I went to Delhi at a speed of 200 km/hr but suddenly I returned to the same place at a speed of 600 km/hr. What is my average speed :

Options

A. 300 km/hr
B. 400 km/hr
C. 366.66 km/hr
D. none of these

Answer

To find the average speed for the entire journey, we can follow these steps:

### Step 1: Define the distance
Let the distance to Delhi be \( x \) km.

### Step 2: Calculate the total distance
Since the journey involves going to Delhi and returning back to the starting point, the total distance covered is:
\[ \text{Total Distance} = x + x = 2x \text{ km} \]

### Step 3: Calculate the time taken for each part of the journey
1. **Time taken to go to Delhi**:
- Speed = 200 km/hr
- Time = Distance / Speed = \( \frac{x}{200} \) hours

2. **Time taken to return from Delhi**:
- Speed = 600 km/hr
- Time = Distance / Speed = \( \frac{x}{600} \) hours

### Step 4: Calculate the total time taken
The total time for the entire journey is the sum of the time taken to go and the time taken to return:
\[ \text{Total Time} = \frac{x}{200} + \frac{x}{600} \]

### Step 5: Find a common denominator to add the times
The common denominator for 200 and 600 is 600. We can rewrite the times:
\[ \frac{x}{200} = \frac{3x}{600} \]
Thus,
\[ \text{Total Time} = \frac{3x}{600} + \frac{x}{600} = \frac{4x}{600} = \frac{x}{150} \text{ hours} \]

### Step 6: Calculate the average speed
Average speed is defined as the total distance divided by the total time:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2x}{\frac{x}{150}} = 2x \times \frac{150}{x} \]
Here, \( x \) cancels out:
\[ \text{Average Speed} = 2 \times 150 = 300 \text{ km/hr} \]

### Conclusion
Thus, the average speed for the entire journey is:
\[ \text{Average Speed} = 300 \text{ km/hr} \]