Find duplicate ratio of:
(i) `3 : 4`
(ii) `3sqrt(3) : 2sqrt(5)`

Answer

To find the duplicate ratio of the given ratios, we will follow these steps:

### Step 1: Understanding Duplicate Ratio
The duplicate ratio of a ratio \( a : b \) is given by \( a^2 : b^2 \).

### Step 2: Finding Duplicate Ratio for \( 3 : 4 \)
1. Identify \( a = 3 \) and \( b = 4 \).
2. Calculate \( a^2 \):
\[ a^2 = 3^2 = 9 \]
3. Calculate \( b^2 \):
\[ b^2 = 4^2 = 16 \]
4. Therefore, the duplicate ratio of \( 3 : 4 \) is:
\[ 9 : 16 \]

### Step 3: Finding Duplicate Ratio for \( 3\sqrt{3} : 2\sqrt{5} \)
1. Identify \( a = 3\sqrt{3} \) and \( b = 2\sqrt{5} \).
2. Calculate \( a^2 \):
\[ a^2 = (3\sqrt{3})^2 = 3^2 \cdot (\sqrt{3})^2 = 9 \cdot 3 = 27 \]
3. Calculate \( b^2 \):
\[ b^2 = (2\sqrt{5})^2 = 2^2 \cdot (\sqrt{5})^2 = 4 \cdot 5 = 20 \]
4. Therefore, the duplicate ratio of \( 3\sqrt{3} : 2\sqrt{5} \) is:
\[ 27 : 20 \]

### Final Answers:
- The duplicate ratio of \( 3 : 4 \) is \( 9 : 16 \).
- The duplicate ratio of \( 3\sqrt{3} : 2\sqrt{5} \) is \( 27 : 20 \).

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