If `(x-9):(3x+6)` is the duplicate ratio of `4:9`, find the value of x using properties of proportion.

Answer

To solve the problem, we need to find the value of \( x \) given that the ratio \( (x-9):(3x+6) \) is the duplicate ratio of \( 4:9 \).

### Step-by-step Solution:

1. **Understanding Duplicate Ratio**:
The duplicate ratio of \( a:b \) is given by \( \left(\frac{a}{b}\right)^2 \). Therefore, the duplicate ratio of \( 4:9 \) is:
\[ \left(\frac{4}{9}\right)^2 = \frac{16}{81} \]

2. **Setting Up the Equation**:
According to the problem, we have:
\[ \frac{x-9}{3x+6} = \frac{16}{81} \]

3. **Cross-Multiplying**:
We can cross-multiply to eliminate the fractions:
\[ 81(x - 9) = 16(3x + 6) \]

4. **Expanding Both Sides**:
Expanding both sides gives:
\[ 81x - 729 = 48x + 96 \]

5. **Rearranging the Equation**:
Now, we will move all terms involving \( x \) to one side and constant terms to the other side:
\[ 81x - 48x = 729 + 96 \]
Simplifying this gives:
\[ 33x = 825 \]

6. **Solving for \( x \)**:
Now, divide both sides by 33:
\[ x = \frac{825}{33} = 25 \]

### Final Answer:
Thus, the value of \( x \) is \( 25 \).

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