To multiply the mixed numbers \(2 \frac{2}{3}\) and \(1 \frac{1}{5}\), we first need to convert them to improper fractions.
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Convert \(2 \frac{2}{3}\): \[ 2 \frac{2}{3} = 2 \times 3 + 2 = 6 + 2 = 8 \quad \Rightarrow \quad \frac{8}{3} \]
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Convert \(1 \frac{1}{5}\): \[ 1 \frac{1}{5} = 1 \times 5 + 1 = 5 + 1 = 6 \quad \Rightarrow \quad \frac{6}{5} \]
Next, we multiply the two improper fractions: \[ \frac{8}{3} \times \frac{6}{5} = \frac{8 \times 6}{3 \times 5} = \frac{48}{15} \]
Now we simplify \(\frac{48}{15}\):
- The greatest common divisor (GCD) of 48 and 15 is 3.
- Dividing both the numerator and the denominator by 3: \[ \frac{48 \div 3}{15 \div 3} = \frac{16}{5} \]
Next, we convert \(\frac{16}{5}\) to a mixed number:
- Divide 16 by 5, which gives 3 as the whole number and a remainder of 1.
- Thus, \[ \frac{16}{5} = 3 \frac{1}{5} \]
Finally, the answer in mixed number form is: \[ \boxed{3} \quad \boxed{1} \quad / \quad \boxed{5} \]