Find the :
(i) duplicate ratio of `2sqrt(2) : 3sqrt(5)`
(ii) triplicate ratio of `2a : 3b`,
(iii) sub-duplicate ratio of `9x^(2)a^(4) : 25y^(6)b^(2)`
(iv) sub-triplicate ratio of `216 : 343`
(v) reciprocal ratio of `3: 5`
(vi) ratio compounded of the duplicate ratio of `5 : 6`, the reciprocal ratio of `25 : 42` and the sub-duplicate ratio of `36 : 49`.

Answer

Let's solve the given problems step by step.

### (i) Duplicate Ratio of `2√2 : 3√5`

1. **Express the ratio**: Let \( a = 2\sqrt{2} \) and \( b = 3\sqrt{5} \).
2. **Find the duplicate ratio**: The duplicate ratio is given by \( \frac{a^2}{b^2} \).
3. **Calculate \( a^2 \)**:
\[ a^2 = (2\sqrt{2})^2 = 4 \cdot 2 = 8 \]
4. **Calculate \( b^2 \)**:
\[ b^2 = (3\sqrt{5})^2 = 9 \cdot 5 = 45 \]
5. **Form the ratio**:
\[ \text{Duplicate Ratio} = \frac{a^2}{b^2} = \frac{8}{45} \]

### (ii) Triplicate Ratio of `2a : 3b`

1. **Express the ratio**: Let \( x = 2a \) and \( y = 3b \).
2. **Find the triplicate ratio**: The triplicate ratio is given by \( \frac{x^3}{y^3} \).
3. **Calculate \( x^3 \)**:
\[ x^3 = (2a)^3 = 8a^3 \]
4. **Calculate \( y^3 \)**:
\[ y^3 = (3b)^3 = 27b^3 \]
5. **Form the ratio**:
\[ \text{Triplicate Ratio} = \frac{x^3}{y^3} = \frac{8a^3}{27b^3} \]

### (iii) Sub-Duplicate Ratio of `9x^2a^4 : 25y^6b^2`

1. **Express the ratio**: Let \( p = 9x^2a^4 \) and \( q = 25y^6b^2 \).
2. **Find the sub-duplicate ratio**: The sub-duplicate ratio is given by \( \sqrt{\frac{p}{q}} \).
3. **Calculate \( p \)**:
\[ p = 9x^2a^4 = 3^2 \cdot x^2 \cdot a^4 \]
4. **Calculate \( q \)**:
\[ q = 25y^6b^2 = 5^2 \cdot y^6 \cdot b^2 \]
5. **Form the ratio**:
\[ \text{Sub-Duplicate Ratio} = \sqrt{\frac{p}{q}} = \sqrt{\frac{9x^2a^4}{25y^6b^2}} = \frac{3xa^2}{5y^3b} \]

### (iv) Sub-Triplicate Ratio of `216 : 343`

1. **Express the ratio**: Let \( a = 216 \) and \( b = 343 \).
2. **Find the sub-triplicate ratio**: The sub-triplicate ratio is given by \( \sqrt[3]{\frac{a}{b}} \).
3. **Calculate \( a^{1/3} \)**:
\[ a^{1/3} = 6 \quad (\text{since } 6^3 = 216) \]
4. **Calculate \( b^{1/3} \)**:
\[ b^{1/3} = 7 \quad (\text{since } 7^3 = 343) \]
5. **Form the ratio**:
\[ \text{Sub-Triplicate Ratio} = \frac{6}{7} \]

### (v) Reciprocal Ratio of `3 : 5`

1. **Express the ratio**: Let \( a = 3 \) and \( b = 5 \).
2. **Find the reciprocal ratio**: The reciprocal ratio is given by \( \frac{b}{a} \).
3. **Form the ratio**:
\[ \text{Reciprocal Ratio} = \frac{5}{3} \]

### (vi) Compounded Ratio

1. **Find the duplicate ratio of `5 : 6`**:
\[ \text{Duplicate Ratio} = \left(\frac{5}{6}\right)^2 = \frac{25}{36} \]
2. **Find the reciprocal ratio of `25 : 42`**:
\[ \text{Reciprocal Ratio} = \frac{42}{25} \]
3. **Find the sub-duplicate ratio of `36 : 49`**:
\[ \text{Sub-Duplicate Ratio} = \sqrt{\frac{36}{49}} = \frac{6}{7} \]
4. **Combine the ratios**:
\[ \text{Compounded Ratio} = \frac{25}{36} \cdot \frac{42}{25} \cdot \frac{6}{7} \]
5. **Simplify**:
\[ = \frac{42 \cdot 6}{36 \cdot 7} = \frac{252}{252} = 1 \]

### Final Answers:
1. (i) Duplicate Ratio: \( \frac{8}{45} \)
2. (ii) Triplicate Ratio: \( \frac{8a^3}{27b^3} \)
3. (iii) Sub-Duplicate Ratio: \( \frac{3xa^2}{5y^3b} \)
4. (iv) Sub-Triplicate Ratio: \( \frac{6}{7} \)
5. (v) Reciprocal Ratio: \( \frac{5}{3} \)
6. (vi) Compounded Ratio: \( 1 \)