To add the mixed numbers \(2 \frac{2}{3}\) and \(8 \frac{3}{5}\), follow these steps:
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Convert the mixed numbers to improper fractions:
- For \(2 \frac{2}{3}\): \[ 2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3} \]
- For \(8 \frac{3}{5}\): \[ 8 \frac{3}{5} = \frac{8 \times 5 + 3}{5} = \frac{40 + 3}{5} = \frac{43}{5} \]
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Find a common denominator to add the fractions \( \frac{8}{3} \) and \( \frac{43}{5} \). The least common multiple of \(3\) and \(5\) is \(15\).
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Convert each fraction to have the common denominator:
- For \( \frac{8}{3} \): \[ \frac{8}{3} = \frac{8 \times 5}{3 \times 5} = \frac{40}{15} \]
- For \( \frac{43}{5} \): \[ \frac{43}{5} = \frac{43 \times 3}{5 \times 3} = \frac{129}{15} \]
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Add the fractions: \[ \frac{40}{15} + \frac{129}{15} = \frac{40 + 129}{15} = \frac{169}{15} \]
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Convert the improper fraction back to a mixed number:
- Divide \(169\) by \(15\): \[ 169 \div 15 = 11 \quad \text{(remainder } 4\text{)} \]
- So, \( \frac{169}{15} = 11 \frac{4}{15} \).
Thus, \(2 \frac{2}{3} + 8 \frac{3}{5} = 11 \frac{4}{15}\).