The value of $$\frac{\frac{1}{2} \div \frac{1}{2} \text{ of } \frac{1}{2}}{\frac{1}{2} + \frac{1}{2} \text{ of } \frac{1}{2}}$$ is

### Concept & Formula Apply the **BODMAS** rule hierarchy: evaluate the 'of' operator before addition or division. ### Step-by-Step Solution * **Step 1: Simplify the Numerator** $$\text{Numerator} = \frac{1}{2} \div \left(\frac{1}{2} \text{ of } \frac{1}{2}\right)$$ $$\frac{1}{2} \text{ of } \frac{1}{2} = \frac{1}{4}$$ Now evaluate division: $$\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times 4 = 2$$ * **Step 2: Simplify the Denominator** $$\text{Denominator} = \frac{1}{2} + \left(\frac{1}{2} \text{ of } \frac{1}{2}\right)$$ $$\frac{1}{2} \text{ of } \frac{1}{2} = \frac{1}{4}$$ Now evaluate addition: $$\frac{1}{2} + \frac{1}{4} = \frac{3}{4}$$ * **Step 3: Divide Numerator by Denominator** $$\text{Value} = \frac{2}{\frac{3}{4}} = 2 \times \frac{4}{3} = \frac{8}{3} = 2\frac{2}{3}$$ ### Exam Strategy & Shortcut Whenever a whole number is divided by a proper fraction less than 1 (like $\frac{3}{4}$), the result must be greater than the numerator itself ($2$). Since $2\frac{2}{3}$ is the only option greater than 2, you can select it immediately without calculating the final reciprocal fraction. ### Common Pitfall Adding the two terms in the denominator before evaluating the 'of' operator will yield an incorrect denominator of $1 \times \frac{1}{2} = \frac{1}{2}$. Always perform 'of' first. ### Final Answer **Therefore, the correct answer is 2(2/3).**