To divide the fraction \( \frac{5}{6} \) by the mixed number \( 1 \frac{1}{2} \), we first need to convert the mixed number to an improper fraction.
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Convert \( 1 \frac{1}{2} \) into an improper fraction: \[ 1 \frac{1}{2} = \frac{2 \cdot 1 + 1}{2} = \frac{3}{2} \]
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Now we can rewrite the division: \[ \frac{5}{6} \div \frac{3}{2} \]
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Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{5}{6} \div \frac{3}{2} = \frac{5}{6} \times \frac{2}{3} \]
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Multiply the fractions: \[ \frac{5 \cdot 2}{6 \cdot 3} = \frac{10}{18} \]
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Now, simplify \( \frac{10}{18} \):
- The greatest common divisor (GCD) of 10 and 18 is 2. \[ \frac{10 \div 2}{18 \div 2} = \frac{5}{9} \]
Thus, the quotient \( \frac{5}{6} \div 1 \frac{1}{2} \) is \( \frac{5}{9} \).