A person covers half of his journey at 30km/h,one-third at 40km/h and rest of his iourney at 20 km/h. Find his average speed for the whole iourney.

Answer

To find the average speed of the person for the entire journey, we can follow these steps:

### Step 1: Define the total distance
Let the total distance of the journey be \( x \) kilometers.

### Step 2: Calculate the distance covered in each segment
1. **First segment:** Half of the journey is covered at 30 km/h.
- Distance = \( \frac{x}{2} \)
2. **Second segment:** One-third of the journey is covered at 40 km/h.
- Distance = \( \frac{x}{3} \)
3. **Third segment:** The remaining distance is covered at 20 km/h.
- Distance = Total distance - (Distance of first segment + Distance of second segment)
- Remaining distance = \( x - \left(\frac{x}{2} + \frac{x}{3}\right) \)

### Step 3: Find the remaining distance
To find the remaining distance, we need to calculate:
\[ \frac{x}{2} + \frac{x}{3} = \frac{3x}{6} + \frac{2x}{6} = \frac{5x}{6} \]
Thus, the remaining distance is:
\[ x - \frac{5x}{6} = \frac{x}{6} \]

### Step 4: Calculate the time taken for each segment
1. **Time for the first segment:**
\[ t_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{x}{2}}{30} = \frac{x}{60} \text{ hours} \]
2. **Time for the second segment:**
\[ t_2 = \frac{\frac{x}{3}}{40} = \frac{x}{120} \text{ hours} \]
3. **Time for the third segment:**
\[ t_3 = \frac{\frac{x}{6}}{20} = \frac{x}{120} \text{ hours} \]

### Step 5: Calculate the total time taken
Now, we can find the total time taken for the journey:
\[ \text{Total time} = t_1 + t_2 + t_3 = \frac{x}{60} + \frac{x}{120} + \frac{x}{120} \]
To add these fractions, we find a common denominator (which is 120):
\[ \frac{x}{60} = \frac{2x}{120} \]
So,
\[ \text{Total time} = \frac{2x}{120} + \frac{x}{120} + \frac{x}{120} = \frac{4x}{120} = \frac{x}{30} \text{ hours} \]

### Step 6: Calculate the average speed
The average speed is given by the formula:
\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{x}{\frac{x}{30}} = 30 \text{ km/h} \]

### Final Answer
The average speed for the whole journey is **30 km/h**.

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