To divide the mixed numbers \( 5 \frac{4}{9} \) and \( 3 \frac{1}{2} \), we first convert both mixed numbers to improper fractions.
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Convert \( 5 \frac{4}{9} \) to an improper fraction: \[ 5 \frac{4}{9} = \left(5 \times 9 + 4\right) / 9 = \left(45 + 4\right) / 9 = \frac{49}{9} \]
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Convert \( 3 \frac{1}{2} \) to an improper fraction: \[ 3 \frac{1}{2} = \left(3 \times 2 + 1\right) / 2 = \left(6 + 1\right) / 2 = \frac{7}{2} \]
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Now, divide the improper fractions: \[ \frac{49}{9} \div \frac{7}{2} \]
To divide by a fraction, multiply by its reciprocal: \[ \frac{49}{9} \times \frac{2}{7} \]
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Multiply the fractions: \[ \frac{49 \times 2}{9 \times 7} = \frac{98}{63} \]
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Simplify \( \frac{98}{63} \): Both the numerator and denominator can be divided by 7: \[ \frac{98 \div 7}{63 \div 7} = \frac{14}{9} \]
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Convert \( \frac{14}{9} \) back to a mixed number: \[ 14 \div 9 = 1 \quad \text{(remainder 5)} \] So, \( \frac{14}{9} = 1 \frac{5}{9} \).
Therefore, the result of \( 5 \frac{4}{9} \div 3 \frac{1}{2} \) is: \[ \boxed{1 \frac{5}{9}} \]