A bus traveling the first one-third distance at a speed of 10 km/h, the next one-fourth at 20- km/h and the remaining at 40 km/h. The average speed of the bus is nearly

Options

A. 9km/h
B. 16km/h
C. 18km/h
D. 48km/h

Answer

To find the average speed of the bus, we will break down the problem step by step.

### Step 1: Define the total distance
Let the total distance traveled by the bus be \( D \).

### Step 2: Calculate the distances for each segment
1. The first segment is one-third of the total distance:
\[ \text{Distance}_1 = \frac{D}{3} \]
2. The second segment is one-fourth of the total distance:
\[ \text{Distance}_2 = \frac{D}{4} \]
3. The remaining distance is:
\[ \text{Distance}_3 = D - \left(\frac{D}{3} + \frac{D}{4}\right) \]

### Step 3: Find the remaining distance
To find \( \text{Distance}_3 \), we first need a common denominator to add \( \frac{D}{3} \) and \( \frac{D}{4} \):
- The least common multiple of 3 and 4 is 12.
- Thus,
\[ \frac{D}{3} = \frac{4D}{12}, \quad \frac{D}{4} = \frac{3D}{12} \]
- Adding these gives:
\[ \frac{D}{3} + \frac{D}{4} = \frac{4D}{12} + \frac{3D}{12} = \frac{7D}{12} \]
- Therefore, the remaining distance is:
\[ \text{Distance}_3 = D - \frac{7D}{12} = \frac{5D}{12} \]

### Step 4: Calculate the time taken for each segment
1. For the first segment:
\[ \text{Speed}_1 = 10 \text{ km/h} \Rightarrow \text{Time}_1 = \frac{\text{Distance}_1}{\text{Speed}_1} = \frac{\frac{D}{3}}{10} = \frac{D}{30} \text{ hours} \]
2. For the second segment:
\[ \text{Speed}_2 = 20 \text{ km/h} \Rightarrow \text{Time}_2 = \frac{\text{Distance}_2}{\text{Speed}_2} = \frac{\frac{D}{4}}{20} = \frac{D}{80} \text{ hours} \]
3. For the third segment:
\[ \text{Speed}_3 = 40 \text{ km/h} \Rightarrow \text{Time}_3 = \frac{\text{Distance}_3}{\text{Speed}_3} = \frac{\frac{5D}{12}}{40} = \frac{5D}{480} = \frac{D}{96} \text{ hours} \]

### Step 5: Calculate the total time taken
Now, we sum the times for all segments:
\[ \text{Total Time} = \text{Time}_1 + \text{Time}_2 + \text{Time}_3 = \frac{D}{30} + \frac{D}{80} + \frac{D}{96} \]
To add these fractions, we need a common denominator. The least common multiple of 30, 80, and 96 is 240. Therefore:
\[ \frac{D}{30} = \frac{8D}{240}, \quad \frac{D}{80} = \frac{3D}{240}, \quad \frac{D}{96} = \frac{2.5D}{240} \]
Adding these gives:
\[ \text{Total Time} = \frac{8D + 3D + 2.5D}{240} = \frac{13.5D}{240} = \frac{D}{17.78} \text{ hours} \]

### Step 6: Calculate the average speed
The average speed is given by:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{\frac{D}{17.78}} = 17.78 \text{ km/h} \]

### Conclusion
Thus, the average speed of the bus is nearly **18 km/h**.

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