To make it easiest to add the expression \( 18 - 9 \frac{4}{9} + \frac{7}{9} - 8 \), we can first convert the mixed number and the whole numbers into fractions, and then group similar terms.
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Convert \( 18 \) and \( 8 \) into fractions with a common denominator of \( 9 \): \[ 18 = \frac{18 \times 9}{9} = \frac{162}{9}, \quad 8 = \frac{8 \times 9}{9} = \frac{72}{9} \]
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Convert \( 9 \frac{4}{9} \) into fractions: \[ 9 \frac{4}{9} = \frac{9 \times 9 + 4}{9} = \frac{81 + 4}{9} = \frac{85}{9} \]
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Rewrite the original expression with these fractions: \[ \frac{162}{9} - \frac{85}{9} + \frac{7}{9} - \frac{72}{9} \]
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Group the terms: \[ \left( \frac{162}{9} - \frac{85}{9} - \frac{72}{9} \right) + \frac{7}{9} \]
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Simplify within the parentheses: \[ \frac{162 - 85 - 72}{9} = \frac{162 - 157}{9} = \frac{5}{9} \]
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Now add \( \frac{7}{9} \): \[ \frac{5}{9} + \frac{7}{9} = \frac{12}{9} = \frac{4}{3} \]
Therefore, the original expression simplifies to \( \frac{4}{3} \) after grouping and adding the fractions.
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