Two trains leave Varanasi for Lucknow at 11:00 AM and at 11:30 AM respectively and travel at speeds of 110 km/h and 140 km/h, respectively. How many kilometres from Varanasi will both trains meet?

The correct answer is 256 2/3km.

Let's solve the problem step-by-step:

Step 1: Understand the initial positions and speeds of both trains
The first train leaves Varanasi at 11:00 AM at a speed of 110 km/h.
The second train leaves Varanasi at 11:30 AM at a speed of 140 km/h.

Step 2: Find the distance traveled by the first train before the second train starts
The time difference between the departure of the first train and the second train is 30 minutes (which is 0.5 hours).
During this time, the distance covered by the first train is:

$$\mathrm{Distance}=\mathrm{Speed}\times \mathrm{Time}=110\times 0.5=55\text{km}$$
Thus, at 11:30 AM, the first train is 55 km ahead of Varanasi.

Step 3: Calculate the relative speed of the two trains
Since both trains are traveling in the same direction, the relative speed of the second train with respect to the first train is the difference of their speeds:

$$\mathrm{Relative\; Speed}=140-110=30\text{km/h}$$
This means the second train closes the distance gap by 30 km every hour.

Step 4: Find the time taken for the second train to overtake the first train
The time required to cover the initial gap of 55 km at the relative speed of 30 km/h is:

$$\mathrm{Time}=\frac{55}{30}\text{hours}=\frac{11}{6}\text{hours}$$

Step 5: Calculate the distance from Varanasi where they meet
The meeting point is the total distance traveled by the second train during this catch-up time:

$$\mathrm{Distance}=\mathrm{Speed\; of\; second\; train}\times \mathrm{Time}$$
$$\mathrm{Distance}=140\times \frac{11}{6}=\frac{770}{3}\text{km}$$
Converting the improper fraction to a mixed fraction, we get:

$$\frac{770}{3}=256\frac{2}{3}\text{km}$$
Therefore, both trains will meet at a distance of 256 2/3 km from Varanasi.