A car travels the first half of a distance between two places a speed of 30 km/hr and the second half of the distance a 50 km/hr. The average speed of the car for the whole journey is

Options

A. 42.5 km/hr
B. 40.0 km/hr
C. 37.5 km/hr
D. 35.0 km/hr

Answer

To find the average speed of the car for the entire journey, we can follow these steps:

### Step 1: Define the total distance
Let the total distance between the two places be \( S \). Therefore, the first half of the distance is \( \frac{S}{2} \) and the second half is also \( \frac{S}{2} \).

### Step 2: Calculate the time taken for the first half
The speed for the first half of the distance is given as 30 km/hr.
Using the formula for time, \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), we can calculate the time taken for the first half:
\[ \text{Time}_1 = \frac{\frac{S}{2}}{30} = \frac{S}{60} \text{ hours} \]

### Step 3: Calculate the time taken for the second half
The speed for the second half of the distance is given as 50 km/hr.
Using the same formula for time, we can calculate the time taken for the second half:
\[ \text{Time}_2 = \frac{\frac{S}{2}}{50} = \frac{S}{100} \text{ hours} \]

### Step 4: Calculate the total time taken for the journey
Now, we can find the total time taken for the entire journey by adding the time for both halves:
\[ \text{Total Time} = \text{Time}_1 + \text{Time}_2 = \frac{S}{60} + \frac{S}{100} \]

### Step 5: Find a common denominator and simplify
To add the two fractions, we need a common denominator. The least common multiple of 60 and 100 is 300. Therefore, we convert each fraction:
\[ \frac{S}{60} = \frac{5S}{300} \quad \text{and} \quad \frac{S}{100} = \frac{3S}{300} \]
Now, we can add them:
\[ \text{Total Time} = \frac{5S}{300} + \frac{3S}{300} = \frac{8S}{300} = \frac{S}{37.5} \text{ hours} \]

### Step 6: Calculate the average speed
The average speed is defined as the total distance divided by the total time:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{S}{\frac{S}{37.5}} = 37.5 \text{ km/hr} \]

### Final Answer
The average speed of the car for the whole journey is **37.5 km/hr**.

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