Compound Interest — Part 2 MCQs with Explanations

$$\eqalign{ & {\text{Principal = Rs 6000}} \cr & {\text{Rate }}\% {\text{ = 10}}\% \cr & {\text{Time (t) = 1}}\frac{1}{2}{\text{ years}} \cr} $$ $$\eqalign{ & {{\text{2}}^{{\text{nd}}}}{\text{ year CI = 660}} \cr & {\text{6 months }}{{\text{2}}^{{\text{nd}}}}{\text{ year CI}} \cr & {\text{ = }}\frac{{660}}{2}{\text{ = 330}} \cr & {\text{Total CI}} \cr & {\text{ = }}\left( {600 + 330} \right) \cr & \,\, = 930 \cr} $$

$$\eqalign{ & {\text{C}}{\text{.I}}{\text{. earned during }}{{\text{1}}^{{\text{st}}}}{\text{ year}} \cr & {\text{= }}\,{\text{Rs}}{\text{.}}\left[ {5000\left( {1 + \frac{5}{{100}}} \right) - 5000} \right]{\text{ }} \cr & = {\text{Rs}}{\text{. }}\left( {5250 - 5000} \right) \cr & = {\text{Rs}}{\text{. 250}} \cr & {\text{Amount after }}{{\text{1}}^{{\text{st}}}}{\text{ year}} \cr & = {\text{Rs}}{\text{. }}\left( {5250 - 20\% {\text{ of }}250} \right) \cr & = {\text{Rs}}{\text{.}}\left( {5250 - 50} \right){\text{ }} \cr & {\text{= }}\,{\text{Rs}}{\text{.}}\,{\text{5200 }} \cr & {\text{C}}{\text{.I}}{\text{. earned during }}{{\text{2}}^{{\text{nd}}}}{\text{ year}} \cr & = {\text{Rs}}{\text{.}}\left[ {5200\left( {1 + \frac{5}{{100}}} \right) - 5200} \right]{\text{ }} \cr & = {\text{Rs}}{\text{. }}\left( {5460 - 5200} \right) \cr & = {\text{Rs}}{\text{.260 }} \cr & {\text{Amount after }}{{\text{2}}^{{\text{nd}}}}{\text{ year}} \cr & {\text{= Rs}}{\text{. }}\left( {5460 - 20\% {\text{ of }}260} \right) \cr & {\text{= Rs}}{\text{. }}\left( {5460 - 52} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{5408 }} \cr & {\text{C}}{\text{.I}}{\text{. earned during }}{{\text{3}}^{{\text{rd}}}}{\text{ year}} \cr & {\text{= Rs}}{\text{. }}\left[ {5408\left( {1 + \frac{5}{{100}}} \right) - 5408} \right] \cr & = {\text{Rs}}{\text{. }}\left( {5678.40 - 5408} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{270}}{\text{.40 }} \cr & {\text{Amount after }}{{\text{3}}^{{\text{rd}}}}{\text{ year}} \cr & = {\text{Rs}}{\text{. }}\left( {5678.40 - 20\% \,{\text{of }}270.40} \right) \cr & = {\text{Rs}}{\text{. }}\left( {5678.40 - 54.08} \right) \cr & = {\text{Rs}}{\text{. 5624}}{\text{.32}} \cr} $$

$$\eqalign{ & {\text{SI for 1 year}} \cr & {\text{ = Rs 260}} \cr & {\text{SI for 2 year}} \cr & {\text{ = 260}} \times {\text{2}} \cr & {\text{ = Rs}}{\text{. 520 }} \cr & {\text{Difference in (CI}} - {\text{SI)}} \cr & \left( {540.80 - 520} \right){\text{ = Rs 20}}{\text{.8}} \cr & {\text{Required rate % }} \cr & {\text{ = }}\frac{{20.8}}{{260}} \times {\text{100}} \cr & {\text{ = 8% }} \cr} $$

$$\eqalign{ & {\text{Rate }}\% {\text{ = 4}}\% \cr & {\text{Time (}}{{\text{t}}_1}) = 2\,{\text{years}} \cr & {\text{SI for 2 years}} \cr & {\text{ = 4}} \times {\text{2 = 8}}\% \cr & {\text{CI for 2 years}} \cr & {\text{ = 4 + 4 + }}\frac{{4 \times 4}}{{100}} \cr & = 8.16\% \cr & \operatorname{Required} \,CI \cr & = \frac{{80}}{8} \times 8.16 \cr & = Rs.\,81.60 \cr} $$

$$\eqalign{ & {\text{Effective rate of CI for 2 years}} \cr & {\text{= 5 + 5 + }}\frac{{5 \times 5}}{{100}} \cr & = 10.25\% \cr & {\text{Effective rate of SI for 3 years}} \cr & {\text{= 6}} \times {\text{3 = 18% }} \cr & {\text{According to question}} \cr & {\text{Required SI}} \cr & {\text{= }}\frac{{246}}{{10.25}} \times 18 \cr & = {\text{Rs. 432}} \cr} $$

$$\eqalign{ & \Rightarrow P\left[ {{{\left( {\frac{{21}}{{20}}} \right)}^3} - 1 - \frac{3}{{20}}} \right] = 122 \cr & \Rightarrow P\left[ {\frac{{{{21}^3} - {{20}^3} - 3 \times {{20}^2}}}{{{{20}^3}}}} \right] = 122 \cr & \Rightarrow P\left[ {\frac{{9261 - 8000 - 1200}}{{8000}}} \right] = 122 \cr & \Rightarrow P \times \frac{{61}}{{8000}} = 122 \cr & \Rightarrow P = \frac{{8000 \times 122}}{{61}} \cr & \Rightarrow P = {\text{Rs}}{\text{.}}\,16000 \cr} $$

$$\eqalign{ & {\text{Let the rate be R}}\% {\text{ p}}{\text{.a}}{\text{.}} \cr & {\text{Then,}} \cr & {\text{2000}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} = 2226.05 \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} = \frac{{222605}}{{200000}} \cr & \, \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} = \frac{{44521}}{{40000}} \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} = {\left( {\frac{{221}}{{200}}} \right)^2} \cr & \Rightarrow 1 + \frac{{\text{R}}}{{100}} = \frac{{211}}{{200}} \cr & \Rightarrow \frac{{\text{R}}}{{100}} = \frac{{11}}{{200}} \cr & \Rightarrow {\text{R}} = \frac{{11}}{2}\% \cr & \Rightarrow {\text{R}} = 5.5\% \cr} $$

$$\eqalign{ & {\text{Amount after three years}} \cr & {\text{ = Rs. 2662}} \cr & {\text{Amount after two years}} \cr & {\text{ = Rs. 2420}} \cr & {\text{Net interest earned in the }}{{\text{3}}^{{\text{rd}}}}{\text{ year}} \cr & {\text{ = }}\,{\text{2662}} - {\text{2420}} \cr & {\text{ = Rs}}{\text{. 242}} \cr & {\text{Rate of interest (r)}} \cr & {\text{ = }}\frac{{242}}{{2420}} \times {\text{100 = 10% }} \cr} $$ (∴ 2 nd year's amount is principal for 3 rd year)

$$\eqalign{ & {\text{Rate of interest}} \cr & {\text{r}} = {\text{12}}\frac{1}{2}\% = \frac{1}{8} \cr} $$ Year Principal Installment ⇒ I 8 ×9 → 9 ×9 ......(i) ⇒ II 64 → 81 ......(ii) Since, installment is equal hence multiply equation (i) by 9 ⇒ Total principal = 72 + 64 = 136 units 136 units → 6800 1 units → 50 81 units → 4050 ⇒ Each installment = Rs. 4050

$$\eqalign{ & {\text{Le the rate be R }}\% {\text{ p}}{\text{.a}}{\text{.}} \cr & {\text{Then,}} \cr & {\text{4000}}\left( {1 + \frac{{{\text{R }}}}{{100}}} \right) = 4320 \cr & \Rightarrow 1 + \frac{{{\text{R }}}}{{100}} = \frac{{4320}}{{4000}} = \frac{{108}}{{100}} \cr & \Rightarrow \frac{{{\text{R }}}}{{100}} = \frac{8}{{100}} \cr & \Rightarrow {\text{R }} = 8 \cr & \therefore {\text{Amount after 3 yeras}} \cr & {\text{ = Rs}}{\text{. }}\left[ {4000 + {{\left( {1 + \frac{8}{{100}}} \right)}^3}} \right] \cr & {\text{ = Rs}}{\text{. }}\left( {4000 \times \frac{{27}}{{25}} \times \frac{{27}}{{25}} \times \frac{{27}}{{25}}} \right) \cr & {\text{ = Rs}}{\text{. }}\left( {\frac{{629856}}{{125}}} \right) \cr & {\text{ = Rs}}{\text{. }}5038.848 \approx 5039 \cr} $$

$$\eqalign{ & {\text{Rate of interest (r)}} \cr & {\text{ = 8}}\frac{3}{4}\% = \frac{7}{{80}} = \frac{{87 \to {\text{ Installment}}}}{{80 \to {\text{Principal}}}} \cr} $$ ⇒ I 80 ×87 → 87 ×87 ......(i) ⇒ II 6400 → 7569 ......(ii) Since, installment is equal hence multiply equation (i) by 87 ⇒ Total principal = 6960 + 6400 = 13360 ⇒ 13360 units = Rs. 13360 ⇒ 1 units = Rs. 1 ⇒ 7569 units = Rs. 7569 ∴ Each installment = Rs. 7569

$$\eqalign{ & {\text{Let the rate be R% p}}{\text{.a}}{\text{. }} \cr & {\text{Then,}} \cr & {\text{10000}}{\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^4} = 14641 \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{200}}} \right)^4} = \frac{{14641}}{{10000}} = {\left( {\frac{{11}}{{10}}} \right)^4} \cr & \Rightarrow 1 + \frac{{\text{R}}}{{200}} = \frac{{11}}{{10}} \cr & \Rightarrow \frac{{\text{R}}}{{200}} = \frac{1}{{10}} \cr & \Rightarrow {\text{R}} = {\text{20% }} \cr} $$

$$\eqalign{ & {\text{Let the time be }}n{\text{ year}} \cr & {\text{Then,}} \cr & {\text{800}} \times {\left( {1 + \frac{5}{{100}}} \right)^{2n}} = 926.10 \cr & \Leftrightarrow {\left( {1 + \frac{5}{{100}}} \right)^{2n}} = \frac{{9261}}{{8000}} \cr & \Leftrightarrow {\left( {\frac{{21}}{{20}}} \right)^{2n}} = {\left( {\frac{{21}}{{20}}} \right)^3} \cr & \Leftrightarrow 2n = 3 \cr & \Leftrightarrow n = \frac{3}{2} \cr & \therefore n = 1\frac{1}{2}{\text{years}} \cr} $$

$$5\% = \frac{1}{{20}} = \frac{{21 \to {\text{ Installment}}}}{{20 \to {\text{ Principal}}}}$$ Year Principal Installment ⇒ I 20 ×21 → 21 ×21 ......(i) ⇒ II 400→ 441 .....(ii) Since, installment is equal, hence multiply equation (i) by 21 ⇒ Total principal = 420 + 400 = 820 ⇒ 820 units = Rs. 12300 ⇒ 1 units = Rs. 15 ⇒ 441 units = Rs. 6615 ∴ Each installment = Rs. 6615

$$\eqalign{ & {\text{Amount = 6000}}{\left( {1 + \frac{5}{{100}}} \right)^2} \cr & \Rightarrow {\text{Amount = 6000}} \times \frac{{21}}{{20}} \times \frac{{21}}{{20}} \cr & \Rightarrow {\text{Amount = Rs. 6615}} \cr} $$

$$\eqalign{ & {\text{C}}{\text{.I}}{\text{.}} \cr & {\text{ = Rs}}{\text{.}}\left[ {4000 \times {{\left( {1 + \frac{{10}}{{100}}} \right)}^2} - 4000} \right] \cr & = {\text{ Rs}}{\text{.}}\left( {4000 \times \frac{{11}}{{10}} \times \frac{{11}}{{10}} - 4000} \right) \cr & = {\text{Rs}}{\text{. 840}} \cr & \therefore {\text{Sum = Rs}}{\text{.}}\left( {\frac{{420 \times 100}}{{3 \times 8}}} \right) \cr & = {\text{Rs}}{\text{. }}1750 \cr} $$

$$\eqalign{ & {\text{Principal}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Amount}} \cr & \,\,\,\,\,\,\,\,\,{\text{1}}\,\,\,\,\,\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,\,\,\,\,\,{\text{4}} \cr & \Rightarrow 4 = 1{\left( {1 + \frac{r}{{100}}} \right)^2} \cr & \Rightarrow 4 = {\left( {1 + \frac{r}{{100}}} \right)^2} \cr & \Rightarrow r = 100\% \cr & \cr & {\text{Alternate}} \cr & {\text{Principal}}\,\,\,\,\,\,\,\,\,\,\,\,{\text{Amount}} \cr & \,\,\,\,\,\,\,\,\,\root 2 \of 1 \,\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,\,\root 2 \of 4 \cr & \,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,\,\,2 \cr & \Rightarrow {\text{Rate of interest}} \cr & {\text{ = }}\frac{{\left( {2 - 1} \right)}}{1} \times 100 = 100\% \cr} $$

When the money is compounded half yearly the effective rate of interest for 6 months = $$\frac{{16}}{2}$$ = 8% = $$\frac{{2}}{25}$$ Let principal = (25) 2 = 625 ⇒ 4 units → 56 ⇒ 1 unit → 14 ⇒ Principal = 14 × 625 = Rs. 8750

Rate % = 10%, Time = 1 year Case (I) : When interest is calculated yearly, Rate = 10% Case (II) : When interest is calculated half yearly $$\eqalign{ & \Rightarrow {\text{New rate }}\% = \frac{{10}}{2} = 5\% \cr & {\text{Time = 1}} \times {\text{2}} = {\text{2 years}} \cr & {\text{Effective rate}}\% \cr & {\text{ = 5 + 5 + }}\frac{{5 \times 5}}{{100}} = 10.25\% \cr & {\text{Difference in rates}} \cr & {\text{ = }}\left( {10.25 - 10} \right)\% = 0.25\% \cr & {\text{According to the question,}} \cr & {\text{0}}{\text{.25% of sum = Rs 180}} \cr & {\text{Sum = }}\frac{{180}}{{0.25}} \times 100 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs. }}72000 \cr} $$

$$\eqalign{ & {\text{Let the rate be R}}\% {\text{ p}}{\text{.a}}{\text{. }} \cr & {\text{then,}} \cr & {\text{25500}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} \cr & = 25500 + 8440.50 \cr & = 33940.50 \cr} $$ $$ \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} = \frac{{33940.50}}{{25500}} = $$ $$\frac{{1331}}{{1000}} = $$ $${\left( {\frac{{11}}{{10}}} \right)^3}$$ $$\eqalign{ & \Rightarrow 1 + \frac{{\text{R}}}{{100}} = \frac{{11}}{{10}} \cr & \Rightarrow \frac{{\text{R}}}{{100}} = \frac{1}{{10}} \cr & \Rightarrow {\text{R}} = 10\,\% \cr & S.I. = {\text{R}}s.\left( {\frac{{25500 \times 10 \times 3}}{{100}}} \right) \cr & = {\text{Rs}}{\text{.}}\,7650 \cr} $$

Let the R% p.a. Then, $$\left[ {26000 \times {{\left( {1 + \frac{{\text{R}}}{{100}}} \right)}^3} - 26000} \right] - $$ $$\left( {\frac{{26000 \times {\text{R}} \times 3}}{{100}}} \right) = $$ $$2994.134$$ $$ \Rightarrow 26000\left[ {{{\left( {1 + \frac{{\text{R}}}{{100}}} \right)}^3} - 1 - \frac{{3{\text{R}}}}{{100}}} \right] = $$ $$2994.134$$ $$ \Rightarrow 26000$$ $$\left[ {\frac{{{{\left( {100 - {\text{R}}} \right)}^3} - 1000000 - 30000{\text{R}}}}{{1000000}}} \right] = $$ $$2994.134$$ $$ \Rightarrow 26\left[ {\left\{ {1000000 + {{\text{R}}^3} + 300{\text{R}}\left( {100 + {\text{R}}} \right) - 1000000 - 30000{\text{R}}} \right\}} \right] = 2994134$$ $$ \Rightarrow {{\text{R}}^3} + 300{{\text{R}}^2} = \frac{{2994134}}{{26}} = $$ $$115159$$ $$ \Rightarrow {{\text{R}}^2}\left( {{\text{R}} + 300} \right) = 115159$$ $$ \Rightarrow {\text{R = 19}}\% $$

$$\eqalign{ & {\text{Let principal = P}} \cr & {\bf{Case (I)}} \cr & {\text{Time = 3 years,}} \cr & {\text{Amount = 8P}} \cr & \Rightarrow 8{\text{P = P}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} \cr & \Rightarrow {\left( 2 \right)^3} = {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} \cr & {\text{Taking cube root of both sides,}} \cr & \Rightarrow {\text{2 = }}\left( {1 + \frac{{\text{R}}}{{100}}} \right) \cr & \Rightarrow {\text{R = 100 }}\% \cr & {\bf{Case (II)}} \cr & {\text{Let after t years it will be 16 times}} \cr & \Rightarrow 16{\text{P = P}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{\text{t}}} \cr & \Rightarrow 16 = {\left( 2 \right)^{\text{t}}} \cr & \Rightarrow {\left( 2 \right)^4} = {\left( 2 \right)^{\text{t}}} \cr & \Rightarrow {\text{t}} = 4 \cr & {\text{Hence required time}} \cr & {\text{(t) = 4 years}} \cr} $$

$$\eqalign{ & {\text{Let}}, \cr & {\text{Principal}} = Rs.\,100\% \cr & {\text{Amount}} = Rs.\,200 \cr & {\text{Rate}} = r\% \cr & {\text{Time}} = 4\,{\text{years}} \cr & {\text{Now}}, \cr & A = P \times {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^n} \cr & 200 = 100 \times {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^4} \cr & 2 = {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^4} - - - - - - \left( i \right) \cr & {\text{If}}\,{\text{sum}}\,{\text{become}}\,{\text{8}}\,{\text{times}}\,{\text{in}}\,{\text{the}}\,{\text{time}}\,n\,{\text{years}} \cr & {\text{then,}} \cr & 4 = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} \cr & {2^2} = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} - - - - - - \left( {ii} \right) \cr & {\text{Using}}\,{\text{eqn}}\,\left( i \right)in\left( {ii} \right),\,{\text{we}}\,{\text{get}} \cr & {\left( {{{\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]}^4}} \right)^2} = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} \cr & {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^{8}} = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} \cr & {\text{Thus}},\,n = 8\,{\text{years}}. \cr} $$

$$\eqalign{ & {\text{Principal = Rs}}{\text{. 30000}} \cr & {\text{CI = Rs 4347}} \cr & {\text{Rate = 7}}\% \cr & {\text{By using formula, }} \cr & \Rightarrow \left( {30000 + 4347} \right) = 30000{\left( {1 + \frac{7}{{100}}} \right)^{\text{t}}} \cr & \Rightarrow 34347 = 30000{\left( {1 + \frac{7}{{100}}} \right)^{\text{t}}} \cr & \Rightarrow \frac{{34347}}{{30000}} = {\left( {\frac{{107}}{{100}}} \right)^{\text{t}}} \cr & \Rightarrow \left( {\frac{{11449}}{{10000}}} \right) = {\left( {\frac{{107}}{{100}}} \right)^{\text{t}}} \cr & \Rightarrow {\left( {\frac{{107}}{{100}}} \right)^2} = {\left( {\frac{{107}}{{100}}} \right)^{\text{t}}} \cr & \Rightarrow {\text{t}} = 2\,{\text{years}} \cr} $$

$$\eqalign{ & {\text{Let the sum Rs}}{\text{. }}x{\text{ }} \cr & {\text{Then,}} \cr & {\text{C}}{\text{.I}}{\text{. when compounded half yearly}} \cr & {\text{ = Rs}}{\text{.}}\left[ {x \times {{\left( {1 + \frac{3}{{100}}} \right)}^2} - x} \right] \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{10609}}{{10000}}x - x} \right) \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{609x}}{{10000}}} \right) \cr & {\text{C}}{\text{.I}}{\text{. when compounded yearly}} \cr & {\text{ = Rs}}{\text{.}}\left[ {x \times \left( {1 + \frac{4}{{100}}} \right) - x} \right] \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{26x}}{{25}} - x} \right) \cr & = {\text{Rs}}{\text{.}}\frac{x}{{25}} \cr & \therefore \frac{{609x}}{{10000}} - \frac{x}{{25}} = 104.50 \cr & \Rightarrow \frac{{209x}}{{10000}} = 104.50 \cr & \Rightarrow x = \left( {\frac{{104.50 \times 10000}}{{209}}} \right) \cr & \Rightarrow x = 5000 \cr} $$

$$\eqalign{ & {\text{Principal = Rs 8000}} \cr & {\text{Amount = Rs 8820}} \cr & {\text{Let Rate = }}R \cr & {\text{Time = 2 years}} \cr & {\text{By using formula, }} \cr & \Rightarrow 8820 = 8000{\left( {1 + \frac{R}{{100}}} \right)^2} \cr & \Rightarrow \frac{{8820}}{{8000}} = {\left( {1 + \frac{R}{{100}}} \right)^2} \cr & \Rightarrow \frac{{441}}{{400}} = {\left( {1 + \frac{R}{{100}}} \right)^2} \cr & {\text{Taking square root of both sides,}} \cr & \Rightarrow \frac{{21}}{{20}} = \left( {1 + \frac{R}{{100}}} \right) \cr & \Rightarrow R = 5\% \cr} $$

$$\eqalign{ & {\text{Rate = 10}}\% \cr & {\text{Time = 2 years}} \cr & {\text{Effective rate of CI for 2 years}} \cr & {\text{ = 10 + 10 + }}\frac{{10 \times 10}}{{100}} \cr & = 21\% \cr & {\text{Effective rate of SI for 2 years}} \cr & {\text{ = 2}} \times {\text{10 = 20}}\% \cr & {\text{Required SI}} \cr & {\text{ = }}\frac{{420}}{{21}} \times {\text{20 = Rs. 400}} \cr} $$

x becomes 3x in 3 years Therefore 3x also becomes 9x in 3 years Required years = 3 + 3 = 6

Let the share of the younger and elder sons be Rs. x and Rs. (16400 - x) Then, amount of Rs. x after 3 years = Amount of Rs. (16400 - x) after 2 years $$\eqalign{ & \Rightarrow x{\left( {1 + \frac{5}{{100}}} \right)^3} = \left( {16400 - x} \right){\left( {1 + \frac{5}{{100}}} \right)^2} \cr & \Rightarrow x\left( {1 + \frac{5}{{100}}} \right) = \left( {16400 - x} \right) \cr & \Rightarrow \frac{{21x}}{{20}} + x = 16400 \cr & \Rightarrow \frac{{41x}}{{20}} = 16400 \cr & \Rightarrow x = \left( {\frac{{16400 \times 20}}{{41}}} \right) \cr & \Rightarrow x = 8000 \cr} $$

$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. on Rs}}{\text{. 672 for 1 year}} \cr & {\text{ = Rs}}{\text{. }}\left( {714 - 672} \right) \cr & {\text{ = Rs}}{\text{. 42}} \cr & \therefore {\text{Rate = }}\left( {\frac{{100 \times 42}}{{672 \times 1}}} \right){\text{% }} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = 6}}{\text{.25% }} \cr} $$

Here, b – a = 3 – 2 = 1 B = Rs. 1728, A = Rs.1440 $$\eqalign{ & R\% = \left( {\frac{B}{A} - 1 \times 100} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\, = \left( {\frac{{1728}}{{1440}} - 1 \times 100} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\, = \left( {\frac{{288}}{{1440}} \times 100} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\, = 20\% \cr} $$

$$\eqalign{ & {\text{Required rate }}\% \cr & {\text{ = }}\frac{{\left( {238.50 - 225} \right)}}{{225}} \times 100 \cr & = 6\,\% \cr} $$

$$\eqalign{ & {\text{Rate }} \Rightarrow {\text{ 10% = }}\frac{1}{{10}} \cr & {\text{Each installment of 2 years}} \cr & \Rightarrow \frac{{10}}{{11}} \times \frac{{\left( {10 + 11} \right)}}{{11}} \times {\text{ Installment = P}}{\text{.A}} \cr & {\text{P}}{\text{.A = 21000}} \cr & {\text{Each installment = 12100}} \cr} $$

S.I. on Rs. 4624 for 1 year $$\eqalign{ & {\text{ = Rs. }}\left( {4913 - 4624} \right) \cr & {\text{ = Rs. 289}} \cr & \therefore {\text{Rate}} = \left( {\frac{{100 \times 289}}{{4624 \times 1}}} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 6\frac{1}{4}\% \cr & {\text{Now,}} \cr & x{\left( {1 + \frac{{25}}{{400}}} \right)^2} = 4624 \cr & \Rightarrow x \times \frac{{17}}{{16}} \times \frac{{17}}{{16}} = 4624 \cr & \Rightarrow x = \left( {4624 \times \frac{{16}}{{17}} \times \frac{{16}}{{17}}} \right) \cr & \Rightarrow x = 4096 \cr} $$

$$\eqalign{ & 12000 \times {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^5} = 24000 \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^5} = 2 \cr & \therefore {\left[ {{{\left( {1 + \frac{{\text{R}}}{{100}}} \right)}^5}} \right]^4} = {2^4} = 16 \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{20}} = 16 \cr & \Rightarrow {\text{P}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{20}}{\text{ = 16P}} \cr & \Rightarrow 12000{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{20}} = 16 \times 12000 \cr & \Rightarrow 12000{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{20}} = 192000 \cr} $$

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

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

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

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

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