Two stations A and B are 200 km apart. A train starts from station A at 7:00 a.m. at 20 km/h towards B. Another train starts from station B at 8:00 a.m. at 25 km/h towards A. At what time will the two trains meet?

Introduction / Context:
This problem is a classic time and distance question involving two trains starting at different times from opposite ends of a straight track. It checks your understanding of relative speed, staggered starting times, and the ability to express times in clock format after solving the equation.

Given Data / Assumptions:
Distance between station A and station B = 200 km.
Train 1 starts from station A at 7:00 a.m. and travels towards B at 20 km/h.
Train 2 starts from station B at 8:00 a.m. and travels towards A at 25 km/h.
Both trains move along the same straight track without stopping.
We need to find the clock time when the two trains meet.

Concept / Approach:
The idea is to fix a time reference and express how far each train has travelled from that reference moment. We can measure time from 7:00 a.m. onwards. The first train moves for a longer period, while the second starts after 1 hour. At the meeting time, the sum of the distances travelled by both trains must equal the total distance between the stations, which is 200 km. This leads to a simple linear equation in one variable representing the time from 7:00 a.m.

Step-by-Step Solution:
Step 1: Let t be the time in hours after 7:00 a.m. when the trains meet.Step 2: Train 1 travels the full duration t at 20 km/h.Step 3: Train 2 starts at 8:00 a.m., which is 1 hour after 7:00 a.m., so it travels for (t − 1) hours at 25 km/h, provided t is greater than 1.Step 4: Distance covered by Train 1 = 20 * t km.Step 5: Distance covered by Train 2 = 25 * (t − 1) km.Step 6: At the meeting point, the sum of the distances is equal to the total distance between the stations: 20 * t + 25 * (t − 1) = 200.Step 7: Expand and simplify: 20t + 25t − 25 = 200.Step 8: Combine like terms: 45t − 25 = 200.Step 9: Add 25 to both sides: 45t = 225.Step 10: Divide by 45: t = 225 / 45 = 5 hours.Step 11: The meeting time is 5 hours after 7:00 a.m., which is 12:00 p.m. (noon).

Verification / Alternative check:
At 12:00 p.m., Train 1 has been moving from 7:00 a.m. for 5 hours at 20 km/h, so its distance is 20 * 5 = 100 km. Train 2 has been moving from 8:00 a.m. for 4 hours at 25 km/h, so its distance is 25 * 4 = 100 km. Together they cover 100 + 100 = 200 km, exactly the distance between stations A and B. Hence they meet at noon.

Why Other Options Are Wrong:
11 a.m. would correspond to 4 hours after 7:00 a.m., which gives 20 * 4 + 25 * 3 = 80 + 75 = 155 km, less than 200 km.
1 p.m. would be 6 hours after 7:00 a.m., giving 20 * 6 + 25 * 5 = 120 + 125 = 245 km, which exceeds the required 200 km separation.
10 a.m. is only 3 hours after 7:00 a.m., and the second train has been running only 2 hours, so the total distance is far less than 200 km.
12 a.m. is midnight and does not fit the time frame described in the problem.

Common Pitfalls:
Sometimes students forget that the second train starts one hour later and mistakenly use t for both travel times. Others may add or subtract the speeds incorrectly. Always define t clearly, keep track of which train starts when, and remember that the condition for meeting is that the sum of distances equals the total distance between the stations.

Final Answer:
The two trains meet at 12:00 p.m. (noon).