Find the value of `x`, if :
(i) `(2x + 3) : (5x - 38)` is the duplicate ratio of sqrt(5) : sqrt(6).
(ii) `(2x + 1) : (3x + 13)` is the sub-duplicate ratia of `9 : 25`.
(iii) `(3x - 7) : (4x + 3)` is the sub-triplicate ratio of `8 : 27`.

Answer

To solve the given problem, we will break it down into three parts as specified in the question. Let's go through each part step by step.

### Part (i)
We need to find the value of `x` such that:
\[ (2x + 3) : (5x - 38) \text{ is the duplicate ratio of } \sqrt{5} : \sqrt{6} \]

1. **Identify the duplicate ratio**: The duplicate ratio of \( \sqrt{a} : \sqrt{b} \) is given by \( a : b \). Therefore, the duplicate ratio of \( \sqrt{5} : \sqrt{6} \) is:
\[ 5 : 6 \]

2. **Set up the equation**: According to the problem, we can write:
\[ \frac{2x + 3}{5x - 38} = \frac{5}{6} \]

3. **Cross-multiply**:
\[ 6(2x + 3) = 5(5x - 38) \]

4. **Expand both sides**:
\[ 12x + 18 = 25x - 190 \]

5. **Rearrange the equation**:
\[ 12x - 25x = -190 - 18 \]
\[ -13x = -208 \]

6. **Solve for x**:
\[ x = \frac{208}{13} = 16 \]

### Part (ii)
Now, we need to find the value of `x` such that:
\[ (2x + 1) : (3x + 13) \text{ is the sub-duplicate ratio of } 9 : 25 \]

1. **Identify the sub-duplicate ratio**: The sub-duplicate ratio of \( a : b \) is given by \( \sqrt{a} : \sqrt{b} \). Therefore, the sub-duplicate ratio of \( 9 : 25 \) is:
\[ \sqrt{9} : \sqrt{25} = 3 : 5 \]

2. **Set up the equation**:
\[ \frac{2x + 1}{3x + 13} = \frac{3}{5} \]

3. **Cross-multiply**:
\[ 5(2x + 1) = 3(3x + 13) \]

4. **Expand both sides**:
\[ 10x + 5 = 9x + 39 \]

5. **Rearrange the equation**:
\[ 10x - 9x = 39 - 5 \]
\[ x = 34 \]

### Part (iii)
Finally, we need to find the value of `x` such that:
\[ (3x - 7) : (4x + 3) \text{ is the sub-triplicate ratio of } 8 : 27 \]

1. **Identify the sub-triplicate ratio**: The sub-triplicate ratio of \( a : b \) is given by \( \sqrt[3]{a} : \sqrt[3]{b} \). Therefore, the sub-triplicate ratio of \( 8 : 27 \) is:
\[ \sqrt[3]{8} : \sqrt[3]{27} = 2 : 3 \]

2. **Set up the equation**:
\[ \frac{3x - 7}{4x + 3} = \frac{2}{3} \]

3. **Cross-multiply**:
\[ 3(3x - 7) = 2(4x + 3) \]

4. **Expand both sides**:
\[ 9x - 21 = 8x + 6 \]

5. **Rearrange the equation**:
\[ 9x - 8x = 6 + 21 \]
\[ x = 27 \]

### Final Answers
- Part (i): \( x = 16 \)
- Part (ii): \( x = 34 \)
- Part (iii): \( x = 27 \)