To divide the mixed number \( 1 \frac{1}{6} \) by \( \frac{1}{2} \), we first convert the mixed number to an improper fraction.
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Convert \( 1 \frac{1}{6} \): \[ 1 \frac{1}{6} = \frac{6 \cdot 1 + 1}{6} = \frac{6 + 1}{6} = \frac{7}{6} \]
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Now we need to divide \( \frac{7}{6} \) by \( \frac{1}{2} \). When dividing fractions, we multiply by the reciprocal: \[ \frac{7}{6} \div \frac{1}{2} = \frac{7}{6} \times \frac{2}{1} = \frac{7 \cdot 2}{6 \cdot 1} = \frac{14}{6} \]
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Simplify \( \frac{14}{6} \): \[ \frac{14}{6} = \frac{7}{3} \]
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Now, convert \( \frac{7}{3} \) to a mixed number:
- Divide \( 7 \) by \( 3 \), which gives \( 2 \) with a remainder of \( 1 \).
- So, \( \frac{7}{3} = 2 \frac{1}{3} \).
The final answer is: \[ \boxed{2 \frac{1}{3}} \]