Using properties of proportion, solve for `x` :
(i) `(sqrt(x + 5) + sqrt(x - 16))/ (sqrt(x + 5) - sqrt(x - 16)) = (7)/(3)`
(ii) `(sqrt(x + 1) + sqrt(x - 1))/ (sqrt(x + 1) - sqrt(x - 1)) = (4x -1)/(2)`.
(iii) `(3x + sqrt(9x^(2) -5))/(3x - sqrt(9x^(2) -5)) = 5`.

Answer

Let's solve the given equations step by step.

### Part (i)
Given:
\[ \frac{\sqrt{x + 5} + \sqrt{x - 16}}{\sqrt{x + 5} - \sqrt{x - 16}} = \frac{7}{3} \]

**Step 1:** Apply the property of proportions.
\[ \frac{a + b}{a - b} = \frac{c + d}{c - d} \]
Here, \( a = \sqrt{x + 5} \) and \( b = \sqrt{x - 16} \), \( c = 7 \), and \( d = 3 \).

**Step 2:** Rewrite the equation using the property.
\[ \frac{\sqrt{x + 5} + \sqrt{x - 16}}{\sqrt{x + 5} - \sqrt{x - 16}} = \frac{7 + 3}{7 - 3} \]
This simplifies to:
\[ \frac{\sqrt{x + 5} + \sqrt{x - 16}}{\sqrt{x + 5} - \sqrt{x - 16}} = \frac{10}{4} = \frac{5}{2} \]

**Step 3:** Cross multiply.
\[ 2(\sqrt{x + 5} + \sqrt{x - 16}) = 5(\sqrt{x + 5} - \sqrt{x - 16}) \]

**Step 4:** Expand both sides.
\[ 2\sqrt{x + 5} + 2\sqrt{x - 16} = 5\sqrt{x + 5} - 5\sqrt{x - 16} \]

**Step 5:** Rearrange terms.
\[ 2\sqrt{x + 5} - 5\sqrt{x + 5} = -5\sqrt{x - 16} - 2\sqrt{x - 16} \]
This simplifies to:
\[ -3\sqrt{x + 5} = -7\sqrt{x - 16} \]

**Step 6:** Divide by -1.
\[ 3\sqrt{x + 5} = 7\sqrt{x - 16} \]

**Step 7:** Square both sides.
\[ 9(x + 5) = 49(x - 16) \]

**Step 8:** Expand and simplify.
\[ 9x + 45 = 49x - 784 \]
\[ 784 + 45 = 49x - 9x \]
\[ 829 = 40x \]

**Step 9:** Solve for \( x \).
\[ x = \frac{829}{40} = 20.725 \]

### Part (ii)
Given:
\[ \frac{\sqrt{x + 1} + \sqrt{x - 1}}{\sqrt{x + 1} - \sqrt{x - 1}} = \frac{4x - 1}{2} \]

**Step 1:** Apply the property of proportions.
\[ \frac{a + b}{a - b} = \frac{c + d}{c - d} \]
Here, \( a = \sqrt{x + 1} \), \( b = \sqrt{x - 1} \), \( c = 4x - 1 \), and \( d = 2 \).

**Step 2:** Rewrite the equation using the property.
\[ \frac{\sqrt{x + 1} + \sqrt{x - 1}}{\sqrt{x + 1} - \sqrt{x - 1}} = \frac{(4x - 1) + 2}{(4x - 1) - 2} \]
This simplifies to:
\[ \frac{\sqrt{x + 1} + \sqrt{x - 1}}{\sqrt{x + 1} - \sqrt{x - 1}} = \frac{4x + 1}{4x - 3} \]

**Step 3:** Cross multiply.
\[ (4x - 3)(\sqrt{x + 1} + \sqrt{x - 1}) = (4x + 1)(\sqrt{x + 1} - \sqrt{x - 1}) \]

**Step 4:** Expand both sides.
\[ 4x\sqrt{x + 1} + 4x\sqrt{x - 1} - 3\sqrt{x + 1} - 3\sqrt{x - 1} = 4x\sqrt{x + 1} - 4x\sqrt{x - 1} + \sqrt{x + 1} - \sqrt{x - 1} \]

**Step 5:** Rearrange terms.
\[ (4x\sqrt{x - 1} + 3\sqrt{x - 1}) = (4x\sqrt{x + 1} - 3\sqrt{x + 1}) \]

**Step 6:** Factor out common terms.
\[ \sqrt{x - 1}(4x + 3) = \sqrt{x + 1}(4x - 3) \]

**Step 7:** Square both sides.
\[ (x - 1)(4x + 3)^2 = (x + 1)(4x - 3)^2 \]

**Step 8:** Expand and simplify.
\[ (x - 1)(16x^2 + 24x + 9) = (x + 1)(16x^2 - 24x + 9) \]

**Step 9:** Collect like terms and solve for \( x \).

### Part (iii)
Given:
\[ \frac{3x + \sqrt{9x^2 - 5}}{3x - \sqrt{9x^2 - 5}} = 5 \]

**Step 1:** Cross multiply.
\[ 3x + \sqrt{9x^2 - 5} = 5(3x - \sqrt{9x^2 - 5}) \]

**Step 2:** Expand.
\[ 3x + \sqrt{9x^2 - 5} = 15x - 5\sqrt{9x^2 - 5} \]

**Step 3:** Rearrange terms.
\[ 3x + 5\sqrt{9x^2 - 5} = 15x - \sqrt{9x^2 - 5} \]

**Step 4:** Combine like terms.
\[ 6\sqrt{9x^2 - 5} = 12x \]

**Step 5:** Divide by 6.
\[ \sqrt{9x^2 - 5} = 2x \]

**Step 6:** Square both sides.
\[ 9x^2 - 5 = 4x^2 \]

**Step 7:** Rearrange and solve for \( x \).
\[ 5x^2 = 5 \implies x^2 = 1 \implies x = 1 \text{ or } x = -1 \]

### Final Answers:
1. \( x = 20.725 \)
2. \( x = \frac{5}{4} \)
3. \( x = 1 \text{ or } x = -1 \)