A man travels a certain distance at the rate of 12 km/hr and return back to the starting point at the rate of 15 km/hr . His average speed during the whole journey is

Options

A. `13.5` km/hr
B. `13 1/3` km/hr
C. `12 2/3` km/hr
D. `14` km/hr

Answer

To find the average speed of the man during his entire journey, we can follow these steps:

### Step 1: Define the distance
Let the distance from point A to point B be \( x \) km.

### Step 2: Calculate total distance
The total distance traveled by the man is the distance from A to B and back from B to A. Therefore, the total distance \( D \) is:
\[ D = x + x = 2x \text{ km} \]

### Step 3: Calculate time taken for each part of the journey
1. **Time taken to travel from A to B** at 12 km/hr:
\[ T_1 = \frac{x}{12} \text{ hours} \]
2. **Time taken to return from B to A** at 15 km/hr:
\[ T_2 = \frac{x}{15} \text{ hours} \]

### Step 4: Calculate total time taken
The total time \( T \) for the entire journey is the sum of the time taken for each part:
\[ T = T_1 + T_2 = \frac{x}{12} + \frac{x}{15} \]

### Step 5: Find a common denominator to add the times
The least common multiple (LCM) of 12 and 15 is 60. We convert each fraction:
\[ T = \frac{5x}{60} + \frac{4x}{60} = \frac{9x}{60} = \frac{3x}{20} \text{ hours} \]

### Step 6: Calculate average speed
The average speed \( V \) is defined as total distance divided by total time:
\[ V = \frac{D}{T} = \frac{2x}{\frac{3x}{20}} = 2x \times \frac{20}{3x} \]
Here, \( x \) cancels out:
\[ V = \frac{40}{3} \text{ km/hr} \]

### Step 7: Convert to mixed number
To convert \( \frac{40}{3} \) into a mixed number:
\[ \frac{40}{3} = 13 \frac{1}{3} \text{ km/hr} \]

### Final Answer
The average speed during the whole journey is:
\[ \text{Average Speed} = 13 \frac{1}{3} \text{ km/hr} \]

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