Two trains leave Patna for Delhi at 10:00am and 10:30am respectively and travel at 120 km/h and 150 km/h, respectively. How many kilometres from Patna, will the two trains meet?

Answer

To solve the problem of when and where the two trains will meet, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Departure Times and Speeds of the Trains**:
- Train 1 leaves Patna at 10:00 AM and travels at a speed of 120 km/h.
- Train 2 leaves Patna at 10:30 AM and travels at a speed of 150 km/h.

2. **Calculate the Head Start of Train 1**:
- Train 1 has a 30-minute head start before Train 2 departs.
- In 30 minutes (which is 0.5 hours), Train 1 will cover:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 120 \text{ km/h} \times 0.5 \text{ h} = 60 \text{ km} \]
- Therefore, when Train 2 starts, Train 1 is already 60 km ahead.

3. **Determine the Relative Speed of the Trains**:
- Since both trains are moving in the same direction, the relative speed of Train 2 with respect to Train 1 is:
\[ \text{Relative Speed} = \text{Speed of Train 2} - \text{Speed of Train 1} = 150 \text{ km/h} - 120 \text{ km/h} = 30 \text{ km/h} \]

4. **Calculate the Time Taken for Train 2 to Catch Up**:
- To find out how long it takes for Train 2 to cover the 60 km gap at a relative speed of 30 km/h, we use the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{60 \text{ km}}{30 \text{ km/h}} = 2 \text{ hours} \]

5. **Determine the Meeting Time**:
- Since Train 2 starts at 10:30 AM and takes 2 hours to catch up, they will meet at:
\[ 10:30 \text{ AM} + 2 \text{ hours} = 12:30 \text{ PM} \]

6. **Calculate the Distance from Patna at the Meeting Point**:
- Now, we can find out how far Train 2 travels in those 2 hours:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 150 \text{ km/h} \times 2 \text{ h} = 300 \text{ km} \]

### Final Answer:
The two trains will meet **300 km from Patna**.

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