A bus maintains an average speed of 60 km/hr while going from P to Q and maintains an average speed of 90 km/hr while coming back from Q to P. The average speed of the bus is______.

Options

A. 75 km/hr
B. 72 km/hr
C. 70 km/hr
D. 30 km/hr

Answer

To find the average speed of the bus for the entire journey from P to Q and back from Q to P, we can follow these steps:

### Step 1: Define the speeds and distances
Let the distance from P to Q be \( d \) km. The bus travels from P to Q at a speed of 60 km/hr and returns from Q to P at a speed of 90 km/hr.

### Step 2: Calculate the time taken for each part of the journey
- **Time taken from P to Q**:
\[ t_1 = \frac{d}{60} \]

- **Time taken from Q to P**:
\[ t_2 = \frac{d}{90} \]

### Step 3: Calculate the total distance and total time
- **Total distance** for the round trip:
\[ \text{Total Distance} = d + d = 2d \]

- **Total time** for the round trip:
\[ \text{Total Time} = t_1 + t_2 = \frac{d}{60} + \frac{d}{90} \]

To add these fractions, we need a common denominator:
- The least common multiple of 60 and 90 is 180.

Rewriting the times:
\[ t_1 = \frac{d}{60} = \frac{3d}{180} \]
\[ t_2 = \frac{d}{90} = \frac{2d}{180} \]

Now, adding these:
\[ \text{Total Time} = \frac{3d}{180} + \frac{2d}{180} = \frac{5d}{180} = \frac{d}{36} \]

### Step 4: Calculate the average speed
The average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{d}{36}} = 2d \times \frac{36}{d} = 72 \text{ km/hr} \]

Thus, the average speed of the bus for the entire journey is **72 km/hr**.

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