Simplification - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & \frac{{x + 1}}{{x - 1}}{\text{ = }}\frac{a}{b}{\text{ }} \cr & {\text{By componendo and dividendo}} \cr & \frac{{x + 1 + x - 1}}{{x + 1 - x + 1}} = \frac{{a + b}}{{a - b}} \cr & \Rightarrow \frac{{2x}}{2} = \frac{{a + b}}{{a - b}} \cr & \Rightarrow x = \frac{{a + b}}{{a - b}}.....(i) \cr & {\text{Again ,}}\frac{{1 - y}}{{1 + y}}{\text{ = }}\frac{b}{a}{\text{ }} \cr & \Rightarrow \frac{{1 + y}}{{1 - y}} = \frac{a}{b} \cr & \Rightarrow \frac{{1 + y + 1 - y}}{{1 + y - 1 + y}} = \frac{{a + b}}{{a - b}} \cr & \Rightarrow \frac{2}{{2y}} = \frac{{a + b}}{{a - b}} \cr & \Rightarrow \frac{1}{y} = \frac{{a + b}}{{a - b}} \cr & \Rightarrow y = \frac{{a - b}}{{a + b}}.....(ii) \cr & {\text{Subtracting equation (ii) from (i) we get }} \cr & \therefore x - y = \frac{{a + b}}{{a - b}} - \frac{{a - b}}{{a + b}} \cr & \left[ {\because {{\left( {a + b} \right)}^2} - {{\left( {a - b} \right)}^2} = 4ab\,\,\,and\,\,{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)} \right] \cr & = \frac{{{{\left( {a + b} \right)}^2} - {{\left( {a - b} \right)}^2}}}{{\left( {a - b} \right)\left( {a + b} \right)}} \cr & = \frac{{4ab}}{{{a^2} - {b^2}}} \cr & {\text{Multiply equation (i) and (ii) we get}} \cr & xy = \frac{{a + b}}{{a - b}} \times \frac{{a - b}}{{a + b}} = 1 \cr & \therefore {\text{Expression,}} \cr & {\text{ = }}\frac{{x - y}}{{1 + xy}} = \frac{{\frac{{4ab}}{{{a^2} - {b^2}}}}}{{1 + 1}} \cr &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{4ab}}{{2\left( {{a^2} - {b^2}} \right)}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{2ab}}{{{a^2} - {b^2}}} \cr} $$

200 ÷ 25 × 4 + 12 - 3
= $$\frac{{200}}{{25}}$$ × 4 + 12 - 3
= 8 × 4 + 12 - 3
= 32 + 12 - 3
= 44 - 3
= 41

$$\eqalign{ & {\text{Let the number be }}x \cr & {\text{Given,}} \cr & \frac{{\left( {923 - 347} \right)}}{x} = 32{\text{ }} \cr & \Rightarrow x = \frac{{576}}{{32}} = 18 \cr & \Rightarrow x = 18 \cr} $$

Let the number be a
Given,
1559.95 - 7.99 × 24.96 - a² = 1154
⇒ 1560 - 8 × 25 - a² = 1154
⇒ 1360 - 200 - a² = 1154
⇒ a² = 1360 - 1154
⇒ a² = 206
⇒ a ≈ $$\sqrt {206} $$
⇒ a ≈ 14

$$\eqalign{ & {\text{Given, }} \cr & {\text{1}}\frac{4}{5} + 20 - 280 \div 25 \cr & = \frac{9}{5} + 20 - \frac{{280}}{{25}} \cr & = \frac{9}{5} + 20 - \frac{{56}}{5} \cr & = \frac{{9 + 100 - 56}}{5} \cr & = \frac{{53}}{5} \cr & = 10\frac{3}{5} \cr} $$

$$\eqalign{ & {\text{Given}}, \cr & \left\{ {\left( {64 - 38} \right) \times 4} \right\} \div 13 \cr & = \frac{{26 \times 4}}{{13}} \cr & = 8 \cr} $$

$$\eqalign{ & {\text{Given, }} \cr & x + y = 2a \cr & \therefore \frac{a}{{x - a}} + \frac{a}{{y - a}} \cr & = \frac{{a\left( {y - a} \right) + a\left( {x - a} \right)}}{{\left( {x - a} \right)\left( {y - a} \right)}} \cr & = \frac{{ay - {a^2} + ax - {a^2}}}{{\left( {x - a} \right)\left( {y - a} \right)}} \cr & = \frac{{a\left( {x + y} \right) - 2{a^2}}}{{\left( {x - a} \right)\left( {y - a} \right)}} \cr & = \frac{{a.2a - 2{a^2}}}{{\left( {x - a} \right)\left( {y - a} \right)}} \cr & = \frac{{2{a^2} - 2{a^2}}}{{\left( {x - a} \right)\left( {y - a} \right)}} \cr & = 0 \cr} $$

$$\eqalign{ & {\text{Let the number be }}x \cr & {\text{Given }} \cr & 421 \div 35 \times 299.97 \div 25.05 = {x^2} \cr & \approx {\left( x \right)^2} = 420 \div 35 \times 300 \div 25 \cr & \approx 12 \times 300 \div 25 = 12 \times 12 \cr & \approx {x^2} = 12 \times 12 \cr & \therefore x = \sqrt {12 \times 12} = 12 \cr & {\text{Hence the number is 12}} \cr} $$

$$\,\,\,\,\,\,\root 3 \of { - 2197} \times \root 3 \of { - 125} \div \root 3 \of {\frac{{27}}{{512}}} \,$$
  $$ = \root 3 \of {13 \times 13 \times 13 \times - 1} \,\, \times $$     $$\sqrt {5 \times 5 \times 5 \times - 1} \,\,\div $$     $$\root 3 \of {\frac{{27}}{{512}}} $$
$$\eqalign{ & = \left( {13 \times 5\root 3 \of { - 1 \times - 1} } \right) \div \frac{3}{8} \cr & = 13 \times 5 \times \frac{8}{3} \cr & = \frac{{520}}{3} \cr} $$

Given Expression
128.43 + 30.21 + ? = 173
⇒ ? = 173 - 128.43 - 30.21
⇒ ? = 173 - 158.64
⇒ ? = 14.36