Fill in `lt` or `gt` sign in the boxes without actually carrying out the multiplication.
a. i. `(3)/(11) xx (2)/(9) square (2)/(9)" "ii. (3)/(11) xx (2)/(9) square (3)/(11)`

Answer

To solve the problem of comparing the fractions without actually carrying out the multiplication, we can use the property of fractions that states if \( a/b \) and \( c/d \) are two fractions, we can compare them by cross-multiplying.

Let's break down the question step by step.

### Step 1: Analyze the first part of the question
We need to compare:
\[ \frac{3}{11} \times \frac{2}{9} \quad \text{and} \quad \left(\frac{2}{9}\right)^2 \]

### Step 2: Calculate the square of the second fraction
Calculating \(\left(\frac{2}{9}\right)^2\):
\[ \left(\frac{2}{9}\right)^2 = \frac{2 \times 2}{9 \times 9} = \frac{4}{81} \]

### Step 3: Calculate the product of the first fraction
Now calculate \(\frac{3}{11} \times \frac{2}{9}\):
\[ \frac{3}{11} \times \frac{2}{9} = \frac{3 \times 2}{11 \times 9} = \frac{6}{99} \]
We can simplify \(\frac{6}{99}\):
\[ \frac{6}{99} = \frac{2}{33} \]

### Step 4: Compare \(\frac{2}{33}\) and \(\frac{4}{81}\)
Now we need to compare \(\frac{2}{33}\) and \(\frac{4}{81}\). We can cross-multiply:
\[ 2 \times 81 \quad \text{and} \quad 4 \times 33 \]
Calculating both sides:
\[ 2 \times 81 = 162 \]
\[ 4 \times 33 = 132 \]

### Step 5: Determine the relationship
Since \(162 > 132\), we have:
\[ \frac{2}{33} > \frac{4}{81} \]

### Conclusion for part (i)
Thus, we can fill in the sign for part (i):
\[ \frac{3}{11} \times \frac{2}{9} \quad \text{gt} \quad \left(\frac{2}{9}\right)^2 \]

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### Step 6: Analyze the second part of the question
Now we need to compare:
\[ \frac{3}{11} \times \frac{2}{9} \quad \text{and} \quad \frac{3}{11} \]

### Step 7: We already calculated \(\frac{3}{11} \times \frac{2}{9}\)
We know from earlier that:
\[ \frac{3}{11} \times \frac{2}{9} = \frac{2}{33} \]

### Step 8: Compare \(\frac{2}{33}\) and \(\frac{3}{11}\)
Now we need to compare \(\frac{2}{33}\) and \(\frac{3}{11}\). We can cross-multiply:
\[ 2 \times 11 \quad \text{and} \quad 3 \times 33 \]
Calculating both sides:
\[ 2 \times 11 = 22 \]
\[ 3 \times 33 = 99 \]

### Step 9: Determine the relationship
Since \(22 < 99\), we have:
\[ \frac{2}{33} < \frac{3}{11} \]

### Conclusion for part (ii)
Thus, we can fill in the sign for part (ii):
\[ \frac{3}{11} \times \frac{2}{9} \quad \text{lt} \quad \frac{3}{11} \]

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### Final Answers:
1. \( \frac{3}{11} \times \frac{2}{9} \quad \text{gt} \quad \left(\frac{2}{9}\right)^2 \)
2. \( \frac{3}{11} \times \frac{2}{9} \quad \text{lt} \quad \frac{3}{11} \)

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