To find the quotient of \(10 \frac{2}{5} \div 2 \frac{1}{4}\), we first need to convert both mixed numbers into improper fractions.
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Convert \(10 \frac{2}{5}\) to an improper fraction: \[ 10 \frac{2}{5} = 10 \times 5 + 2 = 50 + 2 = 52 \quad \Rightarrow \quad \frac{52}{5} \]
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Convert \(2 \frac{1}{4}\) to an improper fraction: \[ 2 \frac{1}{4} = 2 \times 4 + 1 = 8 + 1 = 9 \quad \Rightarrow \quad \frac{9}{4} \]
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Now divide the two improper fractions: \[ \frac{52}{5} \div \frac{9}{4} \] This is the same as multiplying by the reciprocal: \[ \frac{52}{5} \times \frac{4}{9} \]
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Multiply the fractions: \[ = \frac{52 \times 4}{5 \times 9} = \frac{208}{45} \]
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Convert \(\frac{208}{45}\) back to a mixed number: Divide 208 by 45:
- \(208 \div 45 \approx 4\) (since \(45 \times 4 = 180\))
- Subtract \(180\) from \(208\) to get a remainder of \(28\).
Therefore, we can express \( \frac{208}{45} \) as: \[ 4 \frac{28}{45} \]
Thus, the final answer is: \[ \frac{208}{45} \text{ or } 4 \frac{28}{45} \]