To express the numbers \(2 \frac{2}{5}\) and \(1 \frac{1}{4}\) in a ratio, we first need to convert these mixed numbers into improper fractions.
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Convert \(2 \frac{2}{5}\) to an improper fraction: \[ 2 \frac{2}{5} = \frac{(2 \times 5) + 2}{5} = \frac{10 + 2}{5} = \frac{12}{5} \]
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Convert \(1 \frac{1}{4}\) to an improper fraction: \[ 1 \frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{4 + 1}{4} = \frac{5}{4} \]
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Now we can write the ratio: \[ \frac{12}{5} : \frac{5}{4} \]
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To eliminate the fractions, multiply both sides of the ratio by the least common multiple (LCM) of the denominators, which is 20: \[ 20 \left( \frac{12}{5} \right) : 20 \left( \frac{5}{4} \right) \]
Calculating: \[ 20 \times \frac{12}{5} = 20 \div 5 \times 12 = 4 \times 12 = 48 \] \[ 20 \times \frac{5}{4} = 20 \div 4 \times 5 = 5 \times 5 = 25 \]
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Thus, the ratio of \(2 \frac{2}{5}\) to \(1 \frac{1}{4}\) is: \[ 48 : 25 \]
Hence, the ratio of \(2 \frac{2}{5}\) to \(1 \frac{1}{4}\) is \(48 : 25\).