The simplification of $3.\overline{36} - 2.\overline{05} + 1.\overline{33}$ equals

Concept & Strategy When operating on multiple mixed numbers with pure recurring decimal components of the same length, group the integer parts and the fractional parts separately. This avoids converting large numbers into improper fractions. Step-by-Step Solution Given expression: $3.\overline{36} - 2.\overline{05} + 1.\overline{33}$ 1. Separate the whole numbers from the recurring decimal fractions: $$ (3 + \frac{36}{99}) - (2 + \frac{5}{99}) + (1 + \frac{33}{99}) $$ 2. Group the integer components together and the fractional components together: $$ (3 - 2 + 1) + (\frac{36}{99} - \frac{5}{99} + \frac{33}{99}) $$ 3. Calculate the integer sum: $$ 3 - 2 + 1 = 2 $$ 4. Calculate the fractional sum: $$ \frac{36 - 5 + 33}{99} = \frac{64}{99} $$ 5. Combine the integer and fractional parts back into recurring decimal form: $$ 2 + 0.\overline{64} = 2.\overline{64} $$ Exam Strategy & Shortcut Since all decimal parts have a bar over exactly two digits, you can handle them as whole numbers. Add/subtract the integers: $3 - 2 + 1 = 2$. Add/subtract the recurring sequences: $36 - 5 + 33 = 64$. Combine them directly to get $2.\overline{64}$. This takes less than 10 seconds. Common Pitfall A common trap is misaligning the subtraction, especially treating $05$ as $50$. Always respect the place value of the digits under the bar. Additionally, do not forget the bar on the final answer, which would incorrectly lead you to option (b). Final Answer Therefore, the correct answer is $2.\overline{64}$.