To multiply the mixed numbers \( 2 \frac{1}{3} \) and \( 2 \frac{1}{5} \), we first need to convert them into improper fractions.
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Convert \( 2 \frac{1}{3} \) to an improper fraction: \[ 2 \frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \]
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Convert \( 2 \frac{1}{5} \) to an improper fraction: \[ 2 \frac{1}{5} = \frac{(2 \times 5) + 1}{5} = \frac{10 + 1}{5} = \frac{11}{5} \]
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Now, multiply the two improper fractions: \[ \frac{7}{3} \times \frac{11}{5} = \frac{7 \times 11}{3 \times 5} = \frac{77}{15} \]
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Next, we convert \( \frac{77}{15} \) to a mixed number. Divide 77 by 15:
- \( 77 \div 15 = 5 \) remainder \( 2 \)
Thus, \[ 77 = 15 \times 5 + 2 \]
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Combine the quotient and the remainder: \[ \frac{77}{15} = 5 \frac{2}{15} \]
So the product \( 2 \frac{1}{3} \times 2 \frac{1}{5} \) is: \[ \boxed{5 \frac{2}{15}} \]