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 can follow these steps:

### Step 1: Assume the Total Distance
Let's assume the total distance covered by the bus is \( X \) kilometers.

### Step 2: Calculate Time for Each Segment
1. **First Segment (One-third of the distance at 10 km/h)**:
- Distance = \( \frac{X}{3} \)
- Speed = 10 km/h
- Time taken, \( T_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{X}{3}}{10} = \frac{X}{30} \)

2. **Second Segment (One-fourth of the distance at 20 km/h)**:
- Distance = \( \frac{X}{4} \)
- Speed = 20 km/h
- Time taken, \( T_2 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{X}{4}}{20} = \frac{X}{80} \)

3. **Third Segment (Remaining distance at 40 km/h)**:
- Remaining distance = \( X - \left(\frac{X}{3} + \frac{X}{4}\right) \)
- To find this remaining distance, we first calculate:
\[ \frac{X}{3} + \frac{X}{4} = \frac{4X + 3X}{12} = \frac{7X}{12} \]
- Therefore, the remaining distance is:
\[ X - \frac{7X}{12} = \frac{12X - 7X}{12} = \frac{5X}{12} \]
- Speed = 40 km/h
- Time taken, \( T_3 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{5X}{12}}{40} = \frac{5X}{480} = \frac{X}{96} \)

### Step 3: Calculate Total Time
Now, we can find the total time taken by summing up \( T_1, T_2, \) and \( T_3 \):
\[ \text{Total Time} = T_1 + T_2 + T_3 = \frac{X}{30} + \frac{X}{80} + \frac{X}{96} \]

### Step 4: Find the LCM of the Denominators
To add these fractions, we need to find the least common multiple (LCM) of 30, 80, and 96. The LCM is 480.

### Step 5: Convert Each Time to the Common Denominator
Now we convert each term:
- \( T_1 = \frac{X}{30} = \frac{16X}{480} \)
- \( T_2 = \frac{X}{80} = \frac{6X}{480} \)
- \( T_3 = \frac{X}{96} = \frac{5X}{480} \)

### Step 6: Sum the Times
Now, we can sum them:
\[ \text{Total Time} = \frac{16X}{480} + \frac{6X}{480} + \frac{5X}{480} = \frac{(16 + 6 + 5)X}{480} = \frac{27X}{480} \]

### Step 7: Calculate Average Speed
The average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{X}{\frac{27X}{480}} = \frac{480}{27} \approx 17.78 \text{ km/h} \]

### Step 8: Round to the Nearest Whole Number
Rounding \( 17.78 \) km/h gives approximately \( 18 \) km/h.

### Final Answer
Thus, the average speed of the bus is approximately **18 km/h**.

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