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.