To subtract \( 9 \frac{7}{10} - \frac{6}{14} \), we first need to convert the mixed number to an improper fraction and then find a common denominator for the fractions.
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Convert \( 9 \frac{7}{10} \) to an improper fraction: \[ 9 \frac{7}{10} = \frac{9 \times 10 + 7}{10} = \frac{90 + 7}{10} = \frac{97}{10} \]
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Simplify \( \frac{6}{14} \): \[ \frac{6}{14} = \frac{3}{7} \]
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Find a common denominator for \( \frac{97}{10} \) and \( \frac{3}{7} \). The least common multiple of 10 and 7 is 70.
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Convert both fractions to have the common denominator of 70: \[ \frac{97}{10} = \frac{97 \times 7}{10 \times 7} = \frac{679}{70} \] \[ \frac{3}{7} = \frac{3 \times 10}{7 \times 10} = \frac{30}{70} \]
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Now subtract the two fractions: \[ \frac{679}{70} - \frac{30}{70} = \frac{679 - 30}{70} = \frac{649}{70} \]
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The final difference is: \[ \frac{649}{70} \]
To convert \( \frac{649}{70} \) into a mixed number:
- Perform the division: \( 649 \div 70 = 9 \) with a remainder of 19.
So, \( \frac{649}{70} = 9 \frac{19}{70} \).
Thus, the final answer is: \[ \boxed{9 \frac{19}{70}} \]