To add the mixed numbers \(1 \frac{1}{2}\) and \(1 \frac{1}{5}\), we follow these steps:
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Convert each mixed number to an improper fraction.
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For \(1 \frac{1}{2}\): \[ 1 \frac{1}{2} = \frac{2 \cdot 1 + 1}{2} = \frac{3}{2} \]
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For \(1 \frac{1}{5}\): \[ 1 \frac{1}{5} = \frac{5 \cdot 1 + 1}{5} = \frac{6}{5} \]
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Now, we add the two improper fractions: \[ \frac{3}{2} + \frac{6}{5} \]
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To add these fractions, we need a common denominator. The least common multiple of \(2\) and \(5\) is \(10\).
Convert each fraction:
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For \(\frac{3}{2}\): \[ \frac{3}{2} = \frac{3 \cdot 5}{2 \cdot 5} = \frac{15}{10} \]
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For \(\frac{6}{5}\): \[ \frac{6}{5} = \frac{6 \cdot 2}{5 \cdot 2} = \frac{12}{10} \]
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Now add the two fractions: \[ \frac{15}{10} + \frac{12}{10} = \frac{27}{10} \]
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Convert the improper fraction \(\frac{27}{10}\) back to a mixed number:
- \(27 \div 10 = 2\) remainder \(7\),
- So, \(\frac{27}{10} = 2 \frac{7}{10}\).
Thus, the answer in simplest form is: \[ \boxed{2 \frac{7}{10}} \]
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