To solve \( 8 \frac{1}{2} - 5 \frac{2}{3} \), we first convert the mixed numbers to improper fractions.
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Convert \( 8 \frac{1}{2} \): \[ 8 \frac{1}{2} = \frac{8 \times 2 + 1}{2} = \frac{16 + 1}{2} = \frac{17}{2} \]
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Convert \( 5 \frac{2}{3} \): \[ 5 \frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3} \]
Now, we can rewrite the expression: \[ 8 \frac{1}{2} - 5 \frac{2}{3} = \frac{17}{2} - \frac{17}{3} \]
Next, we find a common denominator for the fractions. The least common multiple of 2 and 3 is 6.
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Convert \( \frac{17}{2} \) to have a denominator of 6: \[ \frac{17}{2} = \frac{17 \times 3}{2 \times 3} = \frac{51}{6} \]
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Convert \( \frac{17}{3} \) to have a denominator of 6: \[ \frac{17}{3} = \frac{17 \times 2}{3 \times 2} = \frac{34}{6} \]
Now, we can subtract the two fractions: \[ \frac{51}{6} - \frac{34}{6} = \frac{51 - 34}{6} = \frac{17}{6} \]
Finally, we can convert \( \frac{17}{6} \) back to a mixed number: \[ \frac{17}{6} = 2 \frac{5}{6} \]
Thus, the result of \( 8 \frac{1}{2} - 5 \frac{2}{3} \) is: \[ \boxed{2 \frac{5}{6}} \]