$\text { Write the continued fraction in the form } \frac { p } { q } \text { reduced to lowest terms. }$
-$4 + \frac { 1 } { 1 + \frac { 1 } { 1 + \frac { 1 } { 2 } } }$
A) $\frac { 23 } { 5 }$
B) $\frac { 13 } { 3 }$
C) $\frac { 14 } { 3 }$
D) $\frac { 22 } { 5 }$

Question

Quizplus · Accepted Answer

Starting from the innermost fraction, we simplify step by step: $1 + \frac{1}{2} = \frac{3}{2}$, then $1 + \frac{1}{\frac{3}{2}} = 1 + \frac{2}{3} = \frac{5}{3}$, and $1 + \frac{1}{\frac{5}{3}} = 1 + \frac{3}{5} = \frac{8}{5}$. Finally, $4 + \frac{1}{\frac{8}{5}} = 4 + \frac{5}{8} = \frac{32}{8} + \frac{5}{8} = \frac{37}{8}$. However, none of the options match $\frac{37}{8}$, indicating a mistake in my calculation. Let's correct this: Starting again from the innermost fraction: $1 + \frac{1}{2} = \frac{3}{2}$, then $1 + \frac{1}{\frac{3}{2}} = 1 + \frac{2}{3} = \frac{5}{3}$, and finally, $1 + \frac{1}{\frac{5}{3}} = 1 + \frac{3}{5} = \frac{8}{5}$. Adding this to 4, we get $4 + \frac{5}{8}$, which was incorrectly simplified. Correctly, it should be $4 + \frac{1}{\frac{8}{5}} = 4 + \frac{5}{8}$. The correct calculation should be $4 + \frac{5}{8} = \frac{32}{8} + \frac{5}{8} = \frac{37}{8}$, but this was a mistake in simplification. Correctly simplifying from the beginning: The correct simplification is: $4 + \frac{1}{1 + \frac{1}{1 + \frac{1}{2}}} = 4 + \frac{1}{1 + \frac{1}{\frac{3}{2}}} = 4 + \frac{1}{1 + \frac{2}{3}} = 4 + \frac{1}{\frac{5}{3}} = 4 + \frac{3}{5} = \frac{20}{5} + \frac{3}{5} = \frac{23}{5}$. My initial explanation contained errors in the simplification process. The correct answer is indeed $\frac{23}{5}$, which matches option A.

Write the continued fraction in the form pq reduced to lowest terms. \text { Write the continued fraction in the form } \frac { p } { q } \text { reduced to lowest terms. } Write the continued fraction in the form qp​ reduced to lowest terms.

$\text { Write the continued fraction in the form } \frac { p } { q } \text { reduced to lowest terms. }$