Question : The value of $4-\frac{1}{1\:+\:\frac{1}{3\:+\:\frac{1}{2\:+\:\frac{1}{2\:+\:\frac{1}{4}}}}}$ is:
Option 1: $\frac{1}{8}$
Option 2: $\frac{1}{64}$
Option 3: $\frac{1}{32}$
Option 4: $\frac{313}{97}$
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Correct Answer: $\frac{313}{97}$
Solution :
Given:
$4-\frac{1}{1\:+\:\frac{1}{3\:+\:\frac{1}{2\:+\:\frac{1}{2\:+\:\frac{1}{4}}}}}$
$=4-\frac{1}{1\:+\:\frac{1}{3\:+\:\frac{1}{2\:+\:\frac{1}{\frac{9}{4}}}}}$
$= 4-\frac{1}{1\:+\:\frac{1}{3\:+\:\frac{1}{2\:+\:\frac{4}{9}}}}$
$= 4-\frac{1}{1\:+\:\frac{1}{3\:+\:\frac{1}{\frac{22}{9}}}}$
$= 4-\frac{1}{1\:+\:\frac{1}{3\:+\:\frac{9}{22}}}$
$= 4-\frac{1}{1\:+\:\frac{1}{\frac{75}{22}}}$
$= 4-\frac{1}{1\:+\:\frac{22}{75}}$
$= 4-\frac{1}{\frac{97}{75}}$
$= 4-\frac{75}{97}$
$= \frac{388\:-\:75}{97}$
$= \frac{313}{97}$
Hence, the correct answer is $\frac{313}{97}$.
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