$$\eqalign{ & {\text{If}} \cr & {\text{I = }}\frac{3}{4} \div \frac{5}{6}{\text{,}} \cr & {\text{II = 3}} \div \left[ {\left( {4 \div 5} \right) \div 6} \right]{\text{,}} \cr & {\text{III = }}\left[ {{\text{3}} \div \left( {4 \div 5} \right)} \right] \div {\text{6,}} \cr & {\text{IV = 3}} \div {\text{4}} \div \left( {5 \div 6} \right), \cr & {\text{Then - }} \cr} $$

Solution (By Examveda Team)

$$\eqalign{ & {\text{I = }}\frac{3}{4} \div \frac{5}{6} \cr & \,\,\,\, = \frac{3}{4} \times \frac{6}{5} \cr & \,\,\,\, = \frac{9}{{10}} \cr & {\text{II = 3}} \div \left[ {\left( {4 \div 5} \right) \div 6} \right] \cr & \,\,\,\,\,\, = 3 \div \left( {\frac{4}{5} \times \frac{1}{6}} \right) \cr & \,\,\,\,\,\, = 3 \div \frac{4}{{30}} \cr & \,\,\,\,\,\, = 3 \times \frac{{30}}{4} \cr & \,\,\,\,\,\, = \frac{{45}}{2} \cr & {\text{III = }}\left[ {3 \div \left( {4 \div 5} \right)} \right] \div {\text{6}} \cr & \,\,\,\,\,\,\,\,{\text{ = }}\left[ {3 \div \frac{4}{5}} \right] \div 6 \cr & \,\,\,\,\,\,\,\, = \left[ {3 \times \frac{5}{4}} \right] \div 6 \cr & \,\,\,\,\,\,\,\, = \frac{{15}}{4} \times \frac{1}{6} \cr & \,\,\,\,\,\,\,\, = \frac{5}{8} \cr & {\text{IV = 3}} \div 4 \div \left( {5 \div 6} \right) \cr & \,\,\,\,\,\,\,\, = 3 \div 4 \div \frac{5}{6} \cr & \,\,\,\,\,\,\,\, = \frac{3}{4} \times \frac{6}{5} \cr & \,\,\,\,\,\,\,\, = \frac{9}{{10}} \cr & {\text{So, I and IV are equal}}{\text{.}} \cr} $$