The simplified value of $$\frac{\frac{1}{3} \div \frac{1}{3} \times \frac{1}{3}}{\frac{1}{3} \div \frac{1}{3} \text{ of } \frac{1}{3}} - \frac{1}{9}$$ is

### Concept & Formula This problem highlights the crucial precedence difference between division ($\div$) and the 'of' operator under standard **BODMAS** rules. 'Of' takes complete precedence over division. ### Step-by-Step Solution * **Step 1: Simplify the Numerator** $$\text{Numerator} = \frac{1}{3} \div \frac{1}{3} \times \frac{1}{3}$$ Division takes priority over multiplication here: $$\left(\frac{1}{3} \div \frac{1}{3}\right) \times \frac{1}{3} = 1 \times \frac{1}{3} = \frac{1}{3}$$ * **Step 2: Simplify the Denominator** $$\text{Denominator} = \frac{1}{3} \div \frac{1}{3} \text{ of } \frac{1}{3}$$ The 'of' operation takes priority over division: $$\frac{1}{3} \text{ of } \frac{1}{3} = \frac{1}{9}$$ Now perform the division: $$\frac{1}{3} \div \frac{1}{9} = \frac{1}{3} \times 9 = 3$$ * **Step 3: Evaluate the Overall Fraction** $$\text{Fraction Block} = \frac{\text{Numerator}}{\text{Denominator}} = \frac{\frac{1}{3}}{3} = \frac{1}{9}$$ * **Step 4: Perform Final Subtraction** $$\frac{1}{9} - \frac{1}{9} = 0$$ ### Exam Strategy & Shortcut Recognize this classic pattern instantly: $\text{Term} \div \text{Term} \times \text{Term}$ evaluates to $\text{Term}$, whereas $\text{Term} \div \text{Term of Term}$ evaluates to $\frac{1}{\text{Term}}$. Recognizing this avoids manual fraction computations entirely. ### Common Pitfall Treating the denominator as identical to the numerator by ignoring the 'of' rule leads to a value of $1 - \frac{1}{9} = \frac{8}{9}$, which is a common miscalculation trap. ### Final Answer **Therefore, the correct answer is 0.**