The value of $\left[35.7 - \left(3 + \frac{1}{3 + \frac{1}{3}}\right) - \left(2 + \frac{1}{2 + \frac{1}{2}}\right)\right]$ is

### Concept & Strategy Solve the nested continued fractions from the bottom upward before carrying out the final decimal subtraction. ### Step-by-Step Solution * Evaluate the first continued fraction block: $$ 3 + \frac{1}{3} = \frac{10}{3} $$ $$ 3 + \frac{1}{\frac{10}{3}} = 3 + \frac{3}{10} = \frac{33}{10} = 3.3 $$ * Evaluate the second continued fraction block: $$ 2 + \frac{1}{2} = \frac{5}{2} $$ $$ 2 + \frac{1}{\frac{5}{2}} = 2 + \frac{2}{5} = \frac{12}{5} = 2.4 $$ * Substitute these clean values back into the primary expression: $$ 35.7 - 3.3 - 2.4 $$ $$ = 32.4 - 2.4 $$ $$ = 30 $$ ### Exam Strategy & Shortcut Convert fractions to explicit decimals immediately when denominators are friendly (like $10$ and $5$). Recognizing that $\frac{3}{10} = 0.3$ and $\frac{2}{5} = 0.4$ turns a messy expression into simple arithmetic: $35.7 - 3.3 - 2.4 = 30$. ### Common Pitfall Forgetting to invert the denominator fraction when resolving the nested layer (e.g., adding $\frac{10}{3}$ directly instead of converting it to $\frac{3}{10}$). ### Final Answer **Therefore, the correct answer is 30.**