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To simplify the expression \(3 \frac{1}{1} \times 5 \frac{1}{1} \div \frac{4}{4}\), we can follow these steps:
1. Convert the mixed numbers to improper fractions:
- \(3 \frac{1}{1} = 3 + 1 = 4\)
- \(5 \frac{1}{1} = 5 + 1 = 6\)
2. Now the expression becomes:
\[
4 \times 6 \div \frac{4}{4}
\]
3. Since \(\frac{4}{4} = 1\), we can simplify:
\[
4 \times 6 \div 1 = 4 \times 6
\]
4. Now calculate \(4 \times 6\):
\[
4 \times 6 = 24
\]
So the simplified result of the expression \(3 \frac{1}{1} \times 5 \frac{1}{1} \div \frac{4}{4}\) is \(24\).