A person travels one - third of the total distance at a speed of 2 km/h, the next one third of the total distance at a speed of 3 km/h and the rest of the total distance at a speed of 6 km/h . Find the average speed of the person for the total trip?

Answer

To find the average speed of the person for the total trip, we can follow these steps:

### Step 1: Define the total distance
Let the total distance be \( D \). According to the problem, the person travels one-third of the distance at different speeds.

### Step 2: Calculate the distance for each segment
- The first segment:
\[ \text{Distance} = \frac{D}{3} \]

- The second segment:
\[ \text{Distance} = \frac{D}{3} \]

- The third segment:
\[ \text{Distance} = \frac{D}{3} \]

### Step 3: Calculate the time taken for each segment
- Time for the first segment at 2 km/h:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{D}{3}}{2} = \frac{D}{6} \text{ hours} \]

- Time for the second segment at 3 km/h:
\[ \text{Time} = \frac{\frac{D}{3}}{3} = \frac{D}{9} \text{ hours} \]

- Time for the third segment at 6 km/h:
\[ \text{Time} = \frac{\frac{D}{3}}{6} = \frac{D}{18} \text{ hours} \]

### Step 4: Calculate the total time taken
Now, we add the time taken for all three segments:
\[ \text{Total Time} = \frac{D}{6} + \frac{D}{9} + \frac{D}{18} \]

To add these fractions, we need a common denominator. The least common multiple of 6, 9, and 18 is 18.

- Convert each term:
\[ \frac{D}{6} = \frac{3D}{18}, \quad \frac{D}{9} = \frac{2D}{18}, \quad \frac{D}{18} = \frac{D}{18} \]

- Now add them:
\[ \text{Total Time} = \frac{3D}{18} + \frac{2D}{18} + \frac{D}{18} = \frac{6D}{18} = \frac{D}{3} \text{ hours} \]

### Step 5: Calculate the average speed
The average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{\frac{D}{3}} = 3 \text{ km/h} \]

### Final Answer
The average speed of the person for the total trip is **3 km/h**.

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