Simplification - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & {81^{\frac{3}{4}}} + \left[ {\left( {20 \div 5\,{\text{of }}3 \times 6} \right) + \left\{ {\left( {8 \div 24\,{\text{of }}3} \right) \times 4} \right\} - 10 \div 5} \right] - {\left( {\frac{1}{{32}}} \right)^{ - \frac{2}{5}}} \cr & = 27 + \left[ {\left( {20 \div 15 \times 6} \right) + \left\{ {\left( {8 \div 72} \right) \times 4} \right\} - 2} \right] - 4 \cr & = 27 + \left[ {8 + \frac{4}{9} - 2} \right] - 4 \cr & = 23 + \frac{{58}}{9} \cr & = \frac{{265}}{9} \cr & = 29\frac{4}{9} \cr} $$

$$\eqalign{ & \frac{1}{{4.263}} = 0.2346 \cr & \frac{{1000}}{{4263}} = 0.2346 \cr & \frac{1}{{0.0004263}} \cr & = \frac{{10000000}}{{4263}} \cr & = 10000 \times 0.2346 \cr & = 2346 \cr} $$

$$\eqalign{ & 7.5 + \left( {5.4 \div 4.5 \times 2} \right) - 8 \times 4 \div 3.2 \cr & = 7.5 + \left( {\frac{6}{5} \times 2} \right) - \frac{8}{{0.8}} \cr & = 7.5 + 2.4 - 10 \cr & = - 0.1 \cr} $$

$$\eqalign{ & 5\left( {1 - \frac{x}{5}} \right) - \left( {5 - x} \right) - \frac{1}{{200}}\,{\text{of }}\left( {20 - x} \right) = 0.08 \cr & \Rightarrow 5 - x - 5 + x - \frac{{20 - x}}{{200}} = 0.08 \cr & \Rightarrow - \frac{{20 - x}}{{200}} = 0.08 \cr & \Rightarrow x - 20 = 0.08 \times 200 \cr & \Rightarrow x = 16 + 20 \cr & \Rightarrow x = 36 \cr} $$

$$\eqalign{ & \left( {2.\overline 4 \times 2.\overline 6 \times 3 \times 0.1\overline 6 } \right) \times \left[ {0.\overline {27} \times \left( {0.8\overline 3 \div 0.1\overline 6 } \right)} \right] \cr & = \left( {\frac{{22}}{9} \times \frac{6}{9} \times 3 \times \frac{{15}}{{90}}} \right) \times \left[ {\frac{{27}}{{99}} \times \left( {\frac{{75}}{{90}} \div \frac{{15}}{{90}}} \right)} \right] \cr & = \frac{{22}}{{27}} \times \left[ {\frac{{27}}{{99}} \times \left( {\frac{{75}}{{90}} \times \frac{{90}}{{15}}} \right)} \right] \cr & = \frac{{22}}{{27}} \times \frac{{27}}{{99}} \times 5 \cr & = \frac{{10}}{9} \cr & = 1.\overline 1 \cr} $$

$$\eqalign{ & 3800 \times \frac{{25}}{9} \times 15\% \times 18\% \cr & = 3800 \times \frac{{25}}{9} \times \frac{3}{{20}} \times \frac{{18}}{{100}} \cr & = 19 \times 15 \cr & = 285 \cr} $$

1 + 3 + 5 + 7 + . . . . . . (2n - 1)
We know that such of N odd number = n² = n × n

$$\eqalign{ & \left( {5\frac{1}{4} \div \frac{3}{7}\,{\text{of }}\frac{1}{2}} \right) \div \left( {5\frac{1}{9} - 7\frac{7}{8} \div 9\frac{9}{{20}}} \right) \times \frac{{11}}{{21}} - \left( {5 \div 2\,{\text{of }}\frac{1}{2}} \right) \cr & = \left( {\frac{{21}}{4} \times \frac{{14}}{3}} \right) \div \left( {\frac{{46}}{9} - \frac{{63}}{8} \times \frac{{20}}{{189}}} \right) \times \frac{{11}}{{21}} - 5 \cr & = \frac{{49}}{2} \times \frac{{18}}{{77}} \times \frac{{11}}{{21}} - 5 \cr & = 3 - 5 \cr & = - 2 \cr} $$

$$\eqalign{ & \frac{{0.75}}{{10}} + \frac{{0.31}}{{100}} + \frac{{0.09}}{{1000}} \cr & = \frac{{75}}{{1000}} + \frac{{3.1}}{{1000}} + \frac{{0.09}}{{1000}} \cr & = \frac{{78.19}}{{10000}} \cr & = 0.07819 \cr} $$

$$\eqalign{ & \frac{3}{{70}} + \frac{1}{{42}} + \frac{1}{{66}} + \frac{3}{{286}} + \frac{1}{{130}} + \frac{1}{{170}} \cr & = \left( {\frac{3}{{70}} + \frac{1}{{42}}} \right) + \left( {\frac{1}{{66}} + \frac{3}{{286}}} \right) + \left( {\frac{1}{{130}} + \frac{1}{{170}}} \right) \cr & = \left( {\frac{{9 + 5}}{{70 \times 3}}} \right) + \left( {\frac{{13 + 9}}{{286 \times 3}}} \right) + \left( {\frac{{17 + 13}}{{130 \times 17}}} \right) \cr & = \left( {\frac{{14}}{{70 \times 3}}} \right) + \left( {\frac{{22}}{{286 \times 3}}} \right)\left( {\frac{{30}}{{130 \times 17}}} \right) \cr & = \left( {\frac{1}{{15}}} \right) + \left( {\frac{1}{{13 \times 3}}} \right) + \left( {\frac{3}{{13 \times 17}}} \right) \cr & = \left( {\frac{1}{{15}}} \right) + \left( {\frac{{17 + 9}}{{13 \times 3 \times 17}}} \right) \cr & = \left( {\frac{1}{{15}}} \right) + \left( {\frac{{26}}{{13 \times 3 \times 17}}} \right) \cr & = \left( {\frac{1}{{15}}} \right) + \left( {\frac{2}{{3 \times 17}}} \right) \cr & = \left( {\frac{{17 + 10}}{{85 \times 3}}} \right) \cr & = \frac{{27}}{{85 \times 3}} \cr & = \frac{9}{{85}} \cr} $$