A bus travelled the first one-third distance at a speed of `10 km//h`, the next one-third at `20 km//h` and the last one-third at `60 km//h`. The average speed of the bus is

Options

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

Answer

To find the average speed of the bus that travels different segments of a journey at varying speeds, we can follow these steps:

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

### Step 2: Divide the distance into segments
The bus travels the distance in three equal segments:
- First segment: \( \frac{s}{3} \) at a speed of \( 10 \, \text{km/h} \)
- Second segment: \( \frac{s}{3} \) at a speed of \( 20 \, \text{km/h} \)
- Third segment: \( \frac{s}{3} \) at a speed of \( 60 \, \text{km/h} \)

### Step 3: Calculate the time taken for each segment
Using the formula \( \text{time} = \frac{\text{distance}}{\text{speed}} \):
- Time for the first segment \( t_1 \):
\[ t_1 = \frac{\frac{s}{3}}{10} = \frac{s}{30} \]
- Time for the second segment \( t_2 \):
\[ t_2 = \frac{\frac{s}{3}}{20} = \frac{s}{60} \]
- Time for the third segment \( t_3 \):
\[ t_3 = \frac{\frac{s}{3}}{60} = \frac{s}{180} \]

### Step 4: Calculate the total time taken
The total time \( T \) is the sum of the times for each segment:
\[ T = t_1 + t_2 + t_3 = \frac{s}{30} + \frac{s}{60} + \frac{s}{180} \]

### Step 5: Find a common denominator and simplify
The least common multiple of \( 30, 60, \) and \( 180 \) is \( 180 \). Rewriting each term with a common denominator:
\[ T = \frac{6s}{180} + \frac{3s}{180} + \frac{s}{180} = \frac{10s}{180} = \frac{s}{18} \]

### Step 6: Calculate the average speed
The average speed \( V_{avg} \) is given by the formula:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{s}{T} = \frac{s}{\frac{s}{18}} = 18 \, \text{km/h} \]

### Final Answer
The average speed of the bus is \( 18 \, \text{km/h} \).

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