To perform the multiplication of the mixed number \(1 \frac{1}{6}\) and the fraction \(-\frac{5}{7}\), we first need to convert the mixed number into an improper fraction.
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Convert \(1 \frac{1}{6}\) to an improper fraction: \[ 1 \frac{1}{6} = \frac{6 \times 1 + 1}{6} = \frac{6 + 1}{6} = \frac{7}{6} \]
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Now multiply \(\frac{7}{6}\) by \(-\frac{5}{7}\): \[ \frac{7}{6} \times -\frac{5}{7} = \frac{7 \times -5}{6 \times 7} = \frac{-35}{42} \]
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Simplify \(\frac{-35}{42}\): The greatest common divisor (GCD) of 35 and 42 is 7. \[ \frac{-35 \div 7}{42 \div 7} = \frac{-5}{6} \]
So, the final answer is: \[ -\frac{5}{6} \]