A man has to be at a certain place at a certain time. He finds that he shall be 20 minutes late if he walks at 3km/hour speed and 10 minutes earlier if he walks at a speed of 4km/hour. The distance he has to walk is

Options

A. 24km/hr
B. 12.5km
C. 10km
D. 6km

Answer

To solve the problem, we need to find the distance the man has to walk based on the information provided about his walking speeds and the time differences.

### Step-by-Step Solution:

1. **Define Variables:**
Let the distance the man has to walk be \( x \) kilometers.

2. **Calculate Time Taken at Different Speeds:**
- When walking at 3 km/h, the time taken is \( \frac{x}{3} \) hours.
- When walking at 4 km/h, the time taken is \( \frac{x}{4} \) hours.

3. **Determine Time Differences:**
- If he walks at 3 km/h, he is 20 minutes late. This means he takes 20 minutes longer than the required time.
- If he walks at 4 km/h, he is 10 minutes early. This means he takes 10 minutes less than the required time.

4. **Convert Time Differences to Hours:**
- 20 minutes = \( \frac{20}{60} = \frac{1}{3} \) hours.
- 10 minutes = \( \frac{10}{60} = \frac{1}{6} \) hours.

5. **Set Up the Equation:**
The time difference between the two scenarios is:
\[ \left(\frac{x}{3} + \frac{1}{3}\right) - \left(\frac{x}{4} - \frac{1}{6}\right) = 0 \]
This simplifies to:
\[ \frac{x}{3} + \frac{1}{3} - \frac{x}{4} + \frac{1}{6} = 0 \]

6. **Finding a Common Denominator:**
The least common multiple of 3, 4, and 6 is 12. We will multiply the entire equation by 12 to eliminate the fractions:
\[ 12\left(\frac{x}{3}\right) + 12\left(\frac{1}{3}\right) - 12\left(\frac{x}{4}\right) + 12\left(\frac{1}{6}\right) = 0 \]
This gives:
\[ 4x + 4 - 3x + 2 = 0 \]

7. **Combine Like Terms:**
\[ (4x - 3x) + (4 + 2) = 0 \implies x + 6 = 0 \]

8. **Solve for x:**
\[ x = -6 \]
Since distance cannot be negative, we need to check our calculations.

9. **Revisiting the Equation:**
The correct equation should be:
\[ \frac{x}{3} - \frac{x}{4} = \frac{1}{3} + \frac{1}{6} \]
Simplifying the right side:
\[ \frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \]
Now we have:
\[ \frac{x}{3} - \frac{x}{4} = \frac{1}{2} \]

10. **Finding a Common Denominator Again:**
The common denominator for 3 and 4 is 12:
\[ \frac{4x - 3x}{12} = \frac{1}{2} \]
This simplifies to:
\[ \frac{x}{12} = \frac{1}{2} \]

11. **Cross-Multiply to Solve for x:**
\[ x = 12 \times \frac{1}{2} = 6 \]

### Final Answer:
The distance the man has to walk is **6 kilometers**.