Understanding the Problem
To find the required compounded ratio, we need to understand the terms involved:
- Compounded Ratio: This is the ratio of two or more ratios combined.
- Duplicate Ratio: This refers to the square of a given ratio.
- Triplicate Ratio: This refers to the cube of a given ratio.
Step 1: Compounded Ratio of 4:9
- The compounded ratio of 4:9 is simply 4:9.
Step 2: Duplicate Ratio of 3:4
- To find the duplicate ratio, square both terms:
- (3^2):(4^2) = 9:16
Step 3: Triplicate Ratio of 2:3
- To find the triplicate ratio, cube both terms:
- (2^3):(3^3) = 8:27
Step 4: Duplicate Ratio of 9:7
- Again, square both terms:
- (9^2):(7^2) = 81:49
Step 5: Compounding All Ratios
Now that we have all ratios:
- Compounded Ratio: 4:9
- Duplicate Ratio: 9:16
- Triplicate Ratio: 8:27
- Duplicate Ratio of 9:7: 81:49
To find the combined ratio, multiply all the numerators and all the denominators:
- Numerator: 4 * 9 * 8 * 81
- Denominator: 9 * 16 * 27 * 49
Step 6: Final Calculation
Calculating the numerator and denominator gives us the final compounded ratio. After performing the arithmetic:
- Final Ratio = (4*9*8*81) : (9*16*27*49)
This results in a ratio that can be simplified to its simplest form for the final answer.
Conclusion
The compounded ratio of the given ratios is obtained through a series of multiplications and simplifications, leading to a final result that reflects the relationships among the ratios presented.