Two trains Ajanta express and Barouni express simultaneously started on two parallet tracks from Meerut to Nagpur, which are 390 km apart. The ratio of the speed of Ajanta express and Barouni express is 6 : 7. After how long (in kms) travelling, Barouni express exchanges the speed with Ajanta express so that both the trains reach at their destination simultaneously :

Options

A. 150 km
B. 190 km
C. 210 km
D. can't be determined

Answer

To solve the problem, we need to determine how far the Barouni Express travels before it exchanges speeds with the Ajanta Express, so that both trains reach their destination simultaneously.

### Step-by-Step Solution:

1. **Determine the total distance and speed ratio**:
- The distance between Meerut and Nagpur is 390 km.
- The speed ratio of Ajanta Express to Barouni Express is 6:7.

2. **Express speeds in terms of a variable**:
- Let the speed of Ajanta Express be \( 6x \) km/h and the speed of Barouni Express be \( 7x \) km/h.

3. **Calculate the time taken by each train to cover 390 km**:
- Time taken by Ajanta Express to cover 390 km:
\[ \text{Time}_{Ajanta} = \frac{390}{6x} = \frac{65}{x} \text{ hours} \]
- Time taken by Barouni Express to cover 390 km:
\[ \text{Time}_{Barouni} = \frac{390}{7x} = \frac{390}{7x} \text{ hours} \]

4. **Set the times equal for simultaneous arrival**:
- For both trains to arrive at the same time, we need:
\[ \frac{65}{x} = \frac{390}{7x} \]

5. **Cross-multiply to solve for \( x \)**:
- Cross-multiplying gives:
\[ 65 \cdot 7x = 390 \cdot x \]
\[ 455x = 390x \]
- Simplifying this gives:
\[ 455 = 390 \quad \text{(This is incorrect, we need to check our steps)} \]

6. **Find the distance traveled before speed exchange**:
- Let’s denote the distance traveled by Barouni Express before the speed exchange as \( d \).
- After traveling distance \( d \), the time taken by Barouni Express is:
\[ \text{Time}_{Barouni} = \frac{d}{7x} \]
- The distance traveled by Ajanta Express in the same time is:
\[ \text{Distance}_{Ajanta} = 6x \cdot \frac{d}{7x} = \frac{6d}{7} \]

7. **Total distance covered by both trains**:
- After speed exchange, the remaining distance for Barouni Express is:
\[ 390 - d \]
- The remaining distance for Ajanta Express is:
\[ 390 - \frac{6d}{7} \]

8. **Set the remaining times equal**:
- The time taken by Barouni Express to cover the remaining distance:
\[ \frac{390 - d}{6x} \]
- The time taken by Ajanta Express to cover the remaining distance:
\[ \frac{390 - \frac{6d}{7}}{7x} \]

9. **Equate the times**:
- Set the two times equal:
\[ \frac{390 - d}{6x} = \frac{390 - \frac{6d}{7}}{7x} \]

10. **Solve for \( d \)**:
- Cross-multiply and simplify to find \( d \).

### Final Calculation:
After solving the equations, we find that \( d = 210 \) km.

### Conclusion:
The Barouni Express travels **210 km** before exchanging speeds with the Ajanta Express.