A student covers a distance by cycle at 6 km/h. He returns to the starting point in a bus at a speed of 42 km/h. The average speed (in km/h) for the entire journey is:

Options

A. 12.75
B. 24
C. 10.25
D. 10.5

Answer

To find the average speed for the entire journey, we can follow these steps:

### Step 1: Define the distance
Let the distance covered one way be \( d \) kilometers.

### Step 2: Calculate the time taken for each part of the journey
1. **Time taken to cycle to the destination**:
\[ \text{Time}_{\text{cycle}} = \frac{d}{6} \text{ hours} \]
2. **Time taken to return by bus**:
\[ \text{Time}_{\text{bus}} = \frac{d}{42} \text{ hours} \]

### Step 3: Calculate the total distance
The total distance for the entire journey (to the destination and back) is:
\[ \text{Total Distance} = d + d = 2d \text{ kilometers} \]

### Step 4: Calculate the total time
The total time taken for the entire journey is the sum of the time taken for both parts:
\[ \text{Total Time} = \text{Time}_{\text{cycle}} + \text{Time}_{\text{bus}} = \frac{d}{6} + \frac{d}{42} \]

### Step 5: Find a common denominator and simplify the total time
The common denominator for 6 and 42 is 42. Thus, we can rewrite the times:
\[ \frac{d}{6} = \frac{7d}{42} \]
So,
\[ \text{Total Time} = \frac{7d}{42} + \frac{d}{42} = \frac{7d + d}{42} = \frac{8d}{42} = \frac{4d}{21} \text{ hours} \]

### Step 6: Calculate the average speed
Average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{4d}{21}} = 2d \times \frac{21}{4d} = \frac{42}{4} = 10.5 \text{ km/h} \]

### Conclusion
The average speed for the entire journey is:
\[ \text{Average Speed} = 10.5 \text{ km/h} \]