If I walk at 5 km/hr. I miss a train by 7 minutes. However, if I walk at 6 km/hr I reach the station 5 minutes before the departure of the train. The distance between my house and the station is.
यदि मैं 5 किमी/घंटा की गति से चलता हूँ, तो मेरी रेलगाड़ी 7 मिनट पहले छूट जाती है। परंतु यदि मैं 6 किमी/घंटा की गति से चलता हूँ तो मैं रेलगाड़ी के रवाना होने से 5 मिनट पहले स्टेशन पहुँच जाता हूँ। मेरे घर और स्टेशन के बीच दूरी कितनी है?

Options

A. 5 किमी
B. 6.5 किमी
C. 7 किमी
D. 6 किमी

Explanation

To solve the problem, we need to find the distance between the house and the station based on the given walking speeds and the time differences.

### Step-by-Step Solution:

1. **Define the Variables**:
Let the distance between the house and the station be \( d \) kilometers.

2. **Calculate Time Taken at Different Speeds**:
- When walking at 5 km/hr, the time taken to reach the station is \( \frac{d}{5} \) hours.
- When walking at 6 km/hr, the time taken to reach the station is \( \frac{d}{6} \) hours.

3. **Set Up the Time Difference**:
According to the problem:
- At 5 km/hr, the person misses the train by 7 minutes, which means they arrive 7 minutes late.
- At 6 km/hr, the person arrives 5 minutes early.
- The total time difference between these two scenarios is \( 7 + 5 = 12 \) minutes.

4. **Convert Minutes to Hours**:
Since we need to work in hours, convert 12 minutes to hours:
\[ 12 \text{ minutes} = \frac{12}{60} \text{ hours} = \frac{1}{5} \text{ hours} \]

5. **Set Up the Equation**:
The time difference between the two speeds can be expressed as:
\[ \frac{d}{5} - \frac{d}{6} = \frac{1}{5} \]

6. **Find a Common Denominator**:
The least common multiple of 5 and 6 is 30. Rewrite the equation:
\[ \frac{6d - 5d}{30} = \frac{1}{5} \]
This simplifies to:
\[ \frac{d}{30} = \frac{1}{5} \]

7. **Solve for \( d \)**:
To find \( d \), cross-multiply:
\[ d = 30 \times \frac{1}{5} = 6 \]

8. **Conclusion**:
The distance between the house and the station is \( 6 \) kilometers.

### Final Answer:
The distance between my house and the station is **6 km**.