Two men start together to walk a certain distance, one at 4 km/h and another at 3 km/h. The former arrives half an hour before the latter. Find the distance.

Options

A. 6 km
B. 9 km
C. 8 km
D. 7 km

Answer

To solve the problem step by step, we will use the information provided about the speeds of the two men and the time difference in their arrival.

### Step 1: Define the Variables
Let the distance they both travel be denoted as \( x \) kilometers.

### Step 2: Write the Time Taken by Each Man
- The time taken by the first man (who walks at 4 km/h) to cover the distance \( x \) is given by:
\[ \text{Time}_{1} = \frac{x}{4} \text{ hours} \]
- The time taken by the second man (who walks at 3 km/h) to cover the same distance \( x \) is given by:
\[ \text{Time}_{2} = \frac{x}{3} \text{ hours} \]

### Step 3: Set Up the Equation
According to the problem, the first man arrives half an hour before the second man. Therefore, we can set up the equation:
\[ \text{Time}_{2} - \text{Time}_{1} = \frac{1}{2} \]
Substituting the expressions for time:
\[ \frac{x}{3} - \frac{x}{4} = \frac{1}{2} \]

### Step 4: Solve the Equation
To solve the equation, we need a common denominator for the fractions on the left side. The least common multiple of 3 and 4 is 12. Thus, we can rewrite the equation as:
\[ \frac{4x}{12} - \frac{3x}{12} = \frac{1}{2} \]
This simplifies to:
\[ \frac{4x - 3x}{12} = \frac{1}{2} \]
\[ \frac{x}{12} = \frac{1}{2} \]

### Step 5: Cross-Multiply to Solve for \( x \)
Cross-multiplying gives us:
\[ x = 12 \times \frac{1}{2} \]
\[ x = 6 \]

### Step 6: Conclusion
The distance that both men walked is \( 6 \) kilometers.