To multiply the mixed numbers \(2 \frac{1}{6}\) and \(1 \frac{3}{4}\), we first convert them into improper fractions.
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Convert \(2 \frac{1}{6}\) to an improper fraction: \[ 2 \frac{1}{6} = 2 \times 6 + 1 = 12 + 1 = 13 \quad \text{so} \quad 2 \frac{1}{6} = \frac{13}{6} \]
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Convert \(1 \frac{3}{4}\) to an improper fraction: \[ 1 \frac{3}{4} = 1 \times 4 + 3 = 4 + 3 = 7 \quad \text{so} \quad 1 \frac{3}{4} = \frac{7}{4} \]
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Multiply the improper fractions: \[ \frac{13}{6} \times \frac{7}{4} = \frac{13 \times 7}{6 \times 4} = \frac{91}{24} \]
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Convert the improper fraction \(\frac{91}{24}\) to a mixed number:
- Divide 91 by 24: \[ 91 \div 24 = 3 \quad \text{(whole number part)} \]
- Calculate the remainder: \[ 91 - (24 \times 3) = 91 - 72 = 19 \]
Therefore, \[ \frac{91}{24} = 3 \frac{19}{24} \]
So, the final answer is: \[ 3 \frac{19}{24} \]