If `(3x - 9) : (5x + 4)` is the triplicate ratio of `3 : 4`, find the value of `x`.

Answer

To solve the problem, we need to find the value of \( x \) such that the ratio \( (3x - 9) : (5x + 4) \) is the triplicate ratio of \( 3 : 4 \).

### Step-by-Step Solution:

1. **Understand the triplicate ratio**: The triplicate ratio of \( a : b \) is given by \( a^3 : b^3 \). Therefore, the triplicate ratio of \( 3 : 4 \) is \( 3^3 : 4^3 \).

\[ 3^3 = 27 \quad \text{and} \quad 4^3 = 64 \]

Thus, the triplicate ratio of \( 3 : 4 \) is \( 27 : 64 \).

2. **Set up the equation**: We can now set up the equation based on the given ratio:

\[ \frac{3x - 9}{5x + 4} = \frac{27}{64} \]

3. **Cross-multiply**: To eliminate the fraction, we cross-multiply:

\[ 64(3x - 9) = 27(5x + 4) \]

4. **Distribute**: Now, distribute both sides:

\[ 64 \cdot 3x - 64 \cdot 9 = 27 \cdot 5x + 27 \cdot 4 \]

This simplifies to:

\[ 192x - 576 = 135x + 108 \]

5. **Rearrange the equation**: Move all terms involving \( x \) to one side and constant terms to the other side:

\[ 192x - 135x = 108 + 576 \]

This simplifies to:

\[ 57x = 684 \]

6. **Solve for \( x \)**: Divide both sides by 57 to find \( x \):

\[ x = \frac{684}{57} \]

Calculating this gives:

\[ x = 12 \]

### Final Answer:
The value of \( x \) is \( 12 \).

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