To multiply the mixed numbers \( 3 \frac{3}{5} \) and \( 8 \frac{3}{4} \), we start by converting them into improper fractions.
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Convert \( 3 \frac{3}{5} \): \[ 3 \frac{3}{5} = 3 \times 5 + 3 = 15 + 3 = 18 \quad \text{(so, } 3\frac{3}{5} = \frac{18}{5}\text{)} \]
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Convert \( 8 \frac{3}{4} \): \[ 8 \frac{3}{4} = 8 \times 4 + 3 = 32 + 3 = 35 \quad \text{(so, } 8\frac{3}{4} = \frac{35}{4}\text{)} \]
Now we can multiply the improper fractions: \[ \frac{18}{5} \times \frac{35}{4} = \frac{18 \times 35}{5 \times 4} \]
Calculating the numerator: \[ 18 \times 35 = 630 \]
Calculating the denominator: \[ 5 \times 4 = 20 \]
So, we have: \[ \frac{630}{20} \]
Next, we simplify this fraction. We can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 10: \[ \frac{630 \div 10}{20 \div 10} = \frac{63}{2} \]
Now, converting \( \frac{63}{2} \) back to a mixed number:
- Divide 63 by 2: \[ 63 \div 2 = 31 \quad \text{(remainder } 1\text{)} \]
Thus, \( \frac{63}{2} = 31 \frac{1}{2} \).
Putting it all together, the answer is: \[ \boxed{31 \frac{1}{2}} \]
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