$\frac{2}{2 + \frac{2}{3 + \frac{2}{3 + \frac{2}{3}}}} \times 0.39$ is simplified to

### Concept & Strategy This problem features a deeply nested continued fraction alongside decimal multiplication. **Important Note on Typography:** In typical aptitude tests, a multiplier placed alongside the main denominator bar (like the $\times 0.39$ here) implicitly applies to that entire evaluated denominator grouping. Thus, the expression we are evaluating is mathematically represented as $\frac{2}{(2 + \dots) \times 0.39}$. The strategy is to simplify the continued fraction first, multiply the denominator by 0.39, and finally divide the numerator. ### Step-by-Step Solution Let's simplify the continued fraction in the denominator from the bottom up. * **Step 1: Lowest level** $$3 + \frac{2}{3} = \frac{9 + 2}{3} = \frac{11}{3}$$ * **Step 2: Second level** Substitute and invert: $$\frac{2}{\frac{11}{3}} = \frac{6}{11}$$ Add to 3: $$3 + \frac{6}{11} = \frac{33 + 6}{11} = \frac{39}{11}$$ * **Step 3: Third level** Substitute and invert: $$\frac{2}{\frac{39}{11}} = \frac{22}{39}$$ Add to 2 (completing the bracketed denominator): $$2 + \frac{22}{39} = \frac{78 + 22}{39} = \frac{100}{39}$$ * **Step 4: Apply the Decimal Multiplication** Now, multiply this evaluated denominator by $0.39$. It is easier to convert $0.39$ to a fraction: $$0.39 = \frac{39}{100}$$ Multiplying them together: $$\frac{100}{39} \times \frac{39}{100} = 1$$ * **Step 5: Final Evaluation** Substitute the final denominator value back under the main numerator: $$\frac{2}{1} = 2$$ ### Exam Strategy & Shortcut Look for cancellation opportunities! Once you see the denominator simplifying to $\frac{100}{39}$ and notice the multiplier is $0.39$, you should immediately recognize that $0.39 = 39/100$. These numbers are perfectly designed to cross-cancel to 1. Spotting these relationships prevents unnecessary and error-prone decimal division. ### Common Pitfall Misinterpreting the position of $\times 0.39$. If you multiply the *entire evaluated fraction* by 0.39 (i.e., treating it as $\frac{2}{100/39} \times 0.39$), you would get $0.78 \times 0.39 = 0.3042$, which isn't an option. Always look closely at the horizontal alignment of the arithmetic operators in poorly typeset exams. ### Final Answer Therefore, the correct answer is **2**.