$$\frac{6}{5 - \frac{1}{3}} + \frac{4 - \frac{2}{4 - \frac{1}{2}}}{5 - \frac{3}{2}} - \frac{2}{5} \text{ of } \left\{\frac{6}{9} + \frac{2}{3} \text{ of } \frac{1}{2}\right\} = x$$

### Concept & Formula Follow the strict **BODMAS** order: Parentheses/Brackets, 'Of' operations, Division and Multiplication, and lastly Addition and Subtraction. ### Step-by-Step Solution Let's split the problem into three main independent components separated by the plus and minus signs: * **Step 1: Simplify Term 1** $$\text{Term 1} = \frac{6}{5 - \frac{1}{3}} = \frac{6}{\frac{14}{3}} = \frac{18}{14} = \frac{9}{7}$$ * **Step 2: Simplify Term 2** $$\text{Numerator} = 4 - \frac{2}{\frac{7}{2}} = 4 - \frac{4}{7} = \frac{24}{7}$$ $$\text{Denominator} = 5 - \frac{3}{2} = \frac{7}{2}$$ $$\text{Term 2} = \frac{\frac{24}{7}}{\frac{7}{2}} = \frac{24}{7} \times \frac{2}{7} = \frac{48}{49}$$ * **Step 3: Simplify Term 3** $$\text{Term 3} = \frac{2}{5} \text{ of } \left\{\frac{6}{9} + \left(\frac{2}{3} \times \frac{1}{2}\right)\right\}$$ $$\text{Inside Bracket} = \frac{2}{3} + \frac{1}{3} = 1$$ $$\text{Term 3} = \frac{2}{5} \times 1 = \frac{2}{5}$$ * **Step 4: Combine all terms** $$x = \frac{9}{7} + \frac{48}{49} - \frac{2}{5}$$ Convert $\frac{9}{7}$ to a denominator of 49: $$\frac{63}{49} + \frac{48}{49} = \frac{111}{49}$$ Now, solve: $$\frac{111}{49} = 2\frac{13}{49}$$ ### Exam Strategy & Shortcut Observe that Term 2 has a denominator of $7 \times 7 = 49$. Since 49 is a prominent unique denominator in the options, checking option (b) first saves significant verification time. ### Common Pitfall Misinterpreting the order of the inner 'of' operator inside the curly braces can completely skew the value of Term 3. Always complete inner operations completely before working outwards. ### Final Answer **Therefore, the correct answer is 2(13/49).**