To find the difference between \( 3 \frac{1}{2} \) and \( 1 \frac{2}{3} \), we first convert these mixed numbers into improper fractions.
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Convert \( 3 \frac{1}{2} \): \[ 3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2} \]
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Convert \( 1 \frac{2}{3} \): \[ 1 \frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{3 + 2}{3} = \frac{5}{3} \]
Now, we need to find the difference: \[ \frac{7}{2} - \frac{5}{3} \]
To subtract these fractions, we need a common denominator. The least common multiple of 2 and 3 is 6.
Convert both fractions to have a denominator of 6:
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For \( \frac{7}{2} \): \[ \frac{7}{2} \times \frac{3}{3} = \frac{21}{6} \]
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For \( \frac{5}{3} \): \[ \frac{5}{3} \times \frac{2}{2} = \frac{10}{6} \]
Now we can subtract: \[ \frac{21}{6} - \frac{10}{6} = \frac{21 - 10}{6} = \frac{11}{6} \]
Finally, we can convert \( \frac{11}{6} \) back to a mixed number: \[ 11 \div 6 = 1 \text{ remainder } 5 \] So, \[ \frac{11}{6} = 1 \frac{5}{6} \]
Thus, the difference between \( 3 \frac{1}{2} \) and \( 1 \frac{2}{3} \) is: \[ \boxed{1 \frac{5}{6}} \]
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