$(0.\overline{09} \times 7.\overline{3})$ is equal to:

Concept & Formula Multiplication involving recurring decimals cannot be done reliably in decimal form. You must strictly convert all recurring decimals into their lowest-term fraction equivalents before multiplying. Step-by-Step Solution Given expression: $(0.\overline{09} \times 7.\overline{3})$ 1. Convert the first decimal ($0.\overline{09}$) to a fraction: The repeating part is $09$, which is two digits long. $$ 0.\overline{09} = \frac{09}{99} = \frac{9}{99} = \frac{1}{11} $$ 2. Convert the second decimal ($7.\overline{3}$) to a fraction: Separate the integer or use the full formula. $$ 7.\overline{3} = 7 + \frac{3}{9} = 7 + \frac{1}{3} $$ Convert the mixed fraction to an improper fraction: $$ \frac{(7 \times 3) + 1}{3} = \frac{22}{3} $$ 3. Multiply the two simplified fractions: $$ \frac{1}{11} \times \frac{22}{3} $$ 4. Cancel out common factors across the numerator and denominator: The $22$ in the numerator and $11$ in the denominator simplify to $2$. $$ \frac{1 \times 2}{1 \times 3} = \frac{2}{3} $$ 5. Convert the final fraction back to a recurring decimal to match the options: $$ \frac{2}{3} = 0.6666... = 0.\overline{6} $$ *(Note: The options present this as $.\overline{6}$)* Exam Strategy & Shortcut Familiarize yourself with common fraction-to-decimal conversions. Seeing $0.\overline{09}$ should instantly trigger $\frac{1}{11}$ in your mind, and $7.\overline{3}$ is obviously $7 \frac{1}{3}$ or $\frac{22}{3}$. The multiplication $\frac{1}{11} \times \frac{22}{3}$ cleanly cancels to $\frac{2}{3}$, and every prepared student knows $\frac{2}{3} = 0.\overline{6}$. Common Pitfall Attempting to multiply $0.09 \times 7.3$ as standard terminating decimals completely ignores the infinite nature of the numbers and leads to chaotic, incorrect results that don't match any options. Final Answer Therefore, the correct answer is $.\overline{6}$.