A man walks from A to B and back in a certain time at the rate of 3.5 km per hour. But if he had walked from A to B at the rate of 3 km per hour and back from B to A at the rate of 4 km a hour, he would have taken 5 minutes longer. Find the distance between A and B:

Options

A. (a) 7 km
B. (b) 8 km
C. (c) 9 km
D. (d) 10 km

Answer

To solve the problem step by step, we will denote the distance between points A and B as \( x \) km.

### Step 1: Calculate the time taken in the first scenario
In the first scenario, the man walks from A to B and back at a speed of 3.5 km/h. The total distance for the round trip is \( 2x \) km.

The time taken \( T_1 \) can be calculated using the formula:
\[ T_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{2x}{3.5} \]

### Step 2: Calculate the time taken in the second scenario
In the second scenario, the man walks from A to B at a speed of 3 km/h and returns from B to A at a speed of 4 km/h.

The time taken \( T_2 \) for this trip can be calculated as:
\[ T_2 = \frac{x}{3} + \frac{x}{4} \]

### Step 3: Set up the equation based on the time difference
According to the problem, \( T_2 \) takes 5 minutes longer than \( T_1 \). We need to convert 5 minutes into hours for consistency in units:
\[ 5 \text{ minutes} = \frac{5}{60} \text{ hours} = \frac{1}{12} \text{ hours} \]

Thus, we can write the equation:
\[ T_2 - T_1 = \frac{1}{12} \]

Substituting the expressions for \( T_1 \) and \( T_2 \):
\[ \left( \frac{x}{3} + \frac{x}{4} \right) - \frac{2x}{3.5} = \frac{1}{12} \]

### Step 4: Simplify the equation
First, we need to find a common denominator for the left-hand side. The common denominator for 3, 4, and 3.5 (which is 7/2) is 84.

Rewriting the fractions:
- For \( \frac{x}{3} \):
\[ \frac{x}{3} = \frac{28x}{84} \]
- For \( \frac{x}{4} \):
\[ \frac{x}{4} = \frac{21x}{84} \]
- For \( \frac{2x}{3.5} \):
\[ \frac{2x}{3.5} = \frac{2x \cdot 2}{7} = \frac{4x}{7} = \frac{48x}{84} \]

Now substituting back into the equation:
\[ \left( \frac{28x + 21x}{84} - \frac{48x}{84} \right) = \frac{1}{12} \]

This simplifies to:
\[ \frac{49x - 48x}{84} = \frac{1}{12} \]
\[ \frac{x}{84} = \frac{1}{12} \]

### Step 5: Solve for \( x \)
Cross-multiplying gives:
\[ x \cdot 12 = 84 \cdot 1 \]
\[ 12x = 84 \]
\[ x = \frac{84}{12} = 7 \]

### Conclusion
The distance between A and B is \( 7 \) km.

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