Understanding Ratio Compounders
To solve the problem, we need to understand the concept of ratio compounders, specifically for duplicate and triplicate ratios.

Step 1: Duplicate Ratio of 4:9
- The duplicate ratio is obtained by squaring the terms of the original ratio.
- \( (4^2):(9^2) = 16:81 \)

Step 2: Triplicate Ratio of 3:4
- The triplicate ratio involves cubing the terms of the original ratio.
- \( (3^3):(4^3) = 27:64 \)

Step 3: Triplicate Ratio of 2:3
- Similarly, for the ratio 2:3, we cube the terms.
- \( (2^3):(3^3) = 8:27 \)

Step 4: Triplicate Ratio of 9:7
- For the ratio 9:7, we also cube the terms.
- \( (9^3):(7^3) = 729:343 \)

Step 5: Compounding the Ratios
Now, we need to compound the ratios we calculated:
- Duplicate ratio \( 16:81 \)
- Triplicate ratio \( 27:64 \)
- Triplicate ratio \( 8:27 \)
- Triplicate ratio \( 729:343 \)

Step 6: Final Calculation
To find the compounded ratio, we multiply the corresponding terms:
- Numerator: \( 16 \times 27 \times 8 \times 729 \)
- Denominator: \( 81 \times 64 \times 27 \times 343 \)

Conclusion
The final compounded ratio can be calculated, and it will represent the overall ratio by combining all the ratios through multiplication. The exact numerical values can be computed for precision, depending on the requirement.
This systematic approach allows for clarity in understanding how to compound various ratios effectively for UPSC and other competitive exams.