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

Answer

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

### Step 1: Define the total distance
Let the total distance be \( s \).

### Step 2: Calculate the distances for each segment
- Distance covered at speed \( v_1 \):
\[ d_1 = \frac{2}{3}s \]
- Distance covered at speed \( v_2 \):
\[ d_2 = \frac{1}{3}s \]

### Step 3: Calculate the time taken for each segment
- Time taken to cover the first segment (\( t_1 \)):
\[ t_1 = \frac{d_1}{v_1} = \frac{\frac{2}{3}s}{v_1} = \frac{2s}{3v_1} \]
- Time taken to cover the second segment (\( t_2 \)):
\[ t_2 = \frac{d_2}{v_2} = \frac{\frac{1}{3}s}{v_2} = \frac{s}{3v_2} \]

### Step 4: Calculate the total time taken
The total time \( T \) is the sum of the two times:
\[ T = t_1 + t_2 = \frac{2s}{3v_1} + \frac{s}{3v_2} \]

### Step 5: Simplify the total time expression
To combine the fractions, we need a common denominator:
\[ T = \frac{2s}{3v_1} + \frac{s}{3v_2} = \frac{2s v_2 + s v_1}{3v_1 v_2} = \frac{s(2v_2 + v_1)}{3v_1 v_2} \]

### Step 6: Calculate the 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 the expression for \( T \):
\[ V_{avg} = \frac{s}{\frac{s(2v_2 + v_1)}{3v_1 v_2}} = \frac{s \cdot 3v_1 v_2}{s(2v_2 + v_1)} \]
Cancelling \( s \) from numerator and denominator:
\[ V_{avg} = \frac{3v_1 v_2}{2v_2 + v_1} \]

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