Gautam goes to office at a speed of 12 kmph and returns home at 10 kmph. His average speed is :

Options

A. 11 kmph
B. 22 kmph
C. 10.9 kmph
D. 12.5 kmph

Answer

To find Gautam's average speed for his round trip to the office, we can use the formula for average speed when the distances are the same but the speeds are different.

### Step-by-step Solution:

1. **Define the Distance**:
Let the distance from Gautam's home to his office be \( D \) kilometers.

2. **Calculate Total Distance**:
Since Gautam travels to the office and returns home, the total distance traveled is:
\[ \text{Total Distance} = D + D = 2D \text{ kilometers} \]

3. **Calculate Time Taken for Each Leg of the Journey**:
- Time taken to go to the office:
\[ \text{Time}_{\text{to office}} = \frac{D}{12} \text{ hours} \]
- Time taken to return home:
\[ \text{Time}_{\text{return home}} = \frac{D}{10} \text{ hours} \]

4. **Calculate Total Time**:
The total time taken for the round trip is:
\[ \text{Total Time} = \text{Time}_{\text{to office}} + \text{Time}_{\text{return home}} = \frac{D}{12} + \frac{D}{10} \]
To add these fractions, find a common denominator (which is 60):
\[ \text{Total Time} = \frac{5D}{60} + \frac{6D}{60} = \frac{11D}{60} \text{ hours} \]

5. **Calculate Average Speed**:
The average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2D}{\frac{11D}{60}} = 2D \times \frac{60}{11D} \]
Simplifying this gives:
\[ \text{Average Speed} = \frac{120}{11} \text{ kmph} \]
Calculating this gives approximately:
\[ \text{Average Speed} \approx 10.91 \text{ kmph} \]

### Final Answer:
Gautam's average speed is approximately **10.91 kmph**.