To evaluate the expression \( 9 \frac{3}{4} - 2 \frac{1}{5} + \frac{1}{2} \), we first convert all mixed numbers to improper fractions.
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Convert \( 9 \frac{3}{4} \) to an improper fraction: \[ 9 \frac{3}{4} = 9 \times 4 + 3 = 36 + 3 = \frac{39}{4} \]
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Convert \( 2 \frac{1}{5} \) to an improper fraction: \[ 2 \frac{1}{5} = 2 \times 5 + 1 = 10 + 1 = \frac{11}{5} \]
Now we rewrite the expression with the improper fractions: \[ \frac{39}{4} - \frac{11}{5} + \frac{1}{2} \]
Next, we need a common denominator to perform the operations. The denominators are 4, 5, and 2. The least common multiple (LCM) of these numbers is 20.
- Convert each fraction to have a denominator of 20:
- For \( \frac{39}{4} \): \[ \frac{39}{4} = \frac{39 \times 5}{4 \times 5} = \frac{195}{20} \]
- For \( \frac{11}{5} \): \[ \frac{11}{5} = \frac{11 \times 4}{5 \times 4} = \frac{44}{20} \]
- For \( \frac{1}{2} \): \[ \frac{1}{2} = \frac{1 \times 10}{2 \times 10} = \frac{10}{20} \]
Now we substitute these back into the expression: \[ \frac{195}{20} - \frac{44}{20} + \frac{10}{20} \]
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Combine the fractions: \[ \frac{195 - 44 + 10}{20} = \frac{195 - 44 = 151 + 10 = 161}{20} \] \[ = \frac{161}{20} \]
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Now convert \( \frac{161}{20} \) to a mixed number:
- Divide 161 by 20:
- \( 161 \div 20 = 8 \) remainder \( 1 \).
- Therefore, \( \frac{161}{20} = 8 \frac{1}{20} \).
- Divide 161 by 20:
Thus, the final answer in its simplest mixed number form is: \[ \boxed{8 \frac{1}{20}} \]
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