If a is inversely proportional to 1/b and bis inversely proportional to c square then find the value of 9a + 5c, when b = 27. Given that when b = 3 then a = 4 and c = 6.

We are given two inverse proportionalities:

1. \( a \) is inversely proportional to \( \frac{1}{b} \), meaning \( a \propto \frac{1}{1/b} \), which implies \( a \propto b \). So, we can express this as:
   \[
   a = k_1 b
   \]
   where \( k_1 \) is a constant.

2. \( b \) is inversely proportional to \( c^2 \), meaning \( b \propto \frac{1}{c^2} \). So, we can express this as:
   \[
   b = \frac{k_2}{c^2}
   \]
   where \( k_2 \) is another constant.

### Step 1: Find the values of \( k_1 \) and \( k_2 \)

We are given the following conditions:
- When \( b = 3 \), \( a = 4 \) and \( c = 6 \).

#### From \( a = k_1 b \):
Using \( a = 4 \) and \( b = 3 \):
\[
4 = k_1 \times 3 \quad \Rightarrow \quad k_1 = \frac{4}{3}
\]
Thus, \( a = \frac{4}{3} b \).

#### From \( b = \frac{k_2}{c^2} \):
Using \( b = 3 \) and \( c = 6 \):
\[
3 = \frac{k_2}{6^2} \quad \Rightarrow \quad k_2 = 3 \times 36 = 108
\]
Thus, \( b = \frac{108}{c^2} \).

### Step 2: Find the value of \( a \) and \( c \) when \( b = 27 \)

Using the equation \( b = \frac{108}{c^2} \), substitute \( b = 27 \):
\[
27 = \frac{108}{c^2} \quad \Rightarrow \quad c^2 = \frac{108}{27} = 4 \quad \Rightarrow \quad c = 2
\]

Now, use the equation \( a = \frac{4}{3} b \) and substitute \( b = 27 \):
\[
a = \frac{4}{3} \times 27 = 36
\]

### Step 3: Find the value of \( 9a + 5c \)

Now that we have \( a = 36 \) and \( c = 2 \), calculate \( 9a + 5c \):
\[
9a + 5c = 9 \times 36 + 5 \times 2 = 324 + 10 = 334
\]

Thus, the value of \( 9a + 5c \) is \( \boxed{334} \).

---

Would you like more details on any of these steps, or do you have further questions?

Here are 5 related questions:
1. How do we derive the relationship for inverse proportionality?
2. What happens if both quantities are inversely proportional to each other?
3. How would the solution change if \( b \) was proportional to \( c^3 \)?
4. Can we have multiple constants of proportionality in a single problem?
5. How does inverse proportionality differ from direct proportionality?

**Tip:** In inverse proportionality, as one quantity increases, the other decreases at a rate dictated by the constant of proportionality.

If a is inversely proportional to 1/b and b is inversely proportional to c^2, then find the value of 9a + 5c, when b = 27. Given that when b = 3 then a = 4 and c = 6.

Learn how to solve a problem involving inverse proportionality where a is inversely proportional to 1/b and b is inversely proportional to c^2. Detailed step-by-step solution provided to find the value of 9a + 5c when b = 27, using the given conditions that a = 4 and c = 6 when b = 3. Suitable for students in Grades 9-12.

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