A man covered a certain distance by a train running at the speed of 50 km/hr and covers the same distance back by walking at the speed of 5km/hr. If the whole journey took 5 hours 30 minutes, then what was the distance (in km) of one side?

Options

A. 30
B. 25
C. 20
D. 18

Answer

To solve the problem step by step, we will first define the variables and then use the relationship between distance, speed, and time.

### Step 1: Define the variables
Let the one-way distance be \( x \) km.

### Step 2: Calculate the time taken for each part of the journey
- The time taken to travel to the destination by train at a speed of 50 km/hr is given by:
\[ \text{Time by train} = \frac{x}{50} \text{ hours} \]
- The time taken to return by walking at a speed of 5 km/hr is given by:
\[ \text{Time by walking} = \frac{x}{5} \text{ hours} \]

### Step 3: Set up the equation for total time
The total time for the journey (both to and fro) is given as 5 hours and 30 minutes. We convert 5 hours and 30 minutes into hours:
\[ 5 \text{ hours } 30 \text{ minutes} = 5 + \frac{30}{60} = 5 + 0.5 = 5.5 \text{ hours} \]
Now we can set up the equation:
\[ \frac{x}{50} + \frac{x}{5} = 5.5 \]

### Step 4: Solve the equation
To solve the equation, we need a common denominator. The least common multiple (LCM) of 50 and 5 is 50. We can rewrite the second term:
\[ \frac{x}{5} = \frac{10x}{50} \]
Now, substituting this back into the equation gives:
\[ \frac{x}{50} + \frac{10x}{50} = 5.5 \]
Combining the fractions:
\[ \frac{11x}{50} = 5.5 \]

### Step 5: Isolate \( x \)
To isolate \( x \), we multiply both sides by 50:
\[ 11x = 5.5 \times 50 \]
Calculating the right side:
\[ 5.5 \times 50 = 275 \]
So we have:
\[ 11x = 275 \]
Now, divide both sides by 11:
\[ x = \frac{275}{11} = 25 \]

### Conclusion
The distance of one side is \( x = 25 \) km.