Simplification - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & \frac{1}{{x + \frac{1}{{y + \frac{2}{{z + \frac{1}{4}}}}}}} = \frac{{29}}{{79}} \cr & \frac{1}{{2 + \frac{1}{{1 + \frac{2}{{5 + \frac{1}{4}}}}}}} = \frac{{29}}{{79}} \cr & x = 2,\,y = 1,\,z = 5 \cr & \left( {2x + 3y - z} \right) \cr & = 2 \times 2 + 3 \times 1 - 5 \cr & = 2 \cr} $$

$$\eqalign{ & x \times \frac{1}{4} \times \frac{1}{5} = 35 \cr & x = 35 \times 20 \cr & x = 700 \cr & 700 \times \frac{7}{8} = \frac{{4900}}{8} = 612.5 \cr} $$

$$\eqalign{ & 9\frac{4}{9} \div 11\frac{1}{3}{\text{ of }}\frac{1}{6} + \left( {1\frac{1}{3} \times 1\frac{4}{5} \div \frac{3}{5}} \right) \times 2\frac{1}{6}{\text{ of }}\frac{2}{3} \div \frac{4}{3}{\text{ of }}\frac{2}{3} \cr & = \frac{{85}}{9} \div \frac{{34}}{3}{\text{ of }}\frac{1}{6} + \left( {\frac{4}{3} \times \frac{9}{5} \div \frac{3}{5}} \right) \times \frac{{13}}{6}{\text{ of }}\frac{2}{3} \div \frac{4}{3}{\text{ of }}\frac{2}{3} \cr & = \frac{{85}}{9} \times \frac{{18}}{{34}} + \left( 4 \right) \times \frac{{13}}{9} \times \frac{9}{8} \cr & = 5 + \frac{{13}}{2} \cr & = \frac{{23}}{2} \cr & {\text{Difference of answer}} = \frac{{77}}{4} - \frac{{46}}{4} \cr & = \frac{{31}}{4} \cr & = 7\frac{3}{4} \cr} $$

$$\eqalign{ & \frac{{9 \div \frac{3}{7}{\text{ of }}\left( {9 + 6 \times \overline {4 - 2} } \right) + \left[ {\frac{1}{5} \div \frac{7}{{25}} - \left\{ {\frac{5}{8} + \frac{6}{{16}}} \right\}} \right]}}{{24 \div \overline {16 - 10} + 36 \div \left( {5 + 20 \div 4 - 1} \right)}} \cr & = \frac{{9 \div \frac{3}{7}{\text{ of }}\left( {9 + 6 \times 2} \right) + \left[ {\frac{1}{5} \times \frac{{25}}{7} - \left\{ {\frac{{10 + 6}}{{16}}} \right\}} \right]}}{{24 \div 6 + 36 \div \left( {5 + 5 - 1} \right)}} \cr & = \frac{{9 \div \frac{3}{7}{\text{ of }}\left( {21} \right) + \left[ {\frac{5}{7} - 1} \right]}}{{4 + 36 \div 9}} \cr & = \frac{{9 + \frac{{63}}{7} + \left( {\frac{{ - 2}}{7}} \right)}}{{4 + 4}} \cr & = \frac{{1 - \frac{2}{7}}}{8} \cr & = \frac{{\frac{5}{7}}}{8} \cr & = \frac{5}{{56}} \cr} $$

$$\eqalign{ & 2.\overline {13} \div 3.\overline {12} \cr & = \frac{{213 - 2}}{{99}} \div \frac{{312 - 3}}{{99}} \cr & = \frac{{211}}{{99}} \times \frac{{99}}{{309}} \cr & = \frac{{211}}{{309}}{\text{ Answer}} \cr} $$

$$\eqalign{ & \left[ {\frac{4}{7}{\text{ of }}2\frac{4}{5} \times 1\frac{2}{3} - \left( {3\frac{1}{2} - 2\frac{1}{6}} \right)} \right] \div \left[ {\left( {3\frac{1}{5} \div 4\frac{1}{2}{\text{ of }}5\frac{1}{3}} \right)} \right] \cr & = \left[ {\frac{4}{7}{\text{ of }}\frac{{14}}{5} \times \frac{5}{3} - \left( {\frac{7}{2} - \frac{{13}}{6}} \right)} \right] \div \left[ {\left( {\frac{{16}}{5} \div \frac{9}{2}{\text{ of }}\frac{{16}}{3}} \right)} \right] \cr & = \left[ {\frac{8}{3} - \left( {\frac{{21 - 13}}{6}} \right)} \right] \div \left[ {\left( {\frac{{16}}{5} \div 24} \right)} \right] \cr & = \left[ {\frac{8}{3} - \frac{8}{6}} \right] \div \left[ {\frac{{16}}{5} \times \frac{1}{{24}}} \right] \cr & = \frac{4}{3} \div \frac{2}{{15}} \cr & = \frac{4}{3} \times \frac{{15}}{2} \cr & = 10 \cr} $$

$$\eqalign{ & 3\frac{1}{5} \div 4\frac{1}{2}{\text{ of }}5\frac{1}{3} + \frac{1}{8} \div \frac{1}{2}{\text{ of }}\frac{1}{4} - \frac{1}{4}\left( {\frac{1}{2} \div \frac{1}{8} \times \frac{1}{4}} \right) \cr & = 3\frac{1}{5} \div \left( {\frac{9}{2} \times \frac{{16}}{3}} \right) + \frac{1}{8} \div \frac{1}{8} - \frac{1}{4} \cr & = \frac{{16}}{5} \times \frac{1}{{24}} + 1 - \frac{1}{4} \cr & = \frac{2}{{15}} + \frac{3}{4} \cr & = \frac{{8 + 45}}{{60}} \cr & = \frac{{53}}{{60}} \cr} $$

$$\eqalign{ & \frac{{1\frac{2}{3} \div \frac{5}{6} \times 6 + \frac{4}{5} \times \frac{1}{2} + \frac{2}{3}}}{{2 - \left[ {1\frac{1}{3} \times \left( { - \frac{3}{5}} \right) - 6\left\{ {\frac{3}{5} - \left( {3 - \frac{3}{{10}}} \right)} \right\}} \right]}} \cr & = \frac{{\frac{5}{3} \times \frac{6}{5} \times 6 + \frac{4}{{10}} + \frac{2}{3}}}{{2 - \left[ {\frac{4}{3} \times \frac{{ - 3}}{5} - 6\left\{ {\frac{3}{5} - \frac{{27}}{{10}}} \right\}} \right]}} \cr & = \frac{{12 + \frac{{6 + 10}}{{15}}}}{{2 - \left( { - \frac{4}{5} + \frac{{21 \times 6}}{{10}}} \right)}} \cr & = \frac{{12 + \frac{{16}}{{15}}}}{{2 - \left( { - \frac{4}{5} + \frac{{63}}{5}} \right)}} \cr & = \frac{{\frac{{196}}{{15}}}}{{2 - \frac{{59}}{5}}} \cr & = \frac{{196}}{{15}} \times \frac{{ - 5}}{{49}} \cr & = \boxed{ - \frac{4}{3}} \cr} $$

$$\eqalign{ & \left[ {5\frac{4}{9} \div {{\left( {\frac{{11}}{4} - \frac{{13}}{6}} \right)}^2}} \right] \div {\left[ {7\frac{3}{{11}}{\text{ of }}8\frac{4}{5} \div 1\frac{5}{7} - \frac{4}{3}} \right]^2} \cr & = \left[ {\frac{{49}}{9} \div {{\left( {\frac{{33 - 26}}{{12}}} \right)}^2}} \right] \div {\left[ {\frac{{80}}{{11}}{\text{ of }}\frac{{44}}{5} \div \frac{{12}}{7} - \frac{4}{3}} \right]^2} \cr & = \left[ {\frac{{49}}{9} \times \frac{{{{12}^2}}}{{49}}} \right] \div {\left[ {64 \times \frac{7}{{12}} - \frac{4}{3}} \right]^2} \cr & = \left[ {\frac{{49}}{9} \times \frac{{144}}{{49}}} \right] \div {\left[ {\frac{{112}}{3} - \frac{4}{3}} \right]^2} \cr & = \frac{{16 \times 3 \times 3}}{{108 \times 108}} \cr & = \frac{{16 \times 9}}{{108 \times 108}} \cr & = \frac{1}{{81}} \cr} $$

$$\eqalign{ & \frac{{40 + \frac{3}{4}{\text{ of }}32}}{{37 + \frac{3}{4}{\text{ of }}\left( {34 - 6} \right)}} \cr & = \frac{{40 + 24}}{{37 + 21}} \cr & = \frac{{64}}{{58}} \cr & = \frac{{32}}{{29}} \cr & = 1\frac{3}{{29}} \cr} $$