A man travels half time of its journey with speed of 5m/s and for remaining half time he moves half of its total distance travelled, with speed of 3 m/s and remaining half distance with speed of 6 m/s. Find out the average speed of the man?

Answer

To find the average speed of the man, we can break down the problem into manageable steps. Let's denote the total time of the journey as \( T \).

### Step 1: Define the total distance and time
Let the total distance traveled by the man be \( D \). According to the problem, he travels half of the journey in half the time at different speeds.

### Step 2: Calculate the distance traveled in the first half of the journey
In the first half of the journey, the man travels for time \( T/2 \) at a speed of \( 5 \, \text{m/s} \). The distance \( D_1 \) covered in this time can be calculated as:
\[ D_1 = \text{Speed} \times \text{Time} = 5 \, \text{m/s} \times \frac{T}{2} = \frac{5T}{2} \, \text{m} \]

### Step 3: Calculate the remaining distance
The remaining distance \( D_2 \) is the total distance minus the distance covered in the first half:
\[ D_2 = D - D_1 = D - \frac{5T}{2} \]

### Step 4: Define the second half of the journey
In the second half of the journey, he travels for the remaining half of the time \( T/2 \). According to the problem, he covers half of the remaining distance at a speed of \( 3 \, \text{m/s} \) and the other half at a speed of \( 6 \, \text{m/s} \).

Let \( D_2 = D - \frac{5T}{2} \). The distance covered at \( 3 \, \text{m/s} \) is:
\[ D_{2a} = \frac{1}{2} D_2 = \frac{1}{2} \left(D - \frac{5T}{2}\right) \]
The time taken to cover this distance at \( 3 \, \text{m/s} \) is:
\[ t_a = \frac{D_{2a}}{3} = \frac{\frac{1}{2} \left(D - \frac{5T}{2}\right)}{3} = \frac{D - \frac{5T}{2}}{6} \]

The remaining distance covered at \( 6 \, \text{m/s} \) is:
\[ D_{2b} = \frac{1}{2} D_2 = \frac{1}{2} \left(D - \frac{5T}{2}\right) \]
The time taken to cover this distance at \( 6 \, \text{m/s} \) is:
\[ t_b = \frac{D_{2b}}{6} = \frac{\frac{1}{2} \left(D - \frac{5T}{2}\right)}{6} = \frac{D - \frac{5T}{2}}{12} \]

### Step 5: Total time for the second half of the journey
The total time for the second half of the journey is:
\[ T/2 = t_a + t_b = \frac{D - \frac{5T}{2}}{6} + \frac{D - \frac{5T}{2}}{12} \]
To combine these fractions, we find a common denominator:
\[ T/2 = \frac{2(D - \frac{5T}{2})}{12} + \frac{D - \frac{5T}{2}}{12} = \frac{3(D - \frac{5T}{2})}{12} = \frac{D - \frac{5T}{2}}{4} \]

### Step 6: Solve for total distance and average speed
From the equation \( T/2 = \frac{D - \frac{5T}{2}}{4} \), we can solve for \( D \):
\[ D - \frac{5T}{2} = 2T \implies D = 2T + \frac{5T}{2} = \frac{4T + 5T}{2} = \frac{9T}{2} \]

### Step 7: Calculate the average speed
The average speed \( V_{avg} \) is defined as:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{T} = \frac{\frac{9T}{2}}{T} = \frac{9}{2} = 4.5 \, \text{m/s} \]

### Final Answer
The average speed of the man is \( 4.5 \, \text{m/s} \).