Simplify: $\frac{1}{1 + \frac{\frac{2}{3}}{1 + \frac{2}{3} + \frac{\frac{8}{9}}{1 - \frac{2}{3}}}}$

### Concept & Strategy This is a multi-level complex fraction. While it looks intimidating, it is solved using the standard bottom-up simplification strategy. The key is to carefully isolate and evaluate the deepest nested terms first before combining them with adjacent terms. ### Step-by-Step Solution Let's break the problem down into manageable chunks. * **Step 1: Simplify the deepest term on the right** Evaluate the fraction $\frac{\frac{8}{9}}{1 - \frac{2}{3}}$: Denominator: $1 - \frac{2}{3} = \frac{1}{3}$ Now divide: $\frac{\frac{8}{9}}{\frac{1}{3}} = \frac{8}{9} \times \frac{3}{1} = \frac{24}{9} = \frac{8}{3}$ * **Step 2: Simplify the denominator of the middle layer** The denominator is $1 + \frac{2}{3} + \text{(Result from Step 1)}$: $$1 + \frac{2}{3} + \frac{8}{3}$$ Since the fractions share a common denominator, combine them: $$1 + \frac{10}{3} = \frac{3 + 10}{3} = \frac{13}{3}$$ * **Step 3: Evaluate the middle fraction** Divide the numerator $\frac{2}{3}$ by the result from Step 2: $$\frac{\frac{2}{3}}{\frac{13}{3}} = \frac{2}{3} \times \frac{3}{13} = \frac{2}{13}$$ * **Step 4: Final calculation** Substitute the result from Step 3 back into the main expression: $$\frac{1}{1 + \frac{2}{13}}$$ Simplify the denominator: $$1 + \frac{2}{13} = \frac{13 + 2}{13} = \frac{15}{13}$$ Invert for the final answer: $$\frac{1}{\frac{15}{13}} = \frac{13}{15}$$ ### Exam Strategy & Shortcut When dividing fractions with the same denominators (like in Step 3 where both have a denominator of 3), you can ignore the denominators entirely and just divide the numerators. So, $\frac{2/3}{13/3}$ immediately becomes $\frac{2}{13}$. This mental shortcut saves valuable seconds during an exam. ### Common Pitfall A frequent mistake is getting lost in the arithmetic when copying terms from one step to the next. Students often miscalculate $1 - \frac{2}{3}$ or forget to invert fractions when dividing. Keep your scratchpad organized by clearly drawing boxes around completed sub-components. ### Final Answer Therefore, the correct answer is **\frac{13}{15}**.