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 use the formula for average speed when two different speeds are involved over the same distance. The formula is:

\[ \text{Average Speed} = \frac{2xy}{x + y} \]

where \(x\) is the speed for the first part of the journey and \(y\) is the speed for the return journey.

### Step-by-Step Solution:

1. **Identify the Speeds**:
- Speed while cycling (x) = 6 km/h
- Speed while returning by bus (y) = 42 km/h

2. **Substitute the Values into the Formula**:
\[ \text{Average Speed} = \frac{2 \times 6 \times 42}{6 + 42} \]

3. **Calculate the Denominator**:
\[ 6 + 42 = 48 \]

4. **Calculate the Numerator**:
\[ 2 \times 6 \times 42 = 504 \]

5. **Divide the Numerator by the Denominator**:
\[ \text{Average Speed} = \frac{504}{48} \]

6. **Simplify the Fraction**:
\[ \frac{504}{48} = 10.5 \text{ km/h} \]

### Final Answer:
The average speed for the entire journey is **10.5 km/h**.