https://file.aiquickdraw.com/mathbot/file/17274739832170jbb660h.jpg

The image contains a mathematical exercise, written partly in English and Bengali.

### Here's a breakdown of what's visible in the image:

1. **Ratio problem involving \( P \), \( Q \), and \( R \):**
   - It shows the ratio \( P : Q : R = 5 : 2 : 9 \).
   - A calculation involving:
     \[
     \frac{3200}{16} = 200
     \]
   - Followed by:
     \[
     2 \times 200 = 400
     \]
     This part seems to be solving for a particular value based on given ratios.

2. **Proportion Rule (a, b, c) in English:**
   - \( a : b = b : c \)
   - Which implies:
     \[
     \frac{a}{b} = \frac{b}{c}
     \]
     - Another derived formula: \( b^2 = ac \).

3. **Bengali text problem:**
   - A word problem stating (translated to English):
     > If \( a = 3c \) and the arithmetic mean of \( a \) and \( b \) is \( b \), then find the value of \( \frac{(a-b)^3}{(b-c)^3} \).

### Solving the problem:

Given:
- \( a = 3c \)
- Arithmetic mean of \( a \) and \( b \) is \( b \), so:
  \[
  \frac{a + b}{2} = b
  \]

From this, solving for \( b \):
\[
a + b = 2b \implies a = 2b - b \implies b = a
\]
Since \( b = a \), the expression simplifies.

Finally, calculating the expression \( \frac{(a-b)^3}{(b-c)^3} \).

Would you like more detailed steps for solving this, or do you have other questions related to this?

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### 5 Related Questions:
1. How is the concept of proportions used in real-world problems?
2. What is the significance of the ratio in solving distribution problems like this?
3. Can you explain the formula \( b^2 = ac \) in more detail with examples?
4. What is the importance of arithmetic mean, and where is it commonly applied?
5. How can you generalize the expression \( \frac{(a-b)^3}{(b-c)^3} \) for other values of \( a \), \( b \), and \( c \)?

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**Tip:** For solving ratio problems, always ensure to first simplify the given ratio before proceeding to distribution or calculation steps.

Solve for the value of \(\frac{(a-b)^3}{(b-c)^3}\) given that a = 3c and the arithmetic mean of a and b is b.

Learn how to solve for \(\frac{(a-b)^3}{(b-c)^3}\) using proportionality and arithmetic mean formulas. This problem is suitable for students in Grades 9-10, covering key concepts like ratio, proportion, and cubic expressions.

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