Understanding the Problem
To solve the problem, we need to analyze the given ratio and its components:
- The ratio compounded of 4.5 and the sub-duplicate of \( a:9 \) is given as \( 8:15 \).
- This means we need to calculate the sub-duplicate of \( a:9 \).

Step 1: Define Sub-Duplicate
- The sub-duplicate of a ratio \( x:y \) is given by \( \sqrt{x}/\sqrt{y} \).
- For our case, \( a:9 \) leads to \( \sqrt{a}/\sqrt{9} = \sqrt{a}/3 \).

Step 2: Set Up the Equation
- The compounded ratio of 4.5 and \( \sqrt{a}/3 \) can be expressed as:
\[
\frac{4.5}{\sqrt{a}/3} = \frac{8}{15}
\]
- Rearranging gives:
\[
4.5 \cdot 15 = 8 \cdot \left(\frac{\sqrt{a}}{3}\right)
\]

Step 3: Simplify the Equation
- Calculate \( 4.5 \cdot 15 \):
\[
4.5 \cdot 15 = 67.5
\]
- Thus, we have:
\[
67.5 = \frac{8\sqrt{a}}{3}
\]
- Multiplying both sides by 3 results in:
\[
202.5 = 8\sqrt{a}
\]

Step 4: Solve for 'a'
- Dividing by 8 gives:
\[
\sqrt{a} = \frac{202.5}{8} = 25.3125
\]
- Squaring both sides yields:
\[
a = (25.3125)^2 = 642.640625
\]

Conclusion
- Therefore, the value of \( a \) is approximately \( 642.64 \).
This detailed breakdown clarifies the steps taken to derive the value of \( a \).