A  covers  1/3  rd  of  his  journey  at  a  speed  of 10km/hr;  half  of  the  remaining  at  a  speed  of 15km/hr   and  the  remaining  at  a   speed  of 30km/hr. What  is  the  average  speed  of A?

To find the average speed of A for the entire journey, we need to calculate the total distance and the total time taken for the journey.

Let's denote the total distance of the journey by \( D \).

1. **First part of the journey**:
   - Distance covered: \( \frac{1}{3}D \)
   - Speed: 10 km/hr
   - Time taken: \( \text{Time}_1 = \frac{\frac{1}{3}D}{10} = \frac{D}{30} \) hours

2. **Second part of the journey**:
   - Distance covered: \( \frac{1}{2} \) of the remaining distance
   - Remaining distance after first part: \( D - \frac{1}{3}D = \frac{2}{3}D \)
   - Distance covered in this part: \( \frac{1}{2} \times \frac{2}{3}D = \frac{1}{3}D \)
   - Speed: 15 km/hr
   - Time taken: \( \text{Time}_2 = \frac{\frac{1}{3}D}{15} = \frac{D}{45} \) hours

3. **Third part of the journey**:
   - Distance covered: \( \frac{1}{2} \) of the remaining distance
   - Remaining distance after second part: \( \frac{2}{3}D - \frac{1}{3}D = \frac{1}{3}D \)
   - Distance covered in this part: \( \frac{1}{3}D \)
   - Speed: 30 km/hr
   - Time taken: \( \text{Time}_3 = \frac{\frac{1}{3}D}{30} = \frac{D}{90} \) hours

4. **Total distance and total time**:
   - Total distance: \( D \)
   - Total time taken: \( \text{Time}_1 + \text{Time}_2 + \text{Time}_3 = \frac{D}{30} + \frac{D}{45} + \frac{D}{90} \)

To find the total time, we need to find a common denominator for the fractions \( \frac{D}{30} \), \( \frac{D}{45} \), and \( \frac{D}{90} \). The least common multiple of 30, 45, and 90 is 90.

\[
\text{Total time} = \frac{D}{30} + \frac{D}{45} + \frac{D}{90}
\]
\[
\text{Total time} = \frac{3D}{90} + \frac{2D}{90} + \frac{D}{90} = \frac{3D + 2D + D}{90} = \frac{6D}{90} = \frac{D}{15} \text{ hours}
\]

5. **Average speed**:
   \[
   \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{D}{\frac{D}{15}} = 15 \text{ km/hr}
   \]

Therefore, the average speed of A is 15 km/hr.

Do you want details or have any questions? Here are 5 numbered questions for you to ask:
1. How to solve a quadratic equation?
2. What is the derivative of \( e^x \)?
3. How do you integrate \( \sin(x) \)?
4. Can you explain the Pythagorean theorem?
5. What is the area of a circle?

**Tip:** When dealing with average speed in different segments of a journey, remember that average speed is not simply the mean of speeds but the total distance divided by the total time taken.

Explore how to calculate the average speed of A for a journey involving different speeds (10 km/hr, 15 km/hr, and 30 km/hr). This problem covers distance-time relationships and average speed calculations. Suitable for Grades 9-12 students.

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