Find the ratio compounded of the duplicate ratio of `5 : 6`, the reciprocal ratio of `25 : 42` and the sub-triplicate ratio of `216 : 343`.

Answer

To solve the problem, we need to find the compounded ratio of three different types of ratios: the duplicate ratio of \(5:6\), the reciprocal ratio of \(25:42\), and the sub-triplicate ratio of \(216:343\).

### Step-by-Step Solution:

1. **Find the Duplicate Ratio of \(5:6\)**:
The duplicate ratio is obtained by squaring both terms of the ratio.
\[ \text{Duplicate Ratio} = 5^2 : 6^2 = 25 : 36 \]

2. **Find the Reciprocal Ratio of \(25:42\)**:
The reciprocal ratio is obtained by swapping the two terms of the ratio.
\[ \text{Reciprocal Ratio} = 42 : 25 \]

3. **Find the Sub-Triplicate Ratio of \(216:343\)**:
The sub-triplicate ratio is obtained by taking the cube root of both terms of the ratio.
\[ \text{Sub-Triplicate Ratio} = 216^{1/3} : 343^{1/3} \]
We know that \(216 = 6^3\) and \(343 = 7^3\), so:
\[ 216^{1/3} = 6 \quad \text{and} \quad 343^{1/3} = 7 \]
Thus, the sub-triplicate ratio is:
\[ \text{Sub-Triplicate Ratio} = 6 : 7 \]

4. **Compounded Ratio**:
Now we need to find the compounded ratio of the three ratios we calculated:
\[ \text{Compounded Ratio} = \frac{25 \times 42 \times 6}{36 \times 25 \times 7} \]

5. **Simplifying the Compounded Ratio**:
Let's simplify the expression:
- The \(25\) in the numerator and denominator cancels out.
- The \(6\) in the numerator can be simplified with \(36\) (since \(36 = 6 \times 6\)):
\[ \frac{6}{36} = \frac{1}{6} \]
Now we have:
\[ \frac{1 \times 42}{6 \times 7} = \frac{42}{42} = 1 \]

### Final Answer:
The compounded ratio is:
\[ \text{Compounded Ratio} = 1 \]