Awadh express and Bokaro express start simultaneously from Lucknow and Jamshedpur towards each other and continuously shuttle between these two places. Every time these trains meet each other, they turn back after exchanging their respective speeds, the initial ratio of their speeds is 2 : 1.
Let these two trains first time meet at Patna, then what is the ratio of distance covered by Awadh express and Bokaro express till they meet for the third time at the same place Patna :

Options

A. A. `1:1`
B. B. `14:13`
C. C. `10:11`
D. D. none of these

Answer

To solve the problem, we need to analyze the distances covered by the two trains, Awadh Express and Bokaro Express, until they meet for the third time at Patna.

### Step-by-Step Solution:

1. **Understanding the Initial Speed Ratio**:
- The initial speed ratio of Awadh Express (A) to Bokaro Express (B) is given as 2:1.
- This means if Awadh Express travels at speed 2x, Bokaro Express travels at speed x.

2. **First Meeting**:
- Let’s assume the total distance between Lucknow and Jamshedpur is D.
- When they first meet, the distance covered by Awadh Express is \( \frac{2}{3}D \) and by Bokaro Express is \( \frac{1}{3}D \) because they meet in the ratio of their speeds (2:1).
- Therefore, Awadh Express covers \( 2D \) and Bokaro covers \( D \) in the time taken to meet.

3. **After First Meeting**:
- After the first meeting, they turn back and exchange their speeds. Now, Awadh Express travels at speed x and Bokaro Express at speed 2x.
- The distance they need to cover to meet again is the same D.

4. **Second Meeting**:
- The time taken for the second meeting can be calculated as:
- Time taken by Awadh Express to cover D at speed x: \( \frac{D}{x} \)
- Time taken by Bokaro Express to cover D at speed 2x: \( \frac{D}{2x} \)
- The time taken to meet again is determined by the slower train, which is Awadh Express.

5. **Distance Covered Until Second Meeting**:
- In this time, Awadh Express covers \( D \) and Bokaro covers \( 2D \).
- So, the total distance covered by Awadh Express until the second meeting is \( 2D + D = 3D \) and by Bokaro is \( D + 2D = 3D \).

6. **After Second Meeting**:
- They again turn back and exchange their speeds. Awadh Express is now at speed 2x and Bokaro at speed x.
- They need to cover D again to meet for the third time.

7. **Third Meeting**:
- The time taken for the third meeting is again determined by the slower train (Bokaro Express).
- The distance covered by Awadh Express in this time is \( 2D \) and by Bokaro is \( D \).

8. **Total Distance Covered**:
- Total distance covered by Awadh Express until the third meeting:
- First meeting: \( 2D \)
- Second meeting: \( 2D \)
- Third meeting: \( 2D \)
- Total = \( 2D + 2D + 2D = 6D \)
- Total distance covered by Bokaro Express until the third meeting:
- First meeting: \( D \)
- Second meeting: \( 2D \)
- Third meeting: \( D \)
- Total = \( D + 2D + D = 4D \)

9. **Final Ratio**:
- The ratio of distances covered by Awadh Express to Bokaro Express until they meet for the third time is:
\[ \text{Ratio} = \frac{6D}{4D} = \frac{6}{4} = \frac{3}{2} \]

### Conclusion:
The ratio of the distance covered by Awadh Express to Bokaro Express until they meet for the third time at Patna is \( 3:2 \).