For a train that travels from one station to another at a uniform speed of `40 kmh^(-1)` and returns to final station at speed of `60kmh^(-1)` then its average speed is

Options

A. `98 km//hr`
B. `0 km//hr`
C. `50 km//hr`
D. `48 km//hr`

Explanation

To find the average speed of the train during its journey from one station to another and back, we can follow these steps:

### Step 1: Define the Distance
Let the distance between the two stations be \( x \) kilometers.

### Step 2: Calculate the Time for Each Leg of the Journey
1. **Time for the first leg (from station A to station B)**:
- Speed = 40 km/h
- Time taken \( t_1 = \frac{x}{40} \) hours

2. **Time for the return leg (from station B to station A)**:
- Speed = 60 km/h
- Time taken \( t_2 = \frac{x}{60} \) hours

### Step 3: Calculate Total Distance and Total Time
1. **Total Distance**:
- The total distance covered during the entire journey is \( 2x \) (going to station B and returning to station A).

2. **Total Time**:
- The total time taken for the journey is the sum of the time taken for both legs:
\[ \text{Total Time} = t_1 + t_2 = \frac{x}{40} + \frac{x}{60} \]

### Step 4: Find a Common Denominator for Total Time
To add \( \frac{x}{40} \) and \( \frac{x}{60} \), we need a common denominator:
- The least common multiple of 40 and 60 is 120.
- Rewriting the times:
\[ t_1 = \frac{x}{40} = \frac{3x}{120} \]
\[ t_2 = \frac{x}{60} = \frac{2x}{120} \]
\[ \text{Total Time} = \frac{3x}{120} + \frac{2x}{120} = \frac{5x}{120} = \frac{x}{24} \]

### Step 5: Calculate Average Speed
The average speed \( v_{avg} \) is given by the formula:
\[ v_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2x}{\frac{x}{24}} = 2x \times \frac{24}{x} = 48 \text{ km/h} \]

### Final Answer
The average speed of the train during its journey is **48 km/h**.

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