Find the ratio compounded of the reciprocal ratio of `15 : 28`, the sub-duplicate ratio of `36 : 49` and the trilicate ratio of `5 : 4`.

Answer

To find the compounded ratio of the reciprocal ratio of `15 : 28`, the sub-duplicate ratio of `36 : 49`, and the trilicate ratio of `5 : 4`, we will follow these steps:

### Step 1: Find the Reciprocal Ratio of `15 : 28`
The reciprocal ratio of `15 : 28` is obtained by swapping the two numbers:
\[ \text{Reciprocal Ratio} = 28 : 15 \]

### Step 2: Find the Sub-Duplicate Ratio of `36 : 49`
The sub-duplicate ratio is found by taking the square root of both numbers:
\[ \text{Sub-Duplicate Ratio} = \sqrt{36} : \sqrt{49} = 6 : 7 \]

### Step 3: Find the Trilicate Ratio of `5 : 4`
The trilicate ratio is found by cubing both numbers:
\[ \text{Trilicate Ratio} = 5^3 : 4^3 = 125 : 64 \]

### Step 4: Compounding the Ratios
Now we need to compound the ratios we have found:
\[ \text{Compounded Ratio} = \frac{(28 \times 6 \times 125)}{(15 \times 7 \times 64)} \]

### Step 5: Calculate the Numerator
Calculating the numerator:
\[ 28 \times 6 = 168 \]
\[ 168 \times 125 = 21000 \]

### Step 6: Calculate the Denominator
Calculating the denominator:
\[ 15 \times 7 = 105 \]
\[ 105 \times 64 = 6720 \]

### Step 7: Form the Ratio
Now we can form the ratio:
\[ \text{Compounded Ratio} = \frac{21000}{6720} \]

### Step 8: Simplifying the Ratio
To simplify \( \frac{21000}{6720} \):
1. Find the GCD of 21000 and 6720.
2. Divide both the numerator and the denominator by their GCD.

Calculating the GCD:
- \( 21000 = 2^2 \times 3 \times 5^3 \times 7 \)
- \( 6720 = 2^4 \times 3 \times 5 \times 7 \)

The GCD is \( 2^2 \times 3 \times 5 \times 7 = 420 \).

Now, divide both by 420:
\[ \frac{21000 \div 420}{6720 \div 420} = \frac{50}{16} \]

Further simplifying \( \frac{50}{16} \):
\[ \frac{50 \div 2}{16 \div 2} = \frac{25}{8} \]

### Final Answer
The required compounded ratio is:
\[ \text{Compounded Ratio} = 25 : 8 \]

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