Compound Interest - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & {\text{Principal = Rs. 5000}} \cr & {\text{Time = 3 years}} \cr & {\text{Rate = 10}}\% {\text{ = }}\frac{1}{{10}} \cr & {\text{Principal}}\,\,\,\,\,\,\,{\text{Amount}} \cr & \,\,\,\,\,\,\,\,\,\,{\text{10}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{11}} \cr & \,\,\,\,\,\,\,\,\,\,{\text{10}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{11}} \cr & \,\,\,\,\,\,\,\,\,\,{\text{10}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{11}} \cr & \underbrace {\overline {\,\,\,\,\,\,1000\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{1331}}\,\,\,\,\,} }_{331{\text{ units}}} \cr & \Rightarrow 1000{\text{ units = Rs 5000}} \cr & \Rightarrow 1{\text{ units = Rs 5}} \cr & \Rightarrow 331{\text{ units = 331}} \times {\text{5}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = Rs. 1655}} \cr} $$

$$\eqalign{ & {\text{Rate = 12}}\% \cr & {\text{Time = 2 years}} \cr & {\text{Effective rate of CI for 2 years}} \cr & {\text{ = 12 + 12 + }}\frac{{12 \times 12}}{{100}} \cr & = 25.44\,\% \cr & {\text{Effective rate of SI for 2 years}} \cr & {\text{ = 12}} \times 2{\text{ = 24}}\,\% \cr & {\text{According to question,}} \cr & {\text{Required SI}} \cr & {\text{ = }}\frac{{2544}}{{25.44}} \times {\text{24}} = {\text{ Rs. 2400}} \cr & {\text{Required sum}} \cr & {\text{ = Rs. 2400}} \cr} $$

$$\eqalign{ & {\text{Amount = Rs}}{\text{. 2916}} \cr & {\text{Time = 2 years }} \cr & {\text{Rate = 8}}\% \cr & {\text{Effective rate }}\% {\text{ CI for 2 years}} \cr & {\text{ = 8 + 8 + }}\frac{{8 \times 8}}{{100}} = 16.64\% \cr & {\text{Required sum}} \cr & {\text{ = }}\frac{{2916}}{{\left( {100 + 16.64} \right)}} \times 100 \cr & = {\text{Rs}}{\text{. }}2500 \cr & {\text{Required simple interest}} \cr & {\text{ = }}\frac{{2500 \times 9 \times 3}}{{100}} \cr & = {\text{Rs}}{\text{. }}675 \cr} $$

$$\eqalign{ & P{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{15}} = 2P \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{15}} = 2 \cr & {\text{Let }}P{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^n} = 8P \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^n} = 8 = {2^3} = {\left\{ {{{\left( {1 + \frac{{\text{R}}}{{100}}} \right)}^{15}}} \right\}^3} \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^n} = {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{45}} \cr & \Rightarrow n = 45 \cr & \therefore {\text{Required time = 45 years}} \cr} $$

$$\eqalign{ & P{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} = 8P \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} = 8 \cr & {\text{Let }}P{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^n} = 16P \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^n} = 16 = {2^4} = {\left( {{2^3}} \right)^{\frac{4}{3}}} \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^n} = {\left( 8 \right)^{\frac{4}{3}}} \cr & \Rightarrow {\left\{ {{{\left( {1 + \frac{{\text{R}}}{{100}}} \right)}^3}} \right\}^{\frac{4}{3}}} = {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^4} \cr & \Rightarrow n = 4 \cr & \therefore {\text{ Required time 4 years}} \cr} $$

$$\eqalign{ & {\text{Principal = Rs 250}} \cr & {{\text{R}}_1} = 4\% ,\,\,\,\,\,\,\,\,\,{{\text{R}}_2} = 8\% \cr & {\text{Amount}}\,{\text{after}}{1^{st}}\,{\text{year}} \cr & = 250\left( {1 + \frac{4}{{100}}} \right) = {\text{Rs}}{\text{. }}260 \cr & {\text{Amount after }}{{\text{2}}^{{\text{nd}}}}{\text{ year}} \cr & {\text{ = }}260\left( {1 + \frac{8}{{100}}} \right) \cr & = {\text{Rs}}{\text{. }}280.80 \cr} $$

For a first result cross check with option (Go with option (C) and check it.)
Principal is Rs. 3200
3200 of 5% for 1^st year = 160
then, principal = 3200 + 160 = 3360
3360 of 5% for 2^nd year = 168
∴ Interest = 160 + 168 = 328

$$\eqalign{ & {\text{Rate of interest 5}}\% \cr & = \frac{1}{{20}} \cr & {\text{Let principal}} \cr & {\text{ = }}{\left( {20} \right)^2}{\text{ = 400 units}} \cr & \Rightarrow {\text{ Total compound interest }} \cr & {\text{41 Units }} \to {\text{Rs. 410 }} \cr & {\text{1 Units }} \to {\text{Rs. 10 }} \cr & {\text{400 Units }} \to {\text{Rs. 400 }} \cr & {\text{Total simple interest}} \cr & {\text{ = Rs. 400}} \cr} $$

Alternate
Total compound interest for 2 years at 5% p.a.
$$\eqalign{ & {\text{ = 5 + 5 + }}\frac{{5 \times 5}}{{100}} = 10.25\% \cr & {\text{Total simple interest}} \cr & = 10\% \cr & \Rightarrow 10.25\% \to 410 \cr & \Rightarrow 10\% \to 400 \cr & {\text{Simple interest}} \cr & {\text{ = Rs}}{\text{. 400}} \cr} $$

$$\eqalign{ & P{\left( {1 + \frac{R}{{100}}} \right)^5} = 150\% \,{\text{of }}P = \frac{3}{2}P \cr & \Rightarrow {\left( {1 + \frac{R}{{100}}} \right)^5} = \frac{3}{2} \cr} $$
  $$x{\left( {1 + \frac{R}{{100}}} \right)^{10}} = y{\left( {1 + \frac{R}{{100}}} \right)^{15}} = $$       $$z{\left( {1 + \frac{R}{{100}}} \right)^{20}}$$
  $$ \Rightarrow x{\left\{ {{{\left( {1 + \frac{R}{{100}}} \right)}^5}} \right\}^2} = $$      $$y{\left\{ {{{\left( {1 + \frac{R}{{100}}} \right)}^5}} \right\}^3} = $$     $$z{\left\{ {{{\left( {1 + \frac{R}{{100}}} \right)}^5}} \right\}^4}$$
$$\eqalign{ & \Rightarrow x \times {\left( {\frac{3}{2}} \right)^2} = y \times {\left( {\frac{3}{2}} \right)^3} = z \times {\left( {\frac{3}{2}} \right)^4} \cr & \Rightarrow \frac{{9x}}{4} = \frac{{27y}}{8} = \frac{{81z}}{{16}} = k({\text{say}}) \cr & \Rightarrow x = \frac{{4k}}{9},y = \frac{{8k}}{{27}},z = \frac{{16k}}{{81}} \cr & \Rightarrow x:y:z = \frac{{4k}}{9}:\frac{{8k}}{{27}}:\frac{{16k}}{{81}} \cr & \Rightarrow x:y:z = 36:24:16 \cr & \Rightarrow x:y:z = 9:6:4 \cr} $$

Let the value of each instalment be Rs. x
Then, (P.W. of Rs. x due 1 year hence) + (P.W. of Rs. x due 2 year hence) + (P.W. of Rs. x due 3 year hence) = 126100
$$\eqalign{ & \Rightarrow \frac{x}{{\left( {1 + \frac{5}{{100}}} \right)}} + \frac{x}{{{{\left( {1 + \frac{5}{{100}}} \right)}^2}}} + \frac{x}{{{{\left( {1 + \frac{5}{{100}}} \right)}^3}}} = 126100 \cr & \Rightarrow \frac{{20x}}{{21}} + \frac{{400x}}{{441}} + \frac{{8000x}}{{9261}} = 126100 \cr & \Rightarrow \frac{{8820x + 8400x + 8000x}}{{9261}} = 126100 \cr & \Rightarrow \frac{{25220x}}{{9261}} = 126100 \cr & \Rightarrow x = \left( {\frac{{126100 \times 9261}}{{25220}}} \right) \cr & \Rightarrow x = 46305 \cr} $$