Understanding Ratios and Their Types
Ratios are a way to compare quantities, and they can be expressed in different forms such as simple, duplicate, and triplicate ratios.

1. Simple Ratio
- The simple ratio of 4:9 compares two quantities directly.
- This means for every 4 parts of the first quantity, there are 9 parts of the second.

2. Duplicate Ratio
- The duplicate ratio of 3:4 is obtained by squaring both terms.
- Calculation:
- (3^2):(4^2) = 9:16

3. Triplicate Ratio
- The triplicate ratio of 2:3 involves cubing both terms.
- Calculation:
- (2^3):(3^3) = 8:27

4. Additional Triplicate Ratio
- For the ratio 9:7, we also find the triplicate ratio.
- Calculation:
- (9^3):(7^3) = 729:343

5. Compounding Ratios
- To find the compound ratio, we multiply the corresponding terms of the ratios.

Final Calculation
- We have the following ratios to compound:
- Simple (4:9), Duplicate (9:16), Triplicate (8:27), Triplicate (729:343).
- Compounding:
- Numerator = 4 * 9 * 8 * 729 = 4 * 9 = 36; 36 * 8 = 288; 288 * 729 = 209088
- Denominator = 9 * 16 * 27 * 343 = 9 * 16 = 144; 144 * 27 = 3888; 3888 * 343 = 1336336

Final Ratio
- Therefore, the compound ratio is 209088:1336336.
- This can be simplified if necessary.
In conclusion, by understanding how to calculate and compound different types of ratios, we can accurately represent and compare complex relationships between quantities.