If Somu walked to his office at 6 kmph, he would have reached his office 4 min early. If he walked at 4 kmph, he would have reached the office 6 min late. Find the distance he has to travel to reach his office (in km)

Answer

To solve the problem, we will use the information given about Somu's walking speeds and the time differences to set up equations.

### Step-by-Step Solution:

1. **Define Variables:**
Let the distance to Somu's office be \( d \) kilometers. Let the time taken to reach the office at the normal speed be \( t \) hours.

2. **Set Up Equations:**
- When Somu walks at 6 km/h, he reaches 4 minutes early. This means he takes \( t - \frac{4}{60} \) hours (since 4 minutes is \( \frac{4}{60} \) hours).
- The distance can be expressed as:
\[ d = 6 \left( t - \frac{4}{60} \right) \quad \text{(Equation 1)} \]
- When Somu walks at 4 km/h, he reaches 6 minutes late. This means he takes \( t + \frac{6}{60} \) hours (since 6 minutes is \( \frac{6}{60} \) hours).
- The distance can also be expressed as:
\[ d = 4 \left( t + \frac{6}{60} \right) \quad \text{(Equation 2)} \]

3. **Equate the Two Expressions for Distance:**
Since both equations represent the same distance \( d \), we can set them equal to each other:
\[ 6 \left( t - \frac{4}{60} \right) = 4 \left( t + \frac{6}{60} \right) \]

4. **Simplify the Equation:**
Expanding both sides:
\[ 6t - \frac{24}{60} = 4t + \frac{24}{60} \]
This simplifies to:
\[ 6t - 4t = \frac{24}{60} + \frac{24}{60} \]
\[ 2t = \frac{48}{60} \]
\[ 2t = \frac{4}{5} \]
\[ t = \frac{4}{10} = \frac{2}{5} \text{ hours} = 24 \text{ minutes} \]

5. **Find the Distance:**
Now, substitute \( t \) back into either equation to find \( d \). Using Equation 1:
\[ d = 6 \left( \frac{2}{5} - \frac{4}{60} \right) \]
Convert \( \frac{4}{60} \) to a fraction with a denominator of 5:
\[ \frac{4}{60} = \frac{1}{15} = \frac{2}{30} = \frac{2}{10} = \frac{1}{5} \]
Thus,
\[ d = 6 \left( \frac{2}{5} - \frac{1}{5} \right) = 6 \left( \frac{1}{5} \right) = \frac{6}{5} = 1.2 \text{ km} \]

### Final Answer:
The distance Somu has to travel to reach his office is **2 km**.