P is going to Delhi from Gurgaon by his car at a speed of 40 km/h. While coming back, he returns with a speed of x km/h. What should be the value of x so that his average speed during the intire journey is 80 km/h?

Options

A. 160 km/h
B. 160 km/h
C. 120 km/h
D. It is not possible

Answer

To solve the problem, we need to find the speed \( x \) at which P should return from Delhi to Gurgaon so that his average speed for the entire journey is 80 km/h.

### Step-by-Step Solution:

1. **Define the Distance**:
Let the distance from Gurgaon to Delhi be \( d \) km.

2. **Calculate Time for Outgoing Journey**:
The speed for the journey to Delhi is 40 km/h. The time taken for this part of the journey can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{d}{40} \text{ hours} \]

3. **Calculate Time for Return Journey**:
The speed for the return journey is \( x \) km/h. The time taken for the return journey is:
\[ \text{Time} = \frac{d}{x} \text{ hours} \]

4. **Total Distance and Total Time**:
The total distance for the entire journey (to Delhi and back) is:
\[ \text{Total Distance} = d + d = 2d \text{ km} \]
The total time for the entire journey is:
\[ \text{Total Time} = \frac{d}{40} + \frac{d}{x} \text{ hours} \]

5. **Average Speed Formula**:
The average speed for the entire journey is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
We know the average speed should be 80 km/h, so we set up the equation:
\[ 80 = \frac{2d}{\frac{d}{40} + \frac{d}{x}} \]

6. **Simplifying the Equation**:
We can simplify the equation by multiplying both sides by the total time:
\[ 80 \left( \frac{d}{40} + \frac{d}{x} \right) = 2d \]
Dividing both sides by \( d \) (assuming \( d \neq 0 \)):
\[ 80 \left( \frac{1}{40} + \frac{1}{x} \right) = 2 \]

7. **Distributing 80**:
Distributing 80 gives:
\[ 2 + \frac{80}{x} = 2 \]

8. **Solving for \( x \)**:
Subtracting 2 from both sides:
\[ \frac{80}{x} = 0 \]
This implies that \( x \) must be infinite, which is not possible in practical terms.

9. **Conclusion**:
Since we reached a conclusion that \( x \) cannot be a finite number, we conclude that it is not possible for P to have an average speed of 80 km/h for the entire journey.

### Final Answer:
It is not possible for P to achieve an average speed of 80 km/h for the entire journey.