Simplification - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & {\text{Given expression,}} \cr & \frac{{28 + 14 + 7 + 4 + 2 + 1}}{{28}} \cr & = \frac{{56}}{{28}} \cr & = 2 \cr} $$

$$\eqalign{ & \frac{1}{x} = 4 - \left( {\frac{1}{3} + \frac{1}{2}} \right) \cr & \,\,\,\,\,\, = 4 - {\frac{{2 + 3}}{6}} \cr & \,\,\,\,\,\, = 4 - \frac{5}{6} \cr & \,\,\,\,\,\, = \frac{{24 - 5}}{6} \cr & \,\,\,\,\,\, = \frac{{19}}{6} \cr & \Rightarrow x = \frac{6}{{19}} \cr} $$

$$\eqalign{ & {\text{According to the question,}} \cr & \frac{{\sqrt {80} - \sqrt {112} }}{{\sqrt {45} - \sqrt {63} }} \cr & \Rightarrow \frac{{4\sqrt 5 - 4\sqrt 7 }}{{3\sqrt 5 - 3\sqrt 7 }} \cr & \Rightarrow \frac{{4\left( {\sqrt 5 - \sqrt 7 } \right)}}{{3\left( {\sqrt 5 - \sqrt 7 } \right)}} \cr & \Rightarrow \frac{4}{3} \cr & \Rightarrow 1\frac{1}{3} \cr} $$

$$\eqalign{ & {\text{According to the question}} \cr & x + y = 22\,......(i) \cr & {x^2} + {y^2} = 404\,......(ii){\text{ }} \cr & \therefore {\left( {x + y} \right)^2} = {x^2} + {y^2} + 2xy \cr & \Rightarrow {\left( {22} \right)^2} = 404 + 2xy \cr & \Rightarrow 484 = 404 + 2xy \cr & \Rightarrow 2xy = 80 \cr & \Rightarrow xy = 40 \cr} $$

$$\eqalign{ & {\text{According to the question,}} \cr & {\text{8}}\frac{1}{2} - \left[ {3\frac{1}{4} + \left\{ {1\frac{1}{4} - \frac{1}{2}\left( {1\frac{1}{2} - \frac{1}{3} - \frac{1}{6}} \right)} \right\}} \right] \cr & \Rightarrow \frac{{17}}{2} - \left[ {\frac{{13}}{4} + \left\{ {\frac{5}{4} - \frac{1}{2}\left( {\frac{3}{2} - \frac{1}{3} - \frac{1}{6}} \right)} \right\}} \right] \cr & \Rightarrow \frac{{17}}{2} - \left[ {\frac{{13}}{4} + \left\{ {\frac{5}{4} - \frac{1}{2}\left( {\frac{{9 - 2 - 1}}{6}} \right)} \right\}} \right] \cr & \Rightarrow \frac{{17}}{2} - \left[ {\frac{{13}}{4} + \left\{ {\frac{5}{4} - \frac{1}{2} \times 1} \right\}} \right] \cr & \Rightarrow \frac{{17}}{2} - \left[ {\frac{{13}}{4} + \left\{ {\frac{{5 - 2}}{4}} \right\}} \right] \cr & \Rightarrow \frac{{17}}{2} - \left[ {\frac{{13}}{4} + \frac{3}{4}} \right] \cr & \Rightarrow \frac{{17}}{2} - \frac{{16}}{4} \cr & \Rightarrow \frac{{34 - 16}}{4} \cr & \Rightarrow \frac{9}{2} \cr & \Rightarrow 4\frac{1}{2} \cr} $$

According to the question,
$$\eqalign{ & \frac{{25 - 5\left[ {2 + 3\left\{ {2 - 2\left( {5 - 3} \right) + 5} \right\} - 10} \right]}}{4} \cr & = \frac{{25 - 5\left[ {2 + 3\left\{ {2 - 2 * 2 + 5} \right\} - 10} \right]}}{4} \cr & = \frac{{25 - 5\left[ {2 + 3\left\{ {2 - 4 + 5} \right\} - 10} \right]}}{4} \cr & = \frac{{25 - 5\left[ {2 + 3 * 3 - 10} \right]}}{4} \cr & = \frac{{25 - 5\left[ {11 - 10} \right]}}{4} \cr & = 25 - \frac{5}{4} \cr & = \frac{{100 - 5}}{4} \cr & = \frac{{95}}{4} \cr & = 23.75 \cr} $$

$$\eqalign{ & {\text{Given Expression,}} \cr & {\frac{3}{7} \times 455 + \frac{5}{8} \times 456} \cr & = \left( {3 \times 65 + 5 \times 57} \right) \cr & = 195 + 285 \cr & = 480 \cr} $$

$$\eqalign{ & {\frac{{41}}{6} \times \frac{{16}}{3} + \frac{{53}}{3} \times \frac{9}{2}} \cr & = {\frac{{328}}{9} + \frac{{159}}{2}} \cr & = \frac{{656 + 1431}}{{18}} \cr & = \frac{{2087}}{{18}} \cr & = 115\frac{{17}}{{18}} \cr} $$

$$\eqalign{ & {\text{Given expression}} \cr & \frac{{24}}{5} \div \frac{{32}}{5} \cr & = \frac{{24}}{5} \times \frac{5}{{32}} \cr & = \frac{3}{4} \cr} $$

$$\eqalign{ & x = \frac{1}{{2 + \frac{1}{2}}} = \frac{2}{5} \cr & \Rightarrow \frac{1}{x} = \frac{5}{2} \cr} $$