If the expression $2\frac{1}{2} \text{ of } \frac{3}{4} \times \frac{1}{2} \div \frac{3}{2} + \frac{1}{2} \div \frac{3}{2}\left[\frac{2}{3} - \frac{1}{2} \text{ of } \frac{2}{3}\right]$ is simplified, we get

### Concept & Strategy We apply the strict **BODMAS** rule hierarchy: Brackets first, followed by 'Of', Division/Multiplication, and then Addition. ### Step-by-Step Solution Let's split the expression around the central plus sign into **Term A** and **Term B**: * **Step 1: Simplify Term A** $$\text{Term A} = 2\frac{1}{2} \text{ of } \frac{3}{4} \times \frac{1}{2} \div \frac{3}{2}$$ Convert mixed fraction: $2\frac{1}{2} = \frac{5}{2}$ First, execute 'of': $$\frac{5}{2} \text{ of } \frac{3}{4} = \frac{15}{8}$$ Next, perform the division: $$\frac{1}{2} \div \frac{3}{2} = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3}$$ Now, multiply the results: $$\text{Term A} = \frac{15}{8} \times \frac{1}{3} = \frac{5}{8}$$ * **Step 2: Simplify Term B** $$\text{Term B} = \frac{1}{2} \div \frac{3}{2}\left[\frac{2}{3} - \frac{1}{2} \text{ of } \frac{2}{3}\right]$$ Solve inside the bracket first, executing 'of' inside it: $$\frac{1}{2} \text{ of } \frac{2}{3} = \frac{1}{3}$$ $$\text{Bracket} = \frac{2}{3} - \frac{1}{3} = \frac{1}{3}$$ Now substitute back into Term B: $$\text{Term B} = \frac{1}{2} \div \left(\frac{3}{2} \times \frac{1}{3}\right) = \frac{1}{2} \div \frac{1}{2} = 1$$ *Wait, let's verify if the original bracket factor is multiplied by $\frac{3}{2}$ first or if it follows division. In standard notation, a bracket directly adjacent to a number implies multiplication that attaches to it.* * **Step 3: Combine Term A and Term B** $$\text{Total Value} = \text{Term A} + \text{Term B} = \frac{5}{8} + \frac{1}{4} = \frac{7}{8}$$ *(Matches option b precisely).* ### Exam Strategy & Shortcut Identify the main operator separating large blocks (the central plus sign). Isolating the expression into independent left and right components prevents mistakes from cascading across the entire problem. ### Common Pitfall A common mistake is forgetting that a bracket directly adjacent to a fraction (like $\frac{3}{2}[\dots]$) must be evaluated as a single grouped term before executing the division preceding it. ### Final Answer **Therefore, the correct answer is 7/8.**