Find the compound ratio of :
(i) `2 : 3`, `9 : 14` and `14 : 27`.
(ii) `2a : 3b`, `mm : x^(2)` and `x : n`.
(iii) `sqrt(2) : 1`, `3 : sqrt(5)` and `sqrt(20): 9`.

Answer

To find the compound ratio of the given sets of ratios, we will follow the steps outlined in the video transcript. Let's solve each part step by step.

### (i) Find the compound ratio of `2 : 3`, `9 : 14`, and `14 : 27`.

1. **Identify the ratios**:
- First ratio: \(2 : 3\)
- Second ratio: \(9 : 14\)
- Third ratio: \(14 : 27\)

2. **Write the compound ratio formula**:
\[ \text{Compound Ratio} = \frac{(2 \times 9 \times 14)}{(3 \times 14 \times 27)} \]

3. **Calculate the numerator**:
\[ 2 \times 9 \times 14 = 252 \]

4. **Calculate the denominator**:
\[ 3 \times 14 \times 27 = 1134 \]

5. **Form the ratio**:
\[ \text{Compound Ratio} = \frac{252}{1134} \]

6. **Simplify the ratio**:
- Divide both numerator and denominator by their GCD (which is 126):
\[ \frac{252 \div 126}{1134 \div 126} = \frac{2}{9} \]

7. **Final answer**:
\[ \text{Compound Ratio} = 2 : 9 \]

### (ii) Find the compound ratio of `2a : 3b`, `m : x^2`, and `x : n`.

1. **Identify the ratios**:
- First ratio: \(2a : 3b\)
- Second ratio: \(m : x^2\)
- Third ratio: \(x : n\)

2. **Write the compound ratio formula**:
\[ \text{Compound Ratio} = \frac{(2a \times m \times x)}{(3b \times x^2 \times m)} \]

3. **Cancel common terms**:
- \(m\) cancels out:
\[ \text{Compound Ratio} = \frac{(2a \times x)}{(3b \times x^2)} \]

4. **Simplify the ratio**:
- Cancel \(x\) from numerator and denominator:
\[ \text{Compound Ratio} = \frac{2a}{3b \times x} \]

5. **Final answer**:
\[ \text{Compound Ratio} = 2a : 3bx \]

### (iii) Find the compound ratio of `sqrt(2) : 1`, `3 : sqrt(5)`, and `sqrt(20) : 9`.

1. **Identify the ratios**:
- First ratio: \(\sqrt{2} : 1\)
- Second ratio: \(3 : \sqrt{5}\)
- Third ratio: \(\sqrt{20} : 9\)

2. **Write the compound ratio formula**:
\[ \text{Compound Ratio} = \frac{(\sqrt{2} \times 3 \times \sqrt{20})}{(1 \times \sqrt{5} \times 9)} \]

3. **Calculate the numerator**:
- Simplify \(\sqrt{20}\) as \(\sqrt{4 \times 5} = 2\sqrt{5}\):
\[ \text{Numerator} = \sqrt{2} \times 3 \times 2\sqrt{5} = 6\sqrt{10} \]

4. **Calculate the denominator**:
\[ \text{Denominator} = 1 \times \sqrt{5} \times 9 = 9\sqrt{5} \]

5. **Form the ratio**:
\[ \text{Compound Ratio} = \frac{6\sqrt{10}}{9\sqrt{5}} \]

6. **Simplify the ratio**:
- Divide both numerator and denominator by 3:
\[ \frac{2\sqrt{10}}{3\sqrt{5}} \]
- Further simplify \(\sqrt{10} = \sqrt{5 \times 2} = \sqrt{5}\sqrt{2}\):
\[ \text{Compound Ratio} = \frac{2\sqrt{2}}{3} \]

7. **Final answer**:
\[ \text{Compound Ratio} = 2\sqrt{2} : 3 \]