To multiply the mixed numbers \(1 \frac{4}{5}\) and \(3 \frac{1}{2}\), first convert them to improper fractions.
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Convert \(1 \frac{4}{5}\): \[ 1 \frac{4}{5} = \frac{1 \times 5 + 4}{5} = \frac{5 + 4}{5} = \frac{9}{5} \]
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Convert \(3 \frac{1}{2}\): \[ 3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2} \]
Now, multiply the two improper fractions: \[ \frac{9}{5} \times \frac{7}{2} = \frac{9 \times 7}{5 \times 2} = \frac{63}{10} \]
Finally, convert \(\frac{63}{10}\) back to a mixed number: \[ \frac{63}{10} = 6 \frac{3}{10} \]
So, \(1 \frac{4}{5} \times 3 \frac{1}{2} = 6 \frac{3}{10}\).