Akbar, Birbal and Chanakya run around a circular track of length 500 m. Akbar and Birbal run with the speeds of 15 m/s and 20 m/s in the same direction respectively and Chanakya being very intelligent run in the opposite direction with a speed of 25 m/s. If all three of them start at the same time, then :

Options

A. Akbar meets Chanakya more frequently than Birbal does
B. Akbar and Chanakya meets as frequently as Birbal and Chanakya
C. Akbar meets Birbal least frequently
D. Nothing can be concluded

Answer

To solve the problem of Akbar, Birbal, and Chanakya running around a circular track of length 500 m, we will calculate the time taken for each pair to meet based on their speeds.

### Step 1: Determine the speeds of each runner.
- Akbar's speed = 15 m/s
- Birbal's speed = 20 m/s
- Chanakya's speed = 25 m/s

### Step 2: Calculate the relative speed between Akbar and Birbal.
Since both Akbar and Birbal are running in the same direction, we find their relative speed by subtracting Akbar's speed from Birbal's speed:
\[ \text{Relative Speed (A with respect to B)} = \text{Speed of Birbal} - \text{Speed of Akbar} = 20 \, \text{m/s} - 15 \, \text{m/s} = 5 \, \text{m/s} \]

### Step 3: Calculate the time taken for Akbar and Birbal to meet.
To find the time taken for them to meet, we use the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} \]
Here, the distance is the length of the track (500 m):
\[ T_{A,B} = \frac{500 \, \text{m}}{5 \, \text{m/s}} = 100 \, \text{s} \]

### Step 4: Calculate the relative speed between Birbal and Chanakya.
Since Birbal and Chanakya are running in opposite directions, we add their speeds to find the relative speed:
\[ \text{Relative Speed (B with respect to C)} = \text{Speed of Birbal} + \text{Speed of Chanakya} = 20 \, \text{m/s} + 25 \, \text{m/s} = 45 \, \text{m/s} \]

### Step 5: Calculate the time taken for Birbal and Chanakya to meet.
Using the same formula as before:
\[ T_{B,C} = \frac{500 \, \text{m}}{45 \, \text{m/s}} \approx 11.11 \, \text{s} \]

### Step 6: Calculate the relative speed between Akbar and Chanakya.
Again, since they are running in opposite directions, we add their speeds:
\[ \text{Relative Speed (A with respect to C)} = \text{Speed of Akbar} + \text{Speed of Chanakya} = 15 \, \text{m/s} + 25 \, \text{m/s} = 40 \, \text{m/s} \]

### Step 7: Calculate the time taken for Akbar and Chanakya to meet.
Using the formula:
\[ T_{A,C} = \frac{500 \, \text{m}}{40 \, \text{m/s}} = 12.5 \, \text{s} \]

### Conclusion:
- Akbar and Birbal meet every 100 seconds.
- Birbal and Chanakya meet every 11.11 seconds.
- Akbar and Chanakya meet every 12.5 seconds.

From the calculated times, we can conclude that:
- Akbar meets Chanakya more frequently than he meets Birbal.
- Birbal meets Chanakya more frequently than he meets Akbar.

### Final Answer:
Thus, the correct option is that Akbar meets Birbal less frequently than he meets Chanakya.

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