A and B are two stations A train goes from A to B at 64 km/hr and returns to A at a slower speed if its average speed for the whole journey is 56 km/hr at what speed did it return?

Options

A. 48 km/hr
B. 49.77km/hr
C. 52 km/hr
D. 47.46km/hr

Answer

To solve the problem, we need to find the speed of the train on its return journey from B to A. Let's denote the speed of the train on the return journey as \( v \) km/hr.

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

### Step 2: Calculate the time taken for each leg of the journey
- Time taken to go from A to B at 64 km/hr:
\[ \text{Time}_{AB} = \frac{d}{64} \]

- Time taken to return from B to A at \( v \) km/hr:
\[ \text{Time}_{BA} = \frac{d}{v} \]

### Step 3: Calculate the total time for the journey
The total time for the round trip is:
\[ \text{Total Time} = \text{Time}_{AB} + \text{Time}_{BA} = \frac{d}{64} + \frac{d}{v} \]

### Step 4: Calculate the total distance for the journey
The total distance for the round trip is:
\[ \text{Total Distance} = d + d = 2d \]

### Step 5: Calculate the average speed
The average speed for the whole journey is given as 56 km/hr. The formula for average speed is:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the known values:
\[ 56 = \frac{2d}{\frac{d}{64} + \frac{d}{v}} \]

### Step 6: Simplify the equation
We can simplify the equation:
\[ 56 = \frac{2d}{\frac{d}{64} + \frac{d}{v}} \implies 56 = \frac{2}{\frac{1}{64} + \frac{1}{v}} \]

### Step 7: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 56 \left( \frac{1}{64} + \frac{1}{v} \right) = 2 \]

### Step 8: Distribute and simplify
Distributing \( 56 \):
\[ \frac{56}{64} + \frac{56}{v} = 2 \]
\[ \frac{7}{8} + \frac{56}{v} = 2 \]

### Step 9: Isolate \( \frac{56}{v} \)
Subtract \( \frac{7}{8} \) from both sides:
\[ \frac{56}{v} = 2 - \frac{7}{8} \]
Convert 2 to a fraction with a denominator of 8:
\[ 2 = \frac{16}{8} \]
Thus:
\[ \frac{56}{v} = \frac{16}{8} - \frac{7}{8} = \frac{9}{8} \]

### Step 10: Solve for \( v \)
Cross-multiply to find \( v \):
\[ 56 \cdot 8 = 9v \implies 448 = 9v \implies v = \frac{448}{9} \approx 49.78 \text{ km/hr} \]

### Conclusion
The speed of the train on its return journey from B to A is approximately \( 49.78 \) km/hr.