To subtract the mixed numbers \( 18 \frac{9}{10} \) and \( 11 \frac{1}{6} \), follow these steps:
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Convert the mixed numbers to improper fractions.
For \( 18 \frac{9}{10} \): \[ 18 \times 10 + 9 = 180 + 9 = 189 \] So, \( 18 \frac{9}{10} = \frac{189}{10} \).
For \( 11 \frac{1}{6} \): \[ 11 \times 6 + 1 = 66 + 1 = 67 \] So, \( 11 \frac{1}{6} = \frac{67}{6} \).
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Now, we want to subtract \( \frac{67}{6} \) from \( \frac{189}{10} \). To do this, we need a common denominator. The least common multiple of 10 and 6 is 30.
Convert \( \frac{189}{10} \) to a denominator of 30: \[ \frac{189}{10} = \frac{189 \times 3}{10 \times 3} = \frac{567}{30} \]
Convert \( \frac{67}{6} \) to a denominator of 30: \[ \frac{67}{6} = \frac{67 \times 5}{6 \times 5} = \frac{335}{30} \]
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Perform the subtraction: \[ \frac{567}{30} - \frac{335}{30} = \frac{567 - 335}{30} = \frac{232}{30} \]
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Simplify \( \frac{232}{30} \). Both the numerator and the denominator can be divided by 2: \[ \frac{232 \div 2}{30 \div 2} = \frac{116}{15} \]
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Convert \( \frac{116}{15} \) back to a mixed number: \[ 116 \div 15 = 7 \quad \text{(with a remainder of 11)} \] So, \( \frac{116}{15} = 7 \frac{11}{15} \).
Putting it all together, the result of \( 18 \frac{9}{10} - 11 \frac{1}{6} \) is: \[ \boxed{7 \frac{11}{15}} \]