Walking 6/7th of his usual speed, a man is 12 minutes too late. The usual time taken by him to cover that distance is

Options

A. 1 hour
B. 1 hour 12 minutes
C. 1 hour 15 minutes
D. 1 hour 20 minutes

Answer

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

### Step 1: Define Variables
Let:
- \( d \) = distance covered (in kilometers)
- \( v \) = usual speed (in kilometers per hour)
- \( t \) = usual time taken to cover the distance (in hours)

### Step 2: Write the Equation for Usual Time
The usual time taken to cover the distance can be expressed as:
\[ t = \frac{d}{v} \]

### Step 3: Determine the New Speed
The man walks at \( \frac{6}{7} \) of his usual speed. Therefore, his new speed is:
\[ \text{New Speed} = \frac{6}{7}v \]

### Step 4: Write the Equation for New Time
When walking at the new speed, the time taken to cover the same distance is:
\[ \text{New Time} = \frac{d}{\frac{6}{7}v} = \frac{7d}{6v} \]

### Step 5: Set Up the Equation for the Delay
According to the problem, the man is 12 minutes late. This means:
\[ \text{New Time} = t + 12 \text{ minutes} \]
Converting 12 minutes into hours:
\[ 12 \text{ minutes} = \frac{12}{60} = \frac{1}{5} \text{ hours} \]
Thus, we can write:
\[ \frac{7d}{6v} = t + \frac{1}{5} \]

### Step 6: Substitute the Usual Time
Substituting \( t \) from Step 2 into the equation:
\[ \frac{7d}{6v} = \frac{d}{v} + \frac{1}{5} \]

### Step 7: Eliminate \( d \) and Solve for \( t \)
To eliminate \( d \), we can multiply through by \( 30v \) (the least common multiple of 6 and 5):
\[ 30v \cdot \frac{7d}{6v} = 30v \cdot \frac{d}{v} + 30v \cdot \frac{1}{5} \]
This simplifies to:
\[ 35d = 30d + 6v \]
Rearranging gives:
\[ 35d - 30d = 6v \implies 5d = 6v \]
Thus:
\[ d = \frac{6}{5}v \]

### Step 8: Substitute Back to Find \( t \)
Substituting \( d \) back into the equation for \( t \):
\[ t = \frac{d}{v} = \frac{\frac{6}{5}v}{v} = \frac{6}{5} \text{ hours} \]

### Step 9: Convert \( t \) to Minutes
To convert \( t \) into minutes:
\[ t = \frac{6}{5} \text{ hours} = 1 \text{ hour} + \frac{1}{5} \text{ hour} \]
Calculating \( \frac{1}{5} \text{ hour} \) in minutes:
\[ \frac{1}{5} \text{ hour} = \frac{1}{5} \times 60 = 12 \text{ minutes} \]
Thus, the total time is:
\[ t = 1 \text{ hour} + 12 \text{ minutes} \]

### Final Answer
The usual time taken by him to cover that distance is **1 hour and 12 minutes**.

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