Train A leaves Ludhiana for Delhi at 11 a.m, running at the speed of 60 km/hr. Train B leaves Ludhiana for Delhi by the same route at 2 p.m. on the same day, running at the speed of 72 km/hr. At what time will the two trains meet each other?

### Concept & Relative Speed When two bodies move in the same direction, their relative speed is the difference of their individual speeds. The time taken to meet is the initial distance separating them divided by this relative speed. $$ \text{Relative Speed} = V_2 - V_1 $$ $$ \text{Time to Catch Up} = \frac{\text{Initial Distance}}{\text{Relative Speed}} $$ ### Step-by-Step Solution 1. **Calculate Initial Distance:** Train A leaves at 11 a.m. and Train B leaves at 2 p.m. Train A travels alone for 3 hours. 2. Distance covered by Train A in 3 hours = $60 \text{ km/hr} \times 3 \text{ hrs} = 180 \text{ km}$. 3. **Calculate Relative Speed:** Both trains are moving in the same direction. Relative Speed = $72 \text{ km/hr} - 60 \text{ km/hr} = 12 \text{ km/hr}$. 4. **Calculate Time to Catch Up:** Time = $\frac{\text{Distance}}{\text{Relative Speed}} = \frac{180 \text{ km}}{12 \text{ km/hr}} = 15 \text{ hours}$. 5. **Calculate Meeting Time:** Train B leaves at 2 p.m. Adding 15 hours to 2 p.m.: $2 \text{ p.m.} + 10 \text{ hours} = 12 \text{ a.m.}$ (Midnight) $12 \text{ a.m.} + 5 \text{ hours} = 5 \text{ a.m.}$ on the next day. ### Exam Strategy & Shortcut Instead of forming algebraic equations $60(t+3) = 72t$, directly think in terms of the "gap". Train A creates a 180 km gap. Train B closes this gap at a rate of 12 km per hour. The mental math $\frac{180}{12} = 15$ hours is significantly faster. ### Common Pitfall A frequent error is adding the 15 hours to Train A's departure time (11 a.m.) instead of Train B's departure time (2 p.m.), which would lead to an incorrect answer of 2 a.m. ### Final Answer Therefore, the correct answer is **5 a.m. on the next day**.