To simplify the expression \( 7 \frac{1}{2} \times \left( \frac{7}{8} - \frac{5}{24} \right) \), we can follow these steps:
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Convert the mixed number to an improper fraction: \[ 7 \frac{1}{2} = \frac{15}{2} \]
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Find a common denominator for the fractions: The common denominator for \( \frac{7}{8} \) and \( \frac{5}{24} \) is \( 24 \).
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Convert \( \frac{7}{8} \) to 24ths: \[ \frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24} \]
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\( \frac{5}{24} \) is already in terms of \( 24 \): \[ \frac{5}{24} = \frac{5}{24} \]
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Subtract the fractions: \[ \frac{21}{24} - \frac{5}{24} = \frac{21 - 5}{24} = \frac{16}{24} \]
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Simplify \( \frac{16}{24} \): \[ \frac{16}{24} = \frac{2}{3} \quad \text{(by dividing numerator and denominator by 8)} \]
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Multiply \( \frac{15}{2} \) by \( \frac{2}{3} \): \[ \frac{15}{2} \times \frac{2}{3} = \frac{15 \times 2}{2 \times 3} = \frac{30}{6} \]
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Simplify \( \frac{30}{6} \): \[ \frac{30}{6} = 5 \]
Thus, the simplified result of \( 7 \frac{1}{2} \times \left( \frac{7}{8} - \frac{5}{24} \right) \) is
\[ \boxed{5} \]
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