Train A travelling at 60 km/hr leaves Mumbai for Delhi at 6 p.m. Train B travelling at 90 km/hr also leaves Mumbai for Delhi at 9 p.m. Train C leaves Delhi for Mumbai at 9 p.m. If all the three trains meet at the same time between Mumbai and Delhi, then what is the speed of train C if the distance between Delhi and Mumbai is 1260 km?

### Concept & Meeting Point Synchronization When multiple bodies meet at the same point and time, you can first determine the exact time and location of the meeting using two known bodies. Then, use that specific time and distance to deduce the properties of the third body. ### Step-by-Step Solution 1. **Find the meeting time of Train A and Train B:** Train A starts at 6 p.m. at 60 km/hr. Train B starts at 9 p.m. at 90 km/hr. By 9 p.m., Train A has travelled for 3 hours. Distance = $60 \times 3 = 180 \text{ km}$. Relative speed of B with respect to A = $90 - 60 = 30 \text{ km/hr}$. Time taken for B to catch A = $\frac{180 \text{ km}}{30 \text{ km/hr}} = 6 \text{ hours}$ (after 9 p.m.). Meeting time = 9 p.m. + 6 hours = 3 a.m. 2. **Find the meeting location:** Distance from Mumbai where A and B meet = Speed of B $\times$ Time of B = $90 \text{ km/hr} \times 6 \text{ hrs} = 540 \text{ km}$. 3. **Calculate Train C's speed:** Train C meets them at the same time (3 a.m.) and place (540 km from Mumbai). Total distance from Delhi to Mumbai = 1260 km. Distance Train C travelled from Delhi = $1260 \text{ km} - 540 \text{ km} = 720 \text{ km}$. Time taken by Train C = 9 p.m. to 3 a.m. = 6 hours. Speed of Train C = $\frac{\text{Distance}}{\text{Time}} = \frac{720 \text{ km}}{6 \text{ hours}} = 120 \text{ km/hr}$. ### Exam Strategy & Shortcut Break the problem into two independent chunks: A & B, then B & C. The meeting of A and B completely locks in the location and time parameters. Once those are established ($t=6$ hours from 9 p.m., $d=540$ km from Mumbai), Train C becomes a basic $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ calculation using the remaining track distance. ### Common Pitfall A common mistake is trying to set up a massive three-variable equation equating distances from both sides simultaneously, which leads to algebraic clutter and high chances of calculation errors. ### Final Answer Therefore, the correct answer is **120 km/hr**.