If I walk at 5 km/hour, I miss a train by 7 minutes. If, however, I walk at 6 km/hour, I reach the station 5 minutes before the departure of the train. The distance (in km) between my house and the station is

Options

A. 6
B. 5
C. 4
D. 3

Answer

To solve the problem step by step, we will use the information provided about the two different speeds and the time differences to find the distance between the house and the station.

### Step 1: Define Variables
Let the distance between the house and the station be \( D \) km. Let the time taken to reach the station at the correct speed be \( T \) hours.

### Step 2: Set Up Equations
1. When walking at 5 km/h, the time taken is \( T + \frac{7}{60} \) hours (since 7 minutes is \( \frac{7}{60} \) hours).
- The equation for distance is:
\[ D = 5 \left( T + \frac{7}{60} \right) \]

2. When walking at 6 km/h, the time taken is \( T - \frac{5}{60} \) hours (since 5 minutes is \( \frac{5}{60} \) hours).
- The equation for distance is:
\[ D = 6 \left( T - \frac{5}{60} \right) \]

### Step 3: Equate the Two Expressions for Distance
Since both expressions represent the same distance \( D \), we can set them equal to each other:
\[ 5 \left( T + \frac{7}{60} \right) = 6 \left( T - \frac{5}{60} \right) \]

### Step 4: Expand Both Sides
Expanding both sides gives:
\[ 5T + \frac{35}{60} = 6T - \frac{30}{60} \]

### Step 5: Simplify the Equation
Rearranging the equation to isolate \( T \):
\[ 5T + \frac{35}{60} + \frac{30}{60} = 6T \]
\[ 5T + \frac{65}{60} = 6T \]
\[ \frac{65}{60} = 6T - 5T \]
\[ \frac{65}{60} = T \]

### Step 6: Calculate \( T \)
Now we can simplify \( T \):
\[ T = \frac{65}{60} \text{ hours} = \frac{13}{12} \text{ hours} \]

### Step 7: Calculate the Distance \( D \)
Using the value of \( T \) in one of the distance equations:
\[ D = 5 \left( \frac{13}{12} + \frac{7}{60} \right) \]
First, convert \( \frac{7}{60} \) to have a common denominator with \( \frac{13}{12} \):
\[ \frac{7}{60} = \frac{7 \times 2}{60 \times 2} = \frac{14}{120} \]
Now convert \( \frac{13}{12} \):
\[ \frac{13}{12} = \frac{13 \times 10}{12 \times 10} = \frac{130}{120} \]
So,
\[ D = 5 \left( \frac{130}{120} + \frac{14}{120} \right) = 5 \left( \frac{144}{120} \right) \]
\[ D = 5 \times \frac{12}{10} = 6 \text{ km} \]

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

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