The value of $\frac{2}{3} \times \frac{3}{\frac{5}{6} \div \frac{2}{3} \text{ of } 1\frac{1}{4}}$ is

Concept & Logic This problem tests the application of the BODMAS rule within complex fractions. The critical rule here is that the 'of' operator (which mathematically means multiplication) has a higher priority than the division symbol ($\div$). You must evaluate the 'of' operation in the denominator before dividing. Step-by-Step Solution * **Step 1: Focus on the denominator's 'of' operation** The denominator is: $$\frac{5}{6} \div \frac{2}{3} \text{ of } 1\frac{1}{4}$$ Convert the mixed fraction to an improper fraction: $$1\frac{1}{4} = \frac{5}{4}$$ Evaluate the 'of' segment first: $$\frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$$ * **Step 2: Simplify the full denominator** Substitute the result back into the denominator: $$\frac{5}{6} \div \frac{5}{6}$$ Any number divided by itself equals $1$. So, the entire denominator reduces to $1$. * **Step 3: Evaluate the final expression** Substitute the simplified denominator back into the original complex fraction: $$\frac{2}{3} \times \frac{3}{1}$$ Multiply the fractions: $$\frac{2 \times 3}{3 \times 1} = \frac{6}{3} = 2$$ Exam Strategy & Shortcut When facing complex fractions, isolate and solve the most heavily nested portion first (in this case, the bottom right). By recognizing that $\frac{2}{3}$ of $1.25$ is exactly $\frac{5}{6}$, the denominator instantly collapses to $1$. The entire problem then visually simplifies to just $\frac{2}{3} \times 3$, which you can solve mentally in seconds. Common Pitfall The most common mistake is reading left-to-right and performing the division before the 'of' operator in the denominator (i.e., calculating $\frac{5}{6} \div \frac{2}{3}$ first). This violates BODMAS and will lead to an entirely incorrect, messy fraction. Final Answer Therefore, the correct answer is **2**.