Find the value of $$ \frac{1}{1 + \frac{1}{3 - \frac{4}{2 + \frac{1}{3 - \frac{1}{2}}}}} + \frac{3}{3 - \frac{4}{3 + \frac{1}{2 - \frac{1}{2}}}} $$

### Concept & Strategy Treat this massive expression as two independent continued fraction problems. Simplify each major fraction independently from the bottom layer up, taking extreme care with subtraction signs, before adding the final two rational numbers together. ### Step-by-Step Solution * **Calculation for the First Term:** 1. Deepest denominator: $3 - \frac{1}{2} = \frac{5}{2}$ 2. Next layer: $2 + \frac{1}{\frac{5}{2}} = 2 + \frac{2}{5} = \frac{12}{5}$ 3. Next layer: $3 - \frac{4}{\frac{12}{5}} = 3 - \left(4 \times \frac{5}{12}\right) = 3 - \frac{5}{3} = \frac{9}{3} - \frac{5}{3} = \frac{4}{3}$ 4. Next layer: $1 + \frac{1}{\frac{4}{3}} = 1 + \frac{3}{4} = \frac{7}{4}$ 5. Final First Term value: $\frac{1}{\frac{7}{4}} = \frac{4}{7}$ * **Calculation for the Second Term:** 1. Deepest denominator: $2 - \frac{1}{2} = \frac{3}{2}$ 2. Next layer: $3 + \frac{1}{\frac{3}{2}} = 3 + \frac{2}{3} = \frac{11}{3}$ 3. Next layer: $3 - \frac{4}{\frac{11}{3}} = 3 - \left(4 \times \frac{3}{11}\right) = 3 - \frac{12}{11} = \frac{33}{11} - \frac{12}{11} = \frac{21}{11}$ 4. Final Second Term value: $\frac{3}{\frac{21}{11}} = 3 \times \frac{11}{21} = \frac{11}{7}$ * **Final Addition:** 1. Sum = First Term + Second Term 2. Sum = $\frac{4}{7} + \frac{11}{7} = \frac{15}{7}$ ### Exam Strategy & Shortcut Break complex problems into distinct modular zones. Write "Left" and "Right" on your scratchpad. Execute the reciprocal flips mentally and only write down the resulting simplified fraction for each vertical step. For example, for the Right side, write: $3/2$ $\rightarrow$ flip add 3 $\rightarrow$ $11/3$ $\rightarrow$ multiply 4, flip, subtract from 3 $\rightarrow$ $3 - 12/11 = 21/11$ $\rightarrow$ multiply 3, flip $\rightarrow$ $33/21 = 11/7$. This keeps your workspace clean and prevents transposition errors. ### Common Pitfall The most dangerous step is $3 - \frac{4}{\frac{12}{5}}$. Students often invert the fraction incorrectly, or multiply $4$ into both the numerator and denominator, resulting in $20/48$ instead of $20/12$. Remember that a whole number multiplied by a fraction only scales the numerator. ### Final Answer **Therefore, the correct answer is $\frac{15}{7}$.**