Simplification - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & {\text{Let the fractions }}a{\text{ and }}b \cr & ab = - 4 \cr & a + b = \frac{{ - 32}}{{15}}{\text{ }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}\left( 1 \right) \cr & {\left( {a + b} \right)^2} = {\left( {a - b} \right)^2} + 4ab \cr & {\left( {\frac{{ - 32}}{{15}}} \right)^2} = {\left( {a - b} \right)^2} + 4\left( { - 4} \right) \cr & {\left( {a - b} \right)^2} = \frac{{1024}}{{225}} + 16 \cr & a - b = \sqrt {\frac{{4624}}{{225}}} \cr & a - b = \frac{{68}}{{15}}{\text{ }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}\left( 2 \right) \cr & {\text{Equate equation }}\left( 1 \right){\text{ and }}\left( 2 \right){\text{,}}\,{\text{we get}} \cr & a = \frac{{18}}{{15}} = \frac{6}{5} \cr & b = \frac{{ - 100}}{{30}} = \frac{{ - 10}}{3} \cr} $$

$$\eqalign{ & 25 - \left[ {16 - \left\{ {14 - \left( {18 - \overline {8 + 3} } \right)} \right\}} \right] \cr & = 25 - \left[ {16 - \left\{ {14 - \left( {18 - 8 - 3} \right)} \right\}} \right] \cr & = 25 - \left[ {16 - \left\{ {14 - \left( {18 - 11} \right)} \right\}} \right] \cr & = 25 - \left[ {16 - \left\{ {14 - 7} \right\}} \right] \cr & = 25 - \left[ {16 - 7} \right] \cr & = 25 - 9 \cr & = 16{\text{ Answer}} \cr} $$

$$\eqalign{ & \frac{{\left( {187 \div 17 \times a - 3 \times 3} \right)}}{{\left( {{8^2} - 9 \times 7 + {a^2}} \right)}} = 1 \cr & 11a - 9 = 64 - 63 + {a^2} \cr & 11a - 9 = 1 + {a^2} \cr & {a^2} - 11a + 10 = 0 \cr & {a^2} - \left( {10 + 1} \right)a + 10 = 0 \cr & {a^2} - 10a - a + 10 = 0 \cr & a\left( {a - 10} \right) - 1\left( {a - 10} \right) = 0 \cr & \left( {a - 10} \right)\left( {a - 1} \right) = 0 \cr & a = 1,\,10{\text{ Answer}} \cr} $$

15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5
= 15.2 + 2 × 2 - 3.5 × 9
= 15.2 + 4 - 14
= 5.2

$$\eqalign{ & \frac{{27 \times {{\left( {0.25} \right)}^3} + 125{{\left( {0.05} \right)}^3}}}{{{{\left( {0.75} \right)}^2} - 0.25 \times 0.5}} \cr & = \frac{{\frac{{27 \times 25 \times 25 \times 25}}{{1000000}} + \frac{{125 \times 5 \times 5 \times 5}}{{1000000}}}}{{\frac{{75 \times 75}}{{10000}} - \frac{{25 \times 5}}{{1000}}}} \cr & = \frac{{\frac{{27 \times 25 \times 25 \times 25}}{{1000}} + \frac{{125 \times 5 \times 5 \times 5}}{{1000}}}}{{\frac{{75 \times 75 - 25 \times 5 \times 10}}{{10}}}} \cr & = \frac{{\frac{{27}}{8} + \frac{1}{8}}}{{\frac{{45 - 10}}{{10}}}} \cr & = \frac{{280}}{{8 \times 35}} \cr & = 1 \cr} $$

$$\eqalign{ & 0.\overline {45} \times 1.\overline {22} \cr & = \frac{{45}}{{99}} \times \frac{{122 - 1}}{{99}} \cr & = \frac{{45}}{{99}} \times \frac{{121}}{{99}} \cr & = \frac{5}{9} \cr & = 0.\overline 5 {\text{ Answer}} \cr} $$

$$\eqalign{ & 22.\overline 4 + 11.5\overline {67} - 33.5\overline 9 \cr & = 22 + \frac{4}{9} + 11 + \frac{{567 - 5}}{{990}} - 33 - \frac{{59 - 5}}{{90}} \cr & = 33 - 33 + \frac{4}{9} + \frac{{562}}{{990}} - \frac{{54}}{{90}} \cr & = \frac{{40}}{{90}} - \frac{{54}}{{90}} + \frac{{562}}{{990}} \cr & = \frac{{ - 14}}{{90}} + \frac{{562}}{{990}} \cr & = \frac{{ - 154 + 562}}{{990}} \cr & = \frac{{408}}{{990}} \cr & = 0.4\overline {12} \cr} $$

$$\eqalign{ & 309 \div \left[ {\left( {\frac{3}{2}} \right){\text{of}}\left( {25 + 35} \right) - 12\frac{3}{4}} \right] \cr & = 309 \div \left[ {\frac{3}{2} \times 60 - \frac{{51}}{4}} \right] \cr & = 309 \div \left[ {90 - \frac{{51}}{4}} \right] \cr & = 309 \times \frac{4}{{360 - 51}} \cr & = 4 \cr} $$

$$\eqalign{ & \frac{{2{\text{abc}}}}{9} \cr & = \frac{{2 \times - 12 \times - 6 \times 18}}{9} \cr & = 288 \cr} $$

$$\eqalign{ & M = \frac{3}{7} \div \frac{6}{5} \times \frac{2}{3} + \frac{1}{5} \times \frac{3}{2} \cr & = \frac{3}{7} \times \frac{5}{6} \times \frac{2}{3} + \frac{3}{{10}} \cr & = \frac{5}{{21}} + \frac{3}{{10}} \cr & = \frac{{50 + 63}}{{210}} \cr & = \frac{{113}}{{210}} \cr & N = \frac{2}{5} \times \frac{5}{6} \div \frac{1}{3} + \frac{3}{5} \times \frac{2}{3} \div \frac{3}{5} \cr & = \frac{2}{5} \times \frac{5}{6} \times \frac{3}{1} + \frac{3}{5} \times \frac{2}{3} \times \frac{5}{3} \cr & = 1 + \frac{2}{3} \cr & = \frac{5}{3} \cr & \frac{M}{N} = \frac{{113}}{{210}} \div \frac{5}{3} \cr & = \frac{{113}}{{210}} \times \frac{3}{5} \cr & = \frac{{113}}{{350}} \cr} $$