Walking at a speed of 5 km/hr, a man reaches his office 6 minutes late. Walking at 6 km/hr, he reaches there 2 minutes early. The distance of his office is

Options

A. 2 km
B. 3 km
C. 4 km
D. 3.5 km

Answer

To solve the problem step-by-step, we can follow these instructions:

### Step 1: Define Variables
Let the time that the man should take to reach his office on time be \( T \) hours.

### Step 2: Convert Late and Early Times to Hours
- When walking at 5 km/hr, he is 6 minutes late. In hours, this is:
\[ 6 \text{ minutes} = \frac{6}{60} = \frac{1}{10} \text{ hours} \]
Therefore, the time taken at 5 km/hr is:
\[ T + \frac{1}{10} \text{ hours} \]

- When walking at 6 km/hr, he is 2 minutes early. In hours, this is:
\[ 2 \text{ minutes} = \frac{2}{60} = \frac{1}{30} \text{ hours} \]
Therefore, the time taken at 6 km/hr is:
\[ T - \frac{1}{30} \text{ hours} \]

### Step 3: Write the Distance Equation
The distance to the office can be expressed using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \).

- For the first scenario (5 km/hr):
\[ \text{Distance} = 5 \left(T + \frac{1}{10}\right) \]

- For the second scenario (6 km/hr):
\[ \text{Distance} = 6 \left(T - \frac{1}{30}\right) \]

### Step 4: Set the Distances Equal
Since the distance is the same in both cases, we can set the two equations equal to each other:
\[ 5 \left(T + \frac{1}{10}\right) = 6 \left(T - \frac{1}{30}\right) \]

### Step 5: Expand Both Sides
Expanding both sides gives:
\[ 5T + \frac{5}{10} = 6T - \frac{6}{30} \]
This simplifies to:
\[ 5T + \frac{1}{2} = 6T - \frac{1}{5} \]

### Step 6: Rearrange the Equation
Rearranging the equation to isolate \( T \):
\[ \frac{1}{2} + \frac{1}{5} = 6T - 5T \]
Finding a common denominator (which is 10):
\[ \frac{5}{10} + \frac{2}{10} = T \]
Thus,
\[ T = \frac{7}{10} \text{ hours} \]

### Step 7: Calculate the Distance
Now that we have \( T \), we can find the distance using either speed. Let's use the speed of 5 km/hr:
\[ \text{Distance} = 5 \left(T + \frac{1}{10}\right) = 5 \left(\frac{7}{10} + \frac{1}{10}\right) = 5 \left(\frac{8}{10}\right) = 5 \times \frac{4}{5} = 4 \text{ km} \]

### Final Answer
The distance to the man's office is **4 km**.

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