$5 - \left[ \frac{3}{4} + \left\{ 2\frac{1}{2} - \left( 0.5 + \overline{\frac{1}{6} - \frac{1}{7}} \right) \right\} \right]$ is equal to

### Concept & Strategy This question tests the strict application of the **BODMAS/VBODMAS** rule, specifically dealing with nested grouping symbols. The order of evaluation for brackets is: 1. Vinculum (Line Bracket): $\overline{a - b}$ 2. Parentheses (Round Brackets): $( )$ 3. Braces (Curly Brackets): $\{ \}$ 4. Square Brackets: $[ ]$ ### Step-by-Step Solution * **Step 1: Evaluate the Vinculum (Line Bracket)** $$\overline{\frac{1}{6} - \frac{1}{7}} = \frac{7 - 6}{42} = \frac{1}{42}$$ * **Step 2: Evaluate the Parentheses** Substitute the vinculum result back into the round brackets: $$\left(0.5 + \frac{1}{42}\right)$$ Convert $0.5$ to $\frac{1}{2}$ to find a common denominator: $$\frac{1}{2} + \frac{1}{42} = \frac{21}{42} + \frac{1}{42} = \frac{22}{42} = \frac{11}{21}$$ * **Step 3: Evaluate the Curly Braces** Substitute the parentheses result into the curly braces. Convert $2\frac{1}{2}$ to an improper fraction: $$\left\{ \frac{5}{2} - \frac{11}{21} \right\}$$ Find the LCM of $2$ and $21$ ($42$): $$\frac{5 \times 21}{42} - \frac{11 \times 2}{42} = \frac{105 - 22}{42} = \frac{83}{42}$$ * **Step 4: Evaluate the Square Brackets** Add the result to $\frac{3}{4}$: $$\left[ \frac{3}{4} + \frac{83}{42} \right]$$ Find the LCM of $4$ and $42$ ($84$): $$\frac{3 \times 21}{84} + \frac{83 \times 2}{84} = \frac{63 + 166}{84} = \frac{229}{84}$$ * **Step 5: Final Subtraction** Subtract the entire bracket result from $5$: $$5 - \frac{229}{84}$$ Convert $5$ into a fraction with a denominator of $84$: $$\frac{420}{84} - \frac{229}{84} = \frac{191}{84}$$ Convert to a mixed number: $$\frac{191}{84} = 2\frac{23}{84}$$ ### Exam Strategy & Shortcut Track the denominators as you peel back the layers. Notice that $6 \times 7 = 42$, and the interaction with $4$ leads to an LCM of $84$. Because all options share the denominator $84$, focus purely on executing the numerator arithmetic flawlessly without worrying about simplifying fractions midway. ### Common Pitfall Ignoring the Vinculum (the bar over the fraction). Many students miss this and simply read the expression from left to right as $\left(0.5 + \frac{1}{6}\right) - \frac{1}{7}$, which completely changes the order of operations and yields an incorrect result. ### Final Answer **Therefore, the correct answer is 2(23/84).**