The value of $(0.\overline{2} + 0.\overline{3} + 0.\overline{32})$ is

Concept & Formula To add recurring decimals, convert each individual recurring decimal into its equivalent vulgar fraction form, find a common denominator to sum them up, and then convert the final fractional sum back into a recurring decimal. Step-by-Step Solution Given expression: $0.\overline{2} + 0.\overline{3} + 0.\overline{32}$ 1. Convert each term to a fraction: - $0.\overline{2} = \frac{2}{9}$ - $0.\overline{3} = \frac{3}{9}$ - $0.\overline{32} = \frac{32}{99}$ 2. Set a common denominator ($99$) to add the fractions: - $\frac{2}{9} = \frac{2 \times 11}{9 \times 11} = \frac{22}{99}$ - $\frac{3}{9} = \frac{3 \times 11}{9 \times 11} = \frac{33}{99}$ 3. Sum the values together: $$ \frac{22}{99} + \frac{33}{99} + \frac{32}{99} = \frac{22 + 33 + 32}{99} = \frac{87}{99} $$ 4. Convert back to decimal form: $$ \frac{87}{99} = 0.\overline{87} $$ Exam Strategy & Shortcut Alternatively, expand the decimals out to a few places and add vertically: $$ 0.222222... $$ $$ 0.333333... $$ $$ 0.323232... $$ $$ \text{Sum} = 0.878787... = 0.\overline{87} $$ This expansion method lets you spot the repeating pattern $87$ in under 5 seconds without calculating fractions. Common Pitfall Do not treat these as normal decimals ($0.2 + 0.3 + 0.32 = 0.82$), which leads directly to the trap option (b). The recurring bar signifies infinite trailing digits that alter the addition outcome. Final Answer Therefore, the correct answer is $0.\overline{87}$.