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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