relative speed = $80 \text{ km/h} - 60 \text{ km/h} = 20 \text{ km/h}$. 4. **Calculate time to catch up:** Time taken for August Kranti to catch Paschim = Head start / Relative speed. Time = $120 \text{ km} / 20 \text{ km/h} = 6 \text{ hours}$. 5. **Calculate meeting distance:** The meeting distance from Delhi is the distance August Kranti Express travels in those 6 hours. Distance = Speed of August Kranti $\times$ Time = $80 \text{ km/h} \times 6 \text{ h} = 480 \text{ km}$. ### Exam Strategy & Shortcut Let $t$ be the time taken by August Kranti to catch up. Paschim Express travels for $t + 2$ hours. Distance is same: $60(t + 2) = 80t \implies 60t + 120 = 80t \implies 20t = 120 \implies t = 6$ hrs. Distance from Delhi = $80 \times 6 = 480 \text{ km}$. ### Common Pitfall Often students correctly find the time it takes the second train to catch the first (6 hours) but then incorrectly multiply this time by the slower train's speed or forget to add the initial 2 hours if they use the slower train for the final calculation ($60 \times (6+2) = 480$). ### Final Answer Therefore, the correct answer is **480 km**.