The expression $$\frac{3\frac{1}{4} - \frac{4}{5} \text{ of } \frac{5}{6}}{4\frac{1}{3} \div \frac{1}{5} - \left(\frac{3}{10} + 21\frac{1}{5}\right)}$$ is equal to

### Concept & Formula Simplify components methodically by applying the **BODMAS** sequence. Convert mixed fractions to improper fractions at the start. ### Step-by-Step Solution * **Step 1: Simplify the Numerator** $$\text{Numerator} = 3\frac{1}{4} - \frac{4}{5} \text{ of } \frac{5}{6}$$ Convert to improper fraction: $3\frac{1}{4} = \frac{13}{4}$ Evaluate the 'of' operation: $$\frac{4}{5} \times \frac{5}{6} = \frac{4}{6} = \frac{2}{3}$$ Subtract the terms: $$\frac{13}{4} - \frac{2}{3} = \frac{39 - 8}{12} = \frac{31}{12}$$ * **Step 2: Simplify the Denominator** $$\text{Denominator} = 4\frac{1}{3} \div \frac{1}{5} - \left(\frac{3}{10} + 21\frac{1}{5}\right)$$ First term of denominator: $$\frac{13}{3} \div \frac{1}{5} = \frac{13}{3} \times 5 = \frac{65}{3}$$ Bracket term of denominator: $$\frac{3}{10} + \frac{106}{5} = \frac{3 + 212}{10} = \frac{215}{10} = \frac{43}{2}$$ Combine the denominator terms: $$\frac{65}{3} - \frac{43}{2} = \frac{130 - 129}{6} = \frac{1}{6}$$ * **Step 3: Compute the Final Value** $$\text{Value} = \frac{\frac{31}{12}}{\frac{1}{6}} = \frac{31}{12} \times 6 = \frac{31}{2} = 15\frac{1}{2}$$ ### Exam Strategy & Shortcut When resolving fractions with simple denominators like 2 and 3 in the final subtraction steps, look for quick subtraction setups (e.g., $130 - 129 = 1$), which simplifies the entire denominator into a single unit fraction ($\frac{1}{6}$) instantly. ### Common Pitfall Forgetting to find a common denominator when adding numbers inside the parentheses ($\frac{3}{10} + \frac{106}{5}$) often introduces early calculation errors. ### Final Answer **Therefore, the correct answer is 15(1/2).**