To subtract the mixed numbers \( 12 \frac{8}{9} \) and \( 10 \frac{1}{6} \), we first convert them to improper fractions.
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Convert \( 12 \frac{8}{9} \): \[ 12 \times 9 + 8 = 108 + 8 = 116 \] So, \( 12 \frac{8}{9} = \frac{116}{9} \).
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Convert \( 10 \frac{1}{6} \): \[ 10 \times 6 + 1 = 60 + 1 = 61 \] So, \( 10 \frac{1}{6} = \frac{61}{6} \).
Next, we need a common denominator to subtract the fractions. The least common multiple (LCM) of 9 and 6 is 18.
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Convert \(\frac{116}{9}\) to have a denominator of 18: \[ \frac{116}{9} \times \frac{2}{2} = \frac{232}{18} \]
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Convert \(\frac{61}{6}\) to have a denominator of 18: \[ \frac{61}{6} \times \frac{3}{3} = \frac{183}{18} \]
Now we can subtract the two fractions: \[ \frac{232}{18} - \frac{183}{18} = \frac{232 - 183}{18} = \frac{49}{18} \]
Finally, convert \(\frac{49}{18}\) back to a mixed number:
- \( 49 \div 18 = 2 \) remainder \( 13 \), so: \[ \frac{49}{18} = 2 \frac{13}{18} \]
Thus, the result of \( 12 \frac{8}{9} - 10 \frac{1}{6} \) is: \[ 2 \frac{13}{18} \]