`A` train travels between two stations `A` and `B.` While going from `A` to `B,` its average speed is `80` km/hr, and when coming back from `B` to `A `. Its average speed is `120` km/hr. find the average speed of the train during the whole journey.

Options

A. `" 96 km/hr"`
B. `" 94 km/hr"`
C. `" 95 km/hr"`
D. `" 90 km/hr"`

Explanation

To find the average speed of the train during the whole journey from station A to station B and back, we can follow these steps:

### Step 1: Define the Distance
Let the distance between stations A and B be \( x \) km.

### Step 2: Calculate the Time Taken for Each Leg of the Journey
- **Time taken from A to B**:
\[ \text{Time}_{A \to B} = \frac{x}{80} \text{ hours} \]

- **Time taken from B to A**:
\[ \text{Time}_{B \to A} = \frac{x}{120} \text{ hours} \]

### Step 3: Calculate the Total Time for the Journey
The total time for the whole journey (A to B and back to A) is:
\[ \text{Total Time} = \text{Time}_{A \to B} + \text{Time}_{B \to A} = \frac{x}{80} + \frac{x}{120} \]

### Step 4: Find a Common Denominator
To add the fractions, we need a common denominator. The least common multiple of 80 and 120 is 240. Thus, we convert the fractions:
\[ \frac{x}{80} = \frac{3x}{240} \quad \text{and} \quad \frac{x}{120} = \frac{2x}{240} \]
So,
\[ \text{Total Time} = \frac{3x}{240} + \frac{2x}{240} = \frac{5x}{240} = \frac{x}{48} \text{ hours} \]

### Step 5: Calculate the Total Distance
The total distance for the whole journey is:
\[ \text{Total Distance} = x + x = 2x \text{ km} \]

### Step 6: Calculate the Average Speed
Average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the values we found:
\[ \text{Average Speed} = \frac{2x}{\frac{5x}{240}} = 2x \times \frac{240}{5x} = \frac{480}{5} = 96 \text{ km/hr} \]

### Conclusion
The average speed of the train during the whole journey is **96 km/hr**.

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