To simplify the expression \(18 - 9 \frac{4}{9} + \frac{7}{9} - 8\), we will proceed step-by-step.
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First, convert \(9 \frac{4}{9}\) to an improper fraction: \[ 9 \frac{4}{9} = \frac{9 \times 9 + 4}{9} = \frac{81 + 4}{9} = \frac{85}{9} \]
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Substitute this improper fraction back into the expression: \[ 18 - \frac{85}{9} + \frac{7}{9} - 8 \]
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Next, convert the whole numbers \(18\) and \(8\) into fractions with a denominator of \(9\): \[ 18 = \frac{18 \times 9}{9} = \frac{162}{9} \] \[ 8 = \frac{8 \times 9}{9} = \frac{72}{9} \]
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Now replace the whole numbers in the expression: \[ \frac{162}{9} - \frac{85}{9} + \frac{7}{9} - \frac{72}{9} \]
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Combine all these fractions: \[ \frac{162 - 85 + 7 - 72}{9} \]
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Calculate the numerator: \[ 162 - 85 = 77 \] \[ 77 + 7 = 84 \] \[ 84 - 72 = 12 \]
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Now, substitute the result back into the fraction: \[ \frac{12}{9} \]
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Finally, simplify \(\frac{12}{9}\) by dividing both the numerator and the denominator by their greatest common divisor, which is \(3\): \[ \frac{12 \div 3}{9 \div 3} = \frac{4}{3} \]
Thus, the simplified expression is: \[ \frac{4}{3} \]
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