A man travels one-third part of his total distance with speed `v_(1)` and rest two-third part with speed `v_(2)`. Find out the average speed of the man?

Options

A. `(3v_(1)v_(2))/(v_(1)+v_(2))`
B. `(2v_(1)v_(2))/(v_(1)+ v_(2))`
C. `(3v_(1)v_(2))/(2v_(1)+v_(2))`
D. `(3v_(1)v_(2))/(v_(1)+2v_(2))`

Explanation

To find the average speed of a man who travels one-third of his total distance with speed \( v_1 \) and the remaining two-thirds with speed \( v_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Define Total Distance**:
Let the total distance traveled by the man be \( s \).

2. **Calculate Distances**:
- The distance traveled at speed \( v_1 \) is \( \frac{1}{3}s \).
- The distance traveled at speed \( v_2 \) is \( \frac{2}{3}s \).

3. **Calculate Time Taken for Each Segment**:
- For the first segment (distance \( \frac{1}{3}s \) at speed \( v_1 \)):
\[ t_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{1}{3}s}{v_1} = \frac{s}{3v_1} \]
- For the second segment (distance \( \frac{2}{3}s \) at speed \( v_2 \)):
\[ t_2 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{2}{3}s}{v_2} = \frac{2s}{3v_2} \]

4. **Calculate Total Time Taken**:
The total time \( T \) taken for the entire journey is the sum of the times for both segments:
\[ T = t_1 + t_2 = \frac{s}{3v_1} + \frac{2s}{3v_2} \]

5. **Combine the Time Expressions**:
To combine the two fractions, we can factor out \( s \):
\[ T = s \left( \frac{1}{3v_1} + \frac{2}{3v_2} \right) = s \left( \frac{v_2 + 2v_1}{3v_1v_2} \right) \]

6. **Calculate Average Speed**:
The average speed \( v_{avg} \) is defined as the total distance divided by the total time:
\[ v_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{s}{T} \]
Substituting for \( T \):
\[ v_{avg} = \frac{s}{s \left( \frac{v_2 + 2v_1}{3v_1v_2} \right)} = \frac{3v_1v_2}{v_2 + 2v_1} \]

### Final Result:
Thus, the average speed of the man is:
\[ v_{avg} = \frac{3v_1v_2}{v_2 + 2v_1} \]