If $\text{I} = \frac{3}{4} \div \frac{5}{6}$, $\text{II} = 3 \div [(4 \div 5) \div 6]$, $\text{III} = [3 \div (4 \div 5)] \div 6$, $\text{IV} = 3 \div 4 \div (5 \div 6)$, then

Concept & Logic This problem evaluates how the placement of brackets alters the outcome of chained division operations. The core principle remains: division must follow BODMAS, executing operations inside brackets first, and transforming division by a fraction into multiplication by its reciprocal. Step-by-Step Solution * **Evaluate Expression I:** $$\text{I} = \frac{3}{4} \div \frac{5}{6} = \frac{3}{4} \times \frac{6}{5} = \frac{18}{20} = \frac{9}{10}$$ * **Evaluate Expression II:** $$\text{II} = 3 \div [(\frac{4}{5}) \div 6]$$ $$\text{II} = 3 \div [\frac{4}{5} \times \frac{1}{6}] = 3 \div [\frac{4}{30}]$$ $$\text{II} = 3 \times \frac{30}{4} = \frac{90}{4} = \frac{45}{2}$$ * **Evaluate Expression III:** $$\text{III} = [3 \div (\frac{4}{5})] \div 6$$ $$\text{III} = [3 \times \frac{5}{4}] \div 6 = \frac{15}{4} \div 6$$ $$\text{III} = \frac{15}{4} \times \frac{1}{6} = \frac{15}{24} = \frac{5}{8}$$ * **Evaluate Expression IV:** $$\text{IV} = 3 \div 4 \div (\frac{5}{6})$$ Evaluate left to right first: $$(3 \div 4) = \frac{3}{4}$$ $$\text{IV} = \frac{3}{4} \div \frac{5}{6} = \frac{3}{4} \times \frac{6}{5} = \frac{18}{20} = \frac{9}{10}$$ * By comparing the simplified results, we see that $\text{I} = \frac{9}{10}$ and $\text{IV} = \frac{9}{10}$. Exam Strategy & Shortcut Notice that Expression IV is $3 \div 4 \div (\frac{5}{6})$. Due to the left-to-right rule, $3 \div 4$ groups naturally into $\frac{3}{4}$. The expression essentially becomes $\frac{3}{4} \div \frac{5}{6}$, which is identical character-for-character to Expression I. Recognizing this structural identicality saves you from having to calculate II and III entirely. Common Pitfall Assuming division is associative is a critical error. Evaluating Expression IV as $3 \div (4 \div \frac{5}{6})$ instead of $(3 \div 4) \div \frac{5}{6}$ will result in a completely incorrect answer. Final Answer Therefore, the correct answer is **I and IV are equal**.