To add \(1 \frac{5}{12}\) and \(1 \frac{2}{5}\), we first convert each mixed number into an improper fraction.
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Convert \(1 \frac{5}{12}\): \[ 1 \frac{5}{12} = \frac{12 \cdot 1 + 5}{12} = \frac{12 + 5}{12} = \frac{17}{12} \]
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Convert \(1 \frac{2}{5}\): \[ 1 \frac{2}{5} = \frac{5 \cdot 1 + 2}{5} = \frac{5 + 2}{5} = \frac{7}{5} \]
Next, we can add the two improper fractions. To do this, we need a common denominator. The least common multiple of \(12\) and \(5\) is \(60\).
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Convert \(\frac{17}{12}\) to a fraction with a denominator of \(60\): \[ \frac{17}{12} = \frac{17 \cdot 5}{12 \cdot 5} = \frac{85}{60} \]
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Convert \(\frac{7}{5}\) to a fraction with a denominator of \(60\): \[ \frac{7}{5} = \frac{7 \cdot 12}{5 \cdot 12} = \frac{84}{60} \]
Now we can add the two fractions: \[ \frac{85}{60} + \frac{84}{60} = \frac{85 + 84}{60} = \frac{169}{60} \]
Finally, we can convert \(\frac{169}{60}\) back to a mixed number:
- Divide \(169\) by \(60\): \[ 169 \div 60 = 2 \quad \text{(since } 60 \cdot 2 = 120\text{)} \] \[ 169 - 120 = 49 \]
So, we have: \[ \frac{169}{60} = 2 \frac{49}{60} \]
Thus, the final answer is: \[ 1 \frac{5}{12} + 1 \frac{2}{5} = 2 \frac{49}{60} \]