To solve the problem, we need to divide \(3 \frac{1}{2}\) by \(2 \frac{1}{4}\).
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Convert mixed numbers to improper fractions:
- \(3 \frac{1}{2} = 3 \cdot 2 + 1 = 6 + 1 = 7\), so \(3 \frac{1}{2} = \frac{7}{2}\).
- \(2 \frac{1}{4} = 2 \cdot 4 + 1 = 8 + 1 = 9\), so \(2 \frac{1}{4} = \frac{9}{4}\).
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Perform the division of fractions: \[ \frac{7}{2} \div \frac{9}{4} = \frac{7}{2} \cdot \frac{4}{9} = \frac{7 \cdot 4}{2 \cdot 9} = \frac{28}{18} \]
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Simplify the fraction \(\frac{28}{18}\):
- Find the greatest common divisor (GCD) of 28 and 18, which is 2. \[ \frac{28 \div 2}{18 \div 2} = \frac{14}{9} \]
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Express \(\frac{14}{9}\) in the form of \( a \frac{b}{c} \): Here, \(a = 1\), \(b = 5\), and \(c = 9\).
From the provided options, we see:
- The simplest form of the answer is \(\frac{14}{9} = 1 \frac{5}{9}\), aligning with the values of \(a = 1\), \(b = 5\), \(c = 9\).
Thus, the correct answer is: a = 1, b = 5, c = 9.