Question

Two buses travel between Jamshedpur and Kolkata in the opposite directions, on the same road. On that road, the maximum allowed speeds are different (but constant) for the opposite directions. Usually, both buses travel at the respective maximum allowed speeds to their respective destinations: the bus from Jamshedpur to Kolkata takes 4 hours, while the bus from Kolkata to Jamshedpur takes 3 hours.
One day, the two buses start at the same time. However, one hour after starting, the bus from Jamshedpur to Kolkata reduces its speed to half of its maximum allowed speed due to congestion on the road.
If both buses do not stop anywhere in between, how many hours after starting do they meet?

• A. 21/11 hours
• B. 12/5 hours
• C. 2 hours
• D. 10/11 hours
• E. 23/12 hours

Hint:

In relative speed problems, always calculate the relative speed after any changes in motion or speed, and remember that after the first hour, the remaining distance is covered at the relative speed.

Answer 1

The Correct Option is C

Step 1: Understand the problem and assign variables.
Let the total distance between Jamshedpur and Kolkata be \( D \). - The bus from Jamshedpur to Kolkata normally takes 4 hours to cover the distance \( D \), so its speed is \( \frac{D}{4} \). - The bus from Kolkata to Jamshedpur normally takes 3 hours to cover the distance \( D \), so its speed is \( \frac{D}{3} \).
Step 2: Analyze the change in speed.
After 1 hour of travel, the bus from Jamshedpur to Kolkata reduces its speed to half. The new speed of the bus from Jamshedpur to Kolkata is \( \frac{D}{8} \).
Step 3: Calculate the distance covered in the first hour.
- The bus from Jamshedpur to Kolkata covers \( \frac{D}{4} \) in the first hour. - The bus from Kolkata to Jamshedpur covers \( \frac{D}{3} \) in the first hour. After 1 hour, the total distance covered by both buses is: \[ \frac{D}{4} + \frac{D}{3} = \frac{3D + 4D}{12} = \frac{7D}{12} \] Thus, after the first hour, the remaining distance between the two buses is \( D - \frac{7D}{12} = \frac{5D}{12} \).
Step 4: Calculate the relative speed after the first hour.
After the first hour, the bus from Jamshedpur to Kolkata travels at \( \frac{D}{8} \), and the bus from Kolkata to Jamshedpur travels at \( \frac{D}{3} \). The relative speed between the two buses is: \[ \frac{D}{3} + \frac{D}{8} = \frac{8D + 3D}{24} = \frac{11D}{24} \]
Step 5: Calculate the time to meet.
The time taken for the buses to meet after the first hour is the remaining distance divided by the relative speed: \[ \text{Time} = \frac{\frac{5D}{12}}{\frac{11D}{24}} = \frac{5D}{12} \times \frac{24}{11D} = \frac{5 \times 2}{11} = \frac{10}{11} \text{ hours} \] Thus, the total time to meet is the first hour plus the time taken to meet after the first hour: \[ 1 + \frac{10}{11} = \frac{21}{11} \text{ hours} \]
Step 6: Conclusion.
The two buses meet after \(\frac{21}{11}\) hours. Therefore, the correct answer is (C) 2 hours.