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Let's assume the distance between Agra and Delhi is D kilometers.
Train X:
Departure time from Agra: 5 pm
Arrival time in Delhi: 10 pm
Travel time: 10 pm - 5 pm = 5 hours
Average speed of Train X = Distance (D) / Travel time (5 hours)
Speed of Train X = D/5 km/hr
Train Y:
Departure time from Agra: 7 pm
Arrival time in Delhi: 11 pm
Travel time: 11 pm - 7 pm = 4 hours
Average speed of Train Y = Distance (D) / Travel time (4 hours)
Speed of Train Y = D/4 km/hr
Relative Speed:
The relative speed between Train X and Train Y is the difference between their individual speeds.
Relative Speed = Speed of Train Y - Speed of Train X
Relative Speed = (D/4) km/hr - (D/5) km/hr
Relative Speed = (5D - 4D) / 20 km/hr
Relative Speed = D/20 km/hr
Now, let's find the time it takes for the trains to meet.
Let's assume they meet at a point C.
Distance between Agra and Point C = Distance Train X travels before meeting
Distance between Agra and Point C = Speed of Train X * Time
Distance between Point C and Delhi = Distance Train Y travels before meeting
Distance between Point C and Delhi = Speed of Train Y * Time
Since the sum of these two distances is equal to the total distance D between Agra and Delhi, we can set up the following equation:
Speed of Train X * Time + Speed of Train Y * Time = D
(D/5) * Time + (D/4) * Time = D
(4D + 5D) / 20 * Time = D
9D / 20 * Time = D
Time = D * 20 / 9D
Time = 20 / 9 hours
Therefore, the two trains will meet 20 / 9 hours after Train X departs from Agra.
Converting this time to hours and minutes:
20 / 9 hours = (20 / 9) * 60 minutes
20 / 9 hours = 11.11 hours ≈ 11 hours 7 minutes
Adding the 11 hours 7 minutes to the departure time of Train X from Agra (5 pm), we find the time at which the trains will meet.
Time of Meeting = 5 pm + 11 hours 7 minutes
Time of Meeting = 4:07 pm
So, the two trains will meet at approximately 4:07 pm.