If I walk at 4 km/h, I miss the bus by 10 minutes. If I walk at 5 km/h, I reach 5 minutes before the arrival of the bus. How far I walk to reach the bus stand ?

Options

A. 5 km
B. 4.5 km
C. `5(1)/(5)` km
D. cannot be determined

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Define the Variables
Let the distance to the bus stand be \( x \) kilometers.

### Step 2: Calculate Time Taken at Different Speeds
1. **At 4 km/h**:
- The time taken to reach the bus stand is \( \frac{x}{4} \) hours.
- Since I miss the bus by 10 minutes, the usual time taken to reach the bus stand is \( \frac{x}{4} - \frac{10}{60} \) hours.

2. **At 5 km/h**:
- The time taken to reach the bus stand is \( \frac{x}{5} \) hours.
- Since I reach 5 minutes early, the usual time taken to reach the bus stand is \( \frac{x}{5} + \frac{5}{60} \) hours.

### Step 3: Set Up the Equation
Since both expressions represent the usual time taken to reach the bus stand, we can set them equal to each other:

\[ \frac{x}{4} - \frac{10}{60} = \frac{x}{5} + \frac{5}{60} \]

### Step 4: Simplify the Equation
1. Convert minutes to hours:
- \( \frac{10}{60} = \frac{1}{6} \) hours
- \( \frac{5}{60} = \frac{1}{12} \) hours

2. Substitute these values into the equation:

\[ \frac{x}{4} - \frac{1}{6} = \frac{x}{5} + \frac{1}{12} \]

### Step 5: Find a Common Denominator
The least common multiple (LCM) of 4, 5, 6, and 12 is 60. We will multiply through by 60 to eliminate the fractions:

\[ 60 \left( \frac{x}{4} \right) - 60 \left( \frac{1}{6} \right) = 60 \left( \frac{x}{5} \right) + 60 \left( \frac{1}{12} \right) \]

This simplifies to:

\[ 15x - 10 = 12x + 5 \]

### Step 6: Solve for \( x \)
1. Rearranging the equation gives:

\[ 15x - 12x = 10 + 5 \]

2. Simplifying further:

\[ 3x = 15 \]

3. Dividing both sides by 3:

\[ x = 5 \]

### Conclusion
The distance to the bus stand is **5 kilometers**.

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