To divide the mixed numbers \(2 \frac{1}{2}\) and \(1 \frac{3}{4}\), first convert them to improper fractions.
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Convert \(2 \frac{1}{2}\): \[ 2 \frac{1}{2} = 2 \times 2 + 1 = 5 \implies \frac{5}{2} \]
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Convert \(1 \frac{3}{4}\): \[ 1 \frac{3}{4} = 1 \times 4 + 3 = 7 \implies \frac{7}{4} \]
Now, divide the fractions: \[ \frac{5}{2} \div \frac{7}{4} \]
When dividing by a fraction, multiply by its reciprocal: \[ \frac{5}{2} \times \frac{4}{7} = \frac{5 \times 4}{2 \times 7} = \frac{20}{14} \]
Next, simplify \(\frac{20}{14}\): \[ \frac{20 \div 2}{14 \div 2} = \frac{10}{7} \]
Now, convert \(\frac{10}{7}\) to a mixed number: \[ 10 \div 7 = 1 \quad \text{remainder} \ 3 \implies 1 \frac{3}{7} \]
Thus, the final answer is: \[ \boxed{1 \frac{3}{7}} \]