To simplify the complex fraction \( \frac{\frac{1}{2}}{\frac{5}{8} + \frac{1}{4}} \), we will follow these steps:
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Find a common denominator for the terms in the denominator \( \frac{5}{8} + \frac{1}{4} \).
The least common multiple of 8 and 4 is 8.
\[ \frac{1}{4} = \frac{2}{8} \]
So we can rewrite the addition:
\[ \frac{5}{8} + \frac{1}{4} = \frac{5}{8} + \frac{2}{8} = \frac{7}{8} \]
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Now substitute this back into the original complex fraction:
\[ \frac{\frac{1}{2}}{\frac{7}{8}} \]
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To divide by a fraction, we multiply by its reciprocal:
\[ \frac{1}{2} \div \frac{7}{8} = \frac{1}{2} \times \frac{8}{7} = \frac{1 \times 8}{2 \times 7} = \frac{8}{14} \]
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Now simplify \( \frac{8}{14} \):
Divide both the numerator and denominator by their greatest common divisor, which is 2:
\[ \frac{8 \div 2}{14 \div 2} = \frac{4}{7} \]
The simplified value of the complex fraction is \( \frac{4}{7} \).
Thus, the answer is:
A. \( \frac{4}{7} \)