Simplification - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & 0.51\overline {345} \cr & = \frac{{51345 - 51}}{{99900}} \cr & = \frac{{51294}}{{99900}} \cr} $$

$$\eqalign{ & {\text{A}} = 3\frac{1}{4} \times 4\frac{1}{4} \div 34 - \frac{{47}}{{32}} + \frac{{47}}{{16}} \cr & {\text{B}} = 2\frac{1}{2} + 5\frac{1}{2} \div 55 - \frac{{11}}{{10}} \cr & \therefore {\text{A}} - {\text{B}} = \left( {3\frac{1}{4} \times 4\frac{1}{4} \div 34 - \frac{{47}}{{32}} + \frac{{47}}{{16}}} \right) - \left( {2\frac{1}{2} + 5\frac{1}{2} \div 55 - \frac{{11}}{{10}}} \right) \cr & = \left( {\frac{{13}}{4} \times \frac{{17}}{4} \times \frac{1}{{34}} - \frac{{47}}{{32}} + \frac{{47}}{{16}}} \right) - \left( {\frac{5}{2} + \frac{{11}}{2} \times \frac{1}{{55}} - \frac{{11}}{{10}}} \right) \cr & = \left( {\frac{{13}}{{32}} - \frac{{47}}{{32}} + \frac{{47}}{{16}}} \right) - \left( {\frac{5}{2} + \frac{1}{{10}} - \frac{{11}}{{10}}} \right) \cr & = \left( {\frac{{13 - 47 + 94}}{{32}}} \right) - \left( {\frac{{25 + 1 - 11}}{{10}}} \right) \cr & = \frac{{60}}{{32}} - \frac{{15}}{{10}} \cr & = \frac{{15}}{8} - \frac{3}{2} \cr & = \frac{{15 - 12}}{8} \cr & = \frac{3}{8} \cr} $$

$$\eqalign{ & 0.\overline {57} - 0.4\overline {32} + 0.3\overline 5 \cr & = 0.\overline {57} - 0.4\overline {32} + 0.3\overline {55} \cr & = \frac{{57}}{{99}} - \frac{{432 - 4}}{{990}} + \frac{{355 - 3}}{{990}} \cr & = \frac{{570 - 428 + 352}}{{990}} \cr & = 0.4\overline {94} \cr} $$

3.352 - (9.759 - x) - 19.64 = 7.052
⇒ 3.352 - 9.759 + x - 19.64 - 7.052 = 0
⇒ x = 36.451 - 3352
∴ x = 33.099

$$\eqalign{ & \frac{8}{9}{\text{ of}}\left( {5\frac{1}{4} \div 2\frac{1}{3}{\text{ of }}4} \right) \div \left( {8 \div \frac{2}{3}{\text{ of }}\frac{4}{5}} \right){\text{of}}\left( {8 \times \frac{2}{3} \div \frac{4}{5}} \right) \cr & = \frac{8}{9}{\text{ of}}\left( {\frac{{21}}{4} \div \frac{7}{3} \times 4} \right) \div \left( {8 \div \frac{8}{{15}}} \right){\text{of}}\left( {8 \times \frac{2}{3} \times \frac{5}{4}} \right) \cr & = \frac{8}{9}{\text{ of}}\left( {\frac{{21}}{4} \times \frac{3}{{28}}} \right) \div \left( {8 \times \frac{{15}}{8}} \right){\text{of}}\left( {\frac{{20}}{3}} \right) \cr & = \left( {\frac{8}{9} \times \frac{9}{{16}}} \right) \div 100 \cr & = \frac{1}{{200}} \cr} $$

$$\eqalign{ & \frac{{\left\{ {\left( {{m^2} + {n^2}} \right)\left( {m - n} \right) - {{\left( {m - n} \right)}^3}} \right\}}}{{\left( {{m^2}n - m{n^2}} \right)}} \cr & = \frac{{\left( {m - n} \right)\left[ {{m^2} + {n^2} - {{\left( {m - n} \right)}^2}} \right]}}{{mn\left( {m - n} \right)}} \cr & = \frac{{{m^2} + {n^2} - {m^2} - {n^2} + 2mn}}{{mn}} \cr & = \frac{{2mn}}{{mn}} \cr & = 2{\text{ Answer}} \cr} $$

56 × 75 × 60 × 84 × 210 = 2^p × 3^q × 5^r × 7^s
7 × 2³ × 5² × 3 × 2² × 3 × 5 × 2² × 3 × 7 × 7 × 3 × 2 × 5 = 2^p × 3^q × 5^r × 7^s
2⁸ × 3⁴ × 5⁴ × 7³ = 2^p × 3^q × 5^r × 7^s
∴ p = 8
q = 4
r = 4
s = 3
$$\eqalign{ & \therefore \left[ {\frac{{\left( {{\text{p}} + {\text{q}}} \right)}}{{\text{s}}}} \right] + {\text{r}} \cr & = \left[ {\frac{{\left( {8 + 4} \right)}}{3}} \right] + 4 \cr & = \frac{{12}}{3} + 4 \cr & = 4 + 4 \cr & = 8 \cr} $$

$$\eqalign{ & \frac{{24 \div \frac{3}{8}{\text{of}}\left( {8 + 2 \times \overline {7 - 3} } \right) + \left[ {\frac{2}{{11}} \div \frac{4}{{55}} - \left\{ {\frac{5}{8} + \frac{6}{{16}}} \right\}} \right]}}{{32 \div \overline {15 - 7} + 75 \div \left( {6 + 15 \div 3 + 4} \right)}} \cr & = \frac{{\left[ {24 \div \frac{3}{8}{\text{of}}\left( {8 + 2 \times 4} \right) + \left( {\frac{2}{{11}} \times \frac{{55}}{4} - \left\{ {\frac{{16}}{{16}}} \right\}} \right)} \right]}}{{32 \div 8 + 75 \div \left( {6 + 5 + 4} \right)}} \cr & = \frac{{\left[ {24 \div 6 + \left( {\frac{5}{2} - 1} \right)} \right]}}{{\left[ {4 + 75 \div 15} \right]}} \cr & = \frac{{\left[ {4 + \frac{3}{2}} \right]}}{{\left[ {4 + 5} \right]}} \cr & = \frac{{\left[ {\frac{{11}}{2}} \right]}}{9} \cr & = \frac{{11}}{{18}} \cr} $$

$$\eqalign{ & \frac{{45}}{{53}} = \frac{1}{{a + \frac{1}{{b + \frac{1}{{c - \frac{2}{5}}}}}}} \cr & \frac{{53}}{{45}} = 1 + \frac{8}{{45}},\,\frac{{45}}{8} = 5 + \frac{5}{8},\,\frac{8}{5} = 2 - \frac{2}{5} \cr & a = 1 \cr & b = 5 \cr & c = 2 \cr & 4a - b + 3c \cr & = 4 \times 1 - 5 + 3 \times 2 \cr & = 4 - 5 + 6 \cr & = 5 \cr} $$

$$\eqalign{ & \left( {1\frac{1}{3} \div 2\frac{6}{7}{\text{ of }}5\frac{3}{5}} \right) \div \left( {6\frac{2}{5} \div 4\frac{1}{2}{\text{ of}}\,5\frac{1}{3}} \right) \times \left( {\frac{3}{4} \times 2\frac{2}{3} \div \frac{5}{9}{\text{ of }}1\frac{1}{5}} \right) = 1 + {\text{k}} \cr & \left( {\frac{4}{3} \div \frac{{20}}{7}{\text{ of}}\frac{{28}}{5}} \right) \div \left( {\frac{{32}}{5} \div \frac{9}{2}{\text{ of}}\,\frac{{16}}{3}} \right) \times \left( {\frac{3}{4} \times \frac{8}{3} \div \frac{5}{9}{\text{ of }}\frac{6}{5}} \right) = 1 + {\text{k}} \cr & \left( {\frac{4}{3} \div 4 \times 4} \right) \div \left( {\frac{{32}}{5} \div 24} \right) \times \left( {\frac{3}{4} \times \frac{8}{3} \div \frac{2}{3}} \right) = 1 + {\text{k}} \cr & \left( {\frac{4}{3} \times \frac{1}{{16}}} \right) \div \left( {\frac{4}{5} \times \frac{1}{3}} \right) = 1 + {\text{k}} \cr & \frac{1}{{12}} \div \frac{4}{{15}} \times 3 = 1 + {\text{k}} \cr & \frac{1}{{12}} \times \frac{{15}}{4} \times 3 = 1 + {\text{k}} \cr & \frac{{15}}{{16}} - 1 = {\text{k}} \cr & \frac{{ - 1}}{{16}} = {\text{k}} \cr & - 0.0625 = {\text{k}} \cr & - 0.07 < {\text{k}} > - 0.0625 \cr} $$