A bus leaves Agra at 5 AM and reaches Delhi at 12 noon. Another bus leaves Delhi at 8 AM and reaches Agra at 3 PM. At what time do they meet?

First bus leaves at 5 am from Agra and reaches Delhi at 12 noon

The first bus takes 7 hours to reach Delhi

The second bus travels from Delhi to Agra from 8am to 3 pm

This second bus also takes 7 hours to cover the same distance. ( The distance between Agra - Delhi and Delhi to Agra is the same; let say x km)

If  each bus covers x km in 7 hours, then in 1 hour each bus covers exactly $$\frac{x}{7}$$ km towards each other.

You can divide this distance in 7 portions and find the position of the two vehicles with respect to time. First bus approaching from $$\frac{x}{7}$$ position and second from $$x$$ position

Distance covered with time for bus 1:

5am - 6am is $$\frac{x}{7}$$

7am -  $$\frac{2x}{7}$$

8am - $$\frac{3x}{7}$$

9am - $$\frac{4x}{7}$$

10am - $$\frac{5x}{7}$$

11am - $$\frac{6x}{7}$$

12pm - $${x}$$

Distance covered with time for bus 2 with respect to first bus:

8am - 9am is $$\frac{6x}{7}$$

10am - $$\frac{5x}{7}$$

11 am - $$\frac{4x}{7}$$

12pm -  $$\frac{3x}{7}$$

1pm - $$\frac{2x}{7}$$

2pm - 3pm is $$\frac{x}{7}$$.

The two vehicles meet at 10 am - after first vehicle covers five hours and the second 2 hours.