To solve the division of mixed numbers \(5 \frac{2}{3} \div 3 \frac{4}{7}\), we first convert the mixed numbers into improper fractions.
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Convert \(5 \frac{2}{3}\): \[ 5 \frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3} \]
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Convert \(3 \frac{4}{7}\): \[ 3 \frac{4}{7} = \frac{3 \times 7 + 4}{7} = \frac{21 + 4}{7} = \frac{25}{7} \]
Now we can perform the division of the two improper fractions: \[ \frac{17}{3} \div \frac{25}{7} \]
Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{17}{3} \times \frac{7}{25} = \frac{17 \times 7}{3 \times 25} = \frac{119}{75} \]
Now, we can convert \(\frac{119}{75}\) to a mixed number:
- Divide 119 by 75: \[ 119 \div 75 = 1 \quad \text{(remainder } 44\text{)} \]
So, we have: \[ \frac{119}{75} = 1 \frac{44}{75} \]
The answer is \(1 \frac{44}{75}\).
Among the responses, the correct answer is: 1 \( \frac{44}{75} \).