To multiply the mixed numbers \(1 \frac{1}{3}\) and \(2 \frac{1}{5}\), we can first rewrite them as improper fractions.
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Convert \(1 \frac{1}{3}\) to an improper fraction: \[ 1 \frac{1}{3} = \frac{3 \times 1 + 1}{3} = \frac{4}{3} \]
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Convert \(2 \frac{1}{5}\) to an improper fraction: \[ 2 \frac{1}{5} = \frac{5 \times 2 + 1}{5} = \frac{11}{5} \]
Now we can multiply the two improper fractions: \[ \frac{4}{3} \times \frac{11}{5} = \frac{4 \times 11}{3 \times 5} = \frac{44}{15} \]
Next, we need to convert \(\frac{44}{15}\) back to a mixed number. We do this by dividing 44 by 15: \[ 44 \div 15 = 2 \quad \text{(the whole number part)} \] with a remainder of \(44 - 15 \times 2 = 14\).
Thus, we can express \(\frac{44}{15}\) as a mixed number: \[ 2 \frac{14}{15} \]
So, the final answer is: \[ \boxed{2 \frac{14}{15}} \]
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