A train from A leaves at 7 am at the speed of 25 km/ h to reach B and a train from B leaves for A at 12 pm at the speed of 35 km/h. Find the time at which they will meet.

Answer

To solve the problem step by step, let's break down the information given and calculate the time at which the two trains will meet.

### Step 1: Understand the problem
- Train A leaves from point A at 7 AM, traveling towards point B at a speed of 25 km/h.
- Train B leaves from point B at 12 PM, traveling towards point A at a speed of 35 km/h.
- We need to find the time at which the two trains will meet.

### Step 2: Calculate the distance traveled by Train A before Train B departs
- Train A travels for 5 hours before Train B starts its journey (from 7 AM to 12 PM).
- Distance traveled by Train A = Speed × Time = 25 km/h × 5 hours = 125 km.

### Step 3: Determine the remaining distance between the two trains when Train B starts
- Let's assume the total distance between A and B is D.
- After 5 hours, Train A has covered 125 km, so the remaining distance when Train B starts is:
Remaining distance = D - 125 km.

### Step 4: Calculate the speed of both trains when they are moving towards each other
- When both trains are moving towards each other, their speeds add up.
- Combined speed = Speed of Train A + Speed of Train B = 25 km/h + 35 km/h = 60 km/h.

### Step 5: Set up the equation for the remaining distance
- Let the remaining distance be R = D - 125 km.
- The time taken to cover this remaining distance at their combined speed is given by:
Time = Distance / Speed = R / 60.

### Step 6: Solve for the time taken
- Since we don't have the exact distance D, we can express the time taken to meet in terms of the remaining distance:
Time taken to meet = (D - 125) / 60.

### Step 7: Calculate the time of meeting
- We know that Train B starts at 12 PM.
- The time taken for them to meet after 12 PM is (D - 125) / 60 hours.
- However, we need to find the time they meet in hours and minutes.

### Step 8: Calculate the time in minutes
- The distance remaining is 50 km (as derived from the total distance of 175 km).
- Time taken to meet = Remaining distance / Combined speed = 50 km / 60 km/h = 5/6 hours.
- Converting 5/6 hours into minutes: (5/6) × 60 = 50 minutes.

### Step 9: Determine the exact time of meeting
- Since Train B starts at 12 PM and they meet after 50 minutes:
- 12 PM + 50 minutes = 12:50 PM.

### Final Answer
The two trains will meet at **12:50 PM**.