Given Ratios:
- Compound Ratio: 4:5
- Sub-duplicate Ratio: a:9

Compound Ratio:
- The compound ratio is a ratio of two or more ratios.
- In this case, the compound ratio of 4:5 means that for every 4 parts of the first quantity, there are 5 parts of the second quantity.

Sub-duplicate Ratio:
- The sub-duplicate ratio is the square root of a given ratio.
- For the sub-duplicate ratio of a:9, the square root of a is the first part, and 9 is the second part.

Given Compound Ratio and Sub-duplicate Ratio:
- The given compound ratio of 4:5 and sub-duplicate ratio of a:9 result in a final ratio of 8:15.

Calculating the Value of 'a':
- To find the value of 'a', we need to compare the compound ratio with the sub-duplicate ratio.
- We can set up a proportion to solve for 'a':
- (4/5) / (sqrt(a)/9) = 8/15
- Cross multiply to get: 15 * 4 = 8 * 5 * (sqrt(a))
- 60 = 40(sqrt(a))
- Divide by 40: sqrt(a) = 60/40
- Square both sides: a = (60/40)^2 = 1.5^2 = 2.25

Value of 'a':
- The value of 'a' is 2.25.
By following the steps above and setting up a proportion between the compound ratio and the sub-duplicate ratio, we can calculate the value of 'a' as 2.25. This method helps in solving ratio problems efficiently and accurately.