A car driver cover same distance between two cities at a speed of 60km/h and on there turn his speed is 40 km/h.He goes again from the 1st to the 2nd city at twice the original speed and returns at half the original return speed. Find his average speed for the entire journey.

Options

A. 55 km/h
B. 50 km/h
C. 48 km/h
D. 40 km/h

Answer

To solve the problem step by step, we will calculate the total distance and total time taken for the entire journey, and then use these values to find the average speed.

### Step 1: Define the distance
Let the distance between the two cities (A and B) be \( d \) km.

### Step 2: Calculate the total distance
The car makes two trips:
1. From A to B
2. From B to A

The total distance covered in the journey is:
\[ \text{Total Distance} = d + d + 2d + 2d = 4d \text{ km} \]

### Step 3: Calculate the time taken for each leg of the journey
1. **From A to B at 60 km/h:**
\[ \text{Time}_{A \to B} = \frac{d}{60} \text{ hours} \]

2. **From B to A at 40 km/h:**
\[ \text{Time}_{B \to A} = \frac{d}{40} \text{ hours} \]

3. **From A to B again at twice the original speed (2 * 60 km/h = 120 km/h):**
\[ \text{Time}_{A \to B \text{ (twice speed)}} = \frac{d}{120} \text{ hours} \]

4. **From B to A again at half the original return speed (0.5 * 40 km/h = 20 km/h):**
\[ \text{Time}_{B \to A \text{ (half speed)}} = \frac{d}{20} \text{ hours} \]

### Step 4: Calculate the total time taken
Now, we can sum all the times:
\[ \text{Total Time} = \text{Time}_{A \to B} + \text{Time}_{B \to A} + \text{Time}_{A \to B \text{ (twice speed)}} + \text{Time}_{B \to A \text{ (half speed)}} \]
\[ = \frac{d}{60} + \frac{d}{40} + \frac{d}{120} + \frac{d}{20} \]

### Step 5: Find a common denominator and simplify
The least common multiple of 60, 40, 120, and 20 is 120. We convert each term:
\[ = \frac{2d}{120} + \frac{3d}{120} + \frac{d}{120} + \frac{6d}{120} \]
\[ = \frac{2d + 3d + d + 6d}{120} = \frac{12d}{120} = \frac{d}{10} \text{ hours} \]

### Step 6: Calculate the average speed
The average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the values we have:
\[ \text{Average Speed} = \frac{4d}{\frac{d}{10}} = 4d \times \frac{10}{d} = 40 \text{ km/h} \]

### Final Answer
The average speed for the entire journey is **40 km/h**.

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