$(8.3\overline{1} + 0.\overline{6} + 0.00\overline{2})$ is equal to

Concept & Logic Adding decimals with different repeating patterns often creates unexpected results. When standard addition yields a terminating decimal (like $0.98$), exam setters might disguise the correct option by expressing it as a recurring decimal ending in a repeating $9$ (e.g., $0.97\overline{9}$), which is mathematically equivalent to $0.98$. Step-by-Step Solution Given expression: $(8.3\overline{1} + 0.\overline{6} + 0.00\overline{2})$ 1. Convert each term to a vulgar fraction: $$ 8.3\overline{1} = 8 + \frac{31 - 3}{90} = 8 + \frac{28}{90} $$ $$ 0.\overline{6} = \frac{6}{9} = \frac{60}{90} $$ $$ 0.00\overline{2} = \frac{2}{900} $$ 2. Find a common denominator ($900$) and sum them up: $$ 8 + \frac{28 \times 10}{90 \times 10} + \frac{60 \times 10}{90 \times 10} + \frac{2}{900} $$ $$ 8 + \frac{280}{900} + \frac{600}{900} + \frac{2}{900} $$ $$ 8 + \frac{882}{900} $$ 3. Simplify the fraction to a decimal: $$ \frac{882}{900} = \frac{882 \div 9}{900 \div 9} = \frac{98}{100} = 0.98 $$ So, the exact sum is $8.98$. 4. Match with the given options: None of the options read exactly $8.98$. However, mathematically, a number ending in an infinite sequence of $9$s is exactly equal to the next terminating decimal. Let's evaluate Option (d): $8.97\overline{9}$ $$ 8.97\overline{9} = 8 + \frac{979 - 97}{900} = 8 + \frac{882}{900} = 8.98 $$ Thus, $8.97\overline{9}$ is mathematically identical to $8.98$. Exam Strategy & Shortcut Expand the decimals to see the trend: $$ \quad 8.311111... $$ $$ \quad 0.666666... $$ $$ + 0.002222... $$ $$ \rule{2.5cm}{0.4pt} $$ $$ \quad 8.979999... $$ You can clearly see the sum produces $8.97\overline{9}$. You don't even need to convert it to $8.98$ to spot the correct option. Common Pitfall Many students reach $8.98$ via fraction calculation, scan the options, panic when they don't see $8.98$, and blindly guess. Recognizing that $0.97\overline{9} = 0.98$ is a crucial advanced arithmetic concept. Final Answer Therefore, the correct answer is $8.97\overline{9}$.