Two trains A and B started to move from Patna to Delhi 6am and 6 : 30 am at the speed of 60 km/h and 90 km/h. How far away from Patna will they meet each other?

Answer

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

### Step-by-Step Solution:

1. **Identify the Details:**
- Train A departs from Patna at 6:00 AM and travels at a speed of 60 km/h.
- Train B departs from Patna at 6:30 AM and travels at a speed of 90 km/h.

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

3. **Determine the Relative Speed:**
- Since both trains are moving in the same direction, we calculate the relative speed:
\[ \text{Relative Speed} = \text{Speed of B} - \text{Speed of A} = 90 \text{ km/h} - 60 \text{ km/h} = 30 \text{ km/h} \]

4. **Calculate the Time Taken for Train B to Catch Up:**
- To find out how long it will take for Train B to catch up with Train A, we use the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{30 \text{ km}}{30 \text{ km/h}} = 1 \text{ hour} \]

5. **Calculate the Distance Covered by Train B in That Time:**
- In the 1 hour it takes for Train B to catch up, it will cover:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 90 \text{ km/h} \times 1 \text{ h} = 90 \text{ km} \]

6. **Conclusion:**
- Therefore, the two trains will meet 90 km away from Patna.

### Final Answer:
The two trains A and B will meet each other 90 km away from Patna.