Simplification - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & 2 = x + \frac{1}{{1 + \frac{1}{{5 + \frac{1}{2}}}}} \cr & 2 = x + \frac{1}{{1 + \frac{1}{{\frac{{11}}{2}}}}} \cr & 2 = x + \frac{1}{{\frac{{11 + 2}}{{11}}}} \cr & 2 = x + \frac{{11}}{{13}} \cr & 2 - \frac{{11}}{{13}} = x \cr & x = \frac{{15}}{{13}} \cr} $$

$$\eqalign{ & \frac{2}{3} \div \frac{3}{{10}}{\text{ of }}\frac{4}{9} - \frac{4}{5} \times 1\frac{1}{9} \div \frac{8}{{15}} + \frac{3}{4} \div \frac{1}{2} \cr & = \frac{2}{3} \div \frac{2}{{15}} - \frac{4}{5} \times \frac{{10}}{9} \times \frac{{15}}{8} + \frac{3}{4} \times 2 \cr & = \frac{2}{3} \times \frac{{15}}{2} - \frac{5}{3} + \frac{3}{2} \cr & = 5 - \frac{5}{3} + \frac{3}{2} \cr & = \frac{{29}}{6} \cr} $$

$$\eqalign{ & 15\frac{3}{5} \times \frac{ * }{3} = 26 \cr & \frac{{78}}{5} \times \frac{ * }{3} = 26 \cr & * = 5{\text{ Answer}} \cr} $$

$$\begin{array}{l}7\frac{1}{2}\times \left(3\frac{1}{5}÷4\frac{1}{2}\text{ of }5\frac{1}{3}\right)+\left[11-\left(\frac{5}{8}+3-1\frac{1}{4}\right)÷5\frac{3}{4}-5÷5\times 5\text{of }5÷25\right]\\ =\frac{15}{2}\left(\frac{16}{5}÷\frac{9}{2}\text{ of }\frac{16}{3}\right)+\left[11-\left(\frac{5}{8}+3-\frac{3}{4}\right)÷\frac{23}{4}-5÷5\times \underset{}{\underbracket{5\text{of }5}}÷25\right]\\ =\frac{15}{2}\left(\frac{16}{5}÷24\right)+\left[11-\left(\frac{5}{8}+\frac{24-10}{8}\right)\right]÷\frac{23}{4}-5÷5\\ =\frac{15}{2}\left(\frac{2}{15}\right)+\left[\frac{88-19}{8}\right]÷\frac{23}{4}-1\\ =1+\frac{69}{8}\times \frac{4}{23}-1\\ =\frac{3}{2}\\ =1\frac{1}{2}\end{array}$$

$$\eqalign{ & \frac{{x + 3}}{{2x + 4 - 3}} = \frac{2}{3} \cr & 3x + 9 = 4x + 2 \cr & x = 7 \cr & {\text{Original fraction}} \cr & \frac{x}{{2x + 4}} = \frac{7}{{18}} > 11 \cr} $$
Differentiate between numerator and denumerator = 11

$$\eqalign{ & 0.4\overline 6 + 0.7\overline {23} - 0.3\overline 9 \times 0.\overline 7 \cr & = \frac{{46 - 4}}{{90}} + \frac{{723 - 7}}{{990}} - \frac{{39 - 3}}{{90}} \times \frac{7}{9} \cr & = \frac{{42}}{{90}} + \frac{{716}}{{990}} - \frac{{36}}{{90}} \times \frac{7}{9} \cr & = \frac{{42}}{{90}} + \frac{{716}}{{990}} - \frac{{28}}{{90}} \cr & = \frac{{14}}{{90}} + \frac{{716}}{{990}} \cr & = \frac{{154 + 716}}{{990}} \cr & = \frac{{870}}{{990}} \cr & = 0.\overline {87} \cr} $$

$$\eqalign{ & \frac{{\left( {\frac{2}{5}{\text{ of }}6\frac{1}{4} \div \frac{3}{7}} \right) \times 1\frac{2}{7}}}{{11\frac{1}{4}}} = \sqrt x \cr & \frac{{\left( {\frac{2}{5}{\text{ of }}\frac{{25}}{4} \div \frac{3}{7}} \right) \times \frac{9}{7}}}{{\frac{{45}}{4}}} = \sqrt x \cr & \frac{{\frac{5}{2} \times \frac{7}{3} \times \frac{9}{7}}}{{\frac{{45}}{4}}} = \sqrt x \cr & \frac{{15}}{2} \times \frac{4}{{45}} = \sqrt x \cr & x = \frac{4}{9} \cr & 81x = 81 \times \frac{4}{9} \cr & 81x = 36 \cr} $$

$$\eqalign{ & \frac{{132 \div 12 \times x - 3 \times 3}}{{{5^2} - 6 \times 4 + {x^2}}} = 1 \cr & 11 \times x - 9 = 25 - 24 + {x^2} \cr & 11x - 9 = 1 + {x^2} \cr & {x^2} + 1 + 9 - 11x = 0 \cr & {x^2} - 11x + 10 = 0 \cr & {x^2} - 10x - x + 10 = 0 \cr & x\left( {x - 10} \right) - 1\left( {x - 10} \right) = 0 \cr & \left( {x - 10} \right)\left( {x - 1} \right) = 0 \cr & x = 10,\,1 \cr & {\text{For the condition}}\left( {x < 10} \right) \cr & x = 1 \cr} $$

$$\eqalign{ & \frac{{\root 3 \of { - 2744} \times \root 3 \of { - 216} }}{{\root 3 \of {\frac{{64}}{{729}}} }} \cr & = \frac{{\left( { - 14} \right) \times \left( { - 6} \right)}}{{\frac{4}{9}}} \cr & = \frac{{84}}{{\frac{4}{9}}} \cr & = 189 \cr} $$

$$\eqalign{ & \frac{{1.6 \times 1.6 \times 1.6 - 0.6 \times 0.6 \times 0.6}}{{1.6 \times 1.6 + 1.6 \times 0.6 + 0.6 \times 0.6}} \cr & = \frac{{{{\left( {1.6} \right)}^3} - {{\left( {0.6} \right)}^3}}}{{{{\left( {1.6} \right)}^2} + 1.6 \times 0.6 + {{\left( {0.6} \right)}^2}}} \cr & = \frac{{\left( {1.6 - 0.6} \right)\left\{ {{{\left( {1.6} \right)}^2} + 1.6 \times 0.6 + {{\left( {0.6} \right)}^2}} \right\}}}{{{{\left( {1.6} \right)}^2} + 1.6 \times 0.6 + {{\left( {0.6} \right)}^2}}} \cr & = 1.6 - 0.6 \cr & = 1 \cr} $$