Compound Interest — Part 1 MCQs with Explanations

$$\eqalign{ & {\text{Amount}} \cr & = {1600 \times {{\left( {1 + \frac{5}{{2 \times 100}}} \right)}^2} + 1600 \times \left( {1 + \frac{5}{{2 \times 100}}} \right)} \cr & = {1600 \times \frac{{41}}{{40}} \times \frac{{41}}{{40}} + 1600 \times \frac{{41}}{{40}}} \cr & = {1600 \times \frac{{41}}{{40}}\left( {\frac{{41}}{{40}} + 1} \right)} \cr & = {\frac{{1600 \times 41 \times 81}}{{40 \times 40}}} \cr & = Rs.\,3321 \cr & \therefore C.I. = Rs.\,\left( {3321 - 3200} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,121 \cr} $$

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{sum}}\,{\text{be}}\,Rs.\,x.\,{\text{Then}}, \cr & {\text{C}}{\text{.I}}{\text{.}} = {x{{\left( {1 + \frac{4}{{100}}} \right)}^2} - x} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\frac{{676}}{{625}}x - x} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{51}}{{625}}x \cr & {\text{S}}{\text{.I}}{\text{.}} = {\frac{{x \times 4 \times 2}}{{100}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{2x}}{{25}} \cr & \therefore \frac{{51x}}{{625}} - \frac{{2x}}{{25}} = 1 \cr & \Rightarrow x = 625 \cr} $$

$$\eqalign{ & {\text{Let}}\,{\text{P = Rs}}{\text{.}}\,{\text{100}}\,{\text{Then}},\, \cr & \,\,\,\,\,{\text{S}}{\text{.I}}{\text{. = }}\,{\text{Rs}}{\text{.}}\,{\text{60}}\,{\text{and}} \cr & \,\,\,\,\,\,\,\,{\text{T = 6}}\,{\text{years}} \cr & \therefore R = {\frac{{100 \times 60}}{{100 \times 6}}} = 10\% \,p.a. \cr & {\text{Now}},\,P = Rs.\,12000 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,T = 3\,{\text{year}}\,{\text{and}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,R = \,10\% \,p.a. \cr & \therefore {\text{C}}{\text{.I}}{\text{.}} = Rs.\,\left[ {12000 \times \left\{ {{{\left( {1 + \frac{{10}}{{100}}} \right)}^3} - 1} \right\}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {12000 \times \frac{{331}}{{1000}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,3972 \cr} $$

$$\eqalign{ & {\text{C}}{\text{.I}}{\text{.}}\,{\text{when}}\,{\text{interest}}\,{\text{compounded}}\,{\text{yearly}} \cr & = Rs.\left[ {5000 \times \left( {1 + \frac{4}{{100}}} \right) \times \left( {1 + \frac{{\frac{1}{2} \times 4}}{{100}}} \right)} \right] \cr & = Rs.\left( {5000 \times \frac{{26}}{{25}} \times \frac{{51}}{{50}}} \right) \cr & = Rs.5304 \cr & {\text{C}}{\text{.I}}{\text{.}}\,{\text{when}}\,{\text{interest}}\,{\text{in}}\,{\text{compounded}}\,{\text{half - yearly}} \cr & = Rs.\,\left[ {5000 \times {{\left( {1 + \frac{2}{{100}}} \right)}^3}} \right] \cr & = Rs.\,\left( {5000 \times \frac{{51}}{{50}} \times \frac{{51}}{{50}} \times \frac{{51}}{{50}}} \right) \cr & = Rs.\,5306.04 \cr & \therefore {\text{Difference}} = Rs.\,\left( {5306.04 - 5304} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,2.04 \cr} $$

$$\eqalign{ & {\text{Amount}} = Rs.\,\left( {30000 + 4347} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,34347 \cr & {\text{Let}}\,{\text{the}}\,{\text{time}}\,{\text{be}}\,n\,{\text{years}} \cr & {\text{Then}},30000\,{\left( {1 + \frac{7}{{100}}} \right)^n} = 34347 \cr & \Rightarrow {\left( {\frac{{107}}{{100}}} \right)^n} = \frac{{34347}}{{30000}} = \frac{{11449}}{{10000}} = {\left( {\frac{{107}}{{100}}} \right)^2} \cr & \therefore n = 2\,{\text{years}} \cr} $$

$$\eqalign{ & {\text{Amount}} = Rs.\,\left[ {25000 \times {{\left( {1 + \frac{{12}}{{100}}} \right)}^3}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {25000 \times \frac{{28}}{{25}} \times \frac{{28}}{{25}} \times \frac{{28}}{{25}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,35123.20 \cr & \therefore {\text{C}}{\text{.I}}{\text{.}} = Rs.\left( {35123.20 - 25000} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,10123.20 \cr} $$

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{rate}}\,{\text{be}}\,R\% \,p.a. \cr & {\text{Then}},\,1200 \times {\left( {1 + \frac{R}{{100}}} \right)^2} = 1348.32 \cr & \Rightarrow {\left( {1 + \frac{R}{{100}}} \right)^2} = \frac{{134832}}{{120000}} = \frac{{11236}}{{10000}} \cr & \therefore {\left( {1 + \frac{R}{{100}}} \right)^2} = {\left( {\frac{{106}}{{100}}} \right)^2} \cr & \Rightarrow 1 + \frac{R}{{100}} = \frac{{106}}{{100}} \cr & \Rightarrow R = 6\% \cr} $$

$$\eqalign{ & P{\left( {1 + \frac{{20}}{{100}}} \right)^n} > 2P\,\,\, \Rightarrow \,\,\,{\left( {\frac{6}{5}} \right)^n} > 2 \cr & {\text{Now}},\left( {\frac{6}{5} \times \frac{6}{5} \times \frac{6}{5} \times \frac{6}{5}} \right) > 2 \cr & So,\,n = 4\,{\text{years}} \cr} $$

$$\eqalign{ & {\text{Amount}} = Rs.\left[ {8000 \times {{\left( {1 + \frac{5}{{100}}} \right)}^2}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {8000 \times \frac{{21}}{{20}} \times \frac{{21}}{{20}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,8820 \cr} $$

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

$$\eqalign{ & {\text{Sum}} = Rs.\,\left( {\frac{{50 \times 100}}{{2 \times 5}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,500 \cr & {\text{Amount}} = Rs.\,\left[ {500 \times {{\left( {1 + \frac{5}{{100}}} \right)}^2}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {500 \times \frac{{21}}{{20}} \times \frac{{21}}{{20}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,551.25 \cr & \therefore {\text{C}}{\text{.I}}{\text{.}} = Rs.\,\left( {551.25 - 500} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,51.25 \cr} $$

$$\eqalign{ & {\text{S}}{\text{.I}}{\text{.}}\, = Rs.\,\left( {\frac{{1200 \times 5 \times 2}}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,120 \cr & {\text{C}}{\text{.I}}{\text{.}} = Rs.\,\left[ {1200 \times {{\left( {1 + \frac{5}{{100}}} \right)}^2} - 1200} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,123 \cr & \therefore {\text{Difference}} = Rs.\,\left( {123 - 120} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,3 \cr} $$

$$\left[ {15000 \times {{\left( {1 + \frac{R}{{100}}} \right)}^2} - 15000} \right]$$ $$ - $$ $$\left( {\frac{{15000 \times R \times 2}}{{100}}} \right)$$ $$ = 96$$ $$ \Rightarrow 15000\left[ {{{\left( {1 + \frac{R}{{100}}} \right)}^2} - 1 - \frac{{2R}}{{100}}} \right] = 96$$ $$ \Rightarrow 15000$$ $$\left[ {\frac{{{{\left( {100 + R} \right)}^2} - 10000 - \left( {200 \times R} \right)}}{{10000}}} \right]$$ $$ = 96$$ $$\eqalign{ & \Rightarrow {R^2} = {\frac{{96 \times 2}}{3}} = 64 \cr & \Rightarrow R = 8 \cr & \therefore {\text{Rate}} = 8\% \cr} $$

$$\eqalign{ & \text{Let the sum be Rs. P} \cr & \text{Then, }\, {P{{\left( {1 + \frac{{10}}{{100}}} \right)}^2} - P} = 525 \cr & \Rightarrow P\left[ {{{\left( {\frac{{11}}{{10}}} \right)}^2} - 1} \right] = 525 \cr & \Rightarrow P = {\frac{{525 \times 100}}{{21}}} = 2500 \cr & \therefore \text{Sum} = Rs.\,2500 \cr & \text{So, S.I.} = Rs.\left( {\frac{{2500 \times 5 \times 4}}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,500 \cr} $$

$$\eqalign{ & {\text{Principal = Rs. 2000}} \cr & {\text{Amount = Rs. 2420}} \cr & {\text{Rate = 10% }} \cr & {\text{By using formula,}} \cr & \Rightarrow 2420 = 2000{\left( {1 + \frac{{10}}{{100}}} \right)^n} \cr & \Rightarrow \frac{{2420}}{{2000}} = {\left( {1 + \frac{{10}}{{100}}} \right)^n} \cr & \Rightarrow \frac{{121}}{{100}} = {\left( {\frac{{11}}{{10}}} \right)^n} \cr & \Rightarrow {\left( {\frac{{11}}{{10}}} \right)^2} = {\left( {\frac{{11}}{{10}}} \right)^n} \cr & \Rightarrow n = 2 \cr & {\text{Hence,}} \cr & {\text{required time = 2 years}} \cr} $$ Alternative Note : In such type of questions to save your valuable time follow the given below method. Principal : Amount Ratio → 2000 : 2420 100 : 121 $${\text{Rate = 10% = }}\frac{1}{{10}}$$ Principal : Amount 1st year → 10 : 11 2nd year → 10 : 11 Ratio → 100 : 121 Note : Now after 2nd year both the principal and amount will be in the same ratio. Hence required time = 2 years

Note : In such type of questions to save your valuable time follow the given below method Rate % = 5% Effective Rate of CI for 3 years = 15.7625% Effective Rate of SI for 3 years = 5 × 3 = 15% According to the question $$\eqalign{ & \left( {15.7625 - 15} \right)\% \,{\text{of sum}} {\text{ = Rs. 36}}{\text{.60}} \cr & {\text{0}}{\text{.7625% of sum}} {\text{ = Rs. 36}}{\text{.60}} \cr & {\text{Sum = }}\frac{{36.60}}{{0.7625}} \times 100 \cr & \,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs. }}4800 \cr} $$

$$\eqalign{ & {\text{Amount}} \cr & {\text{ = Rs}}{\text{.}}\left[ {8400 \times {{\left( {1 + \frac{{25}}{{2 \times 100}}} \right)}^3}} \right] \cr & = {\text{Rs}}{\text{.}}\left( {8400 \times \frac{9}{8} \times \frac{9}{8} \times \frac{9}{8}} \right) \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{382725}}{{32}}} \right) \cr & = {\text{Rs}}.11960.156 \approx {\text{Rs}}.11960.16 \cr & {\text{C}}{\text{.I}}{\text{.}} = {\text{Rs}}{\text{.}}\left( {11960.16 - 8400} \right) \cr & = {\text{Rs}}{\text{.}}\,3560.16 \cr} $$

$$\eqalign{ & {\text{Let Principal = P}} \cr & {\text{Rate = R% }} \cr & {\text{T = 4 years}} \cr & \therefore {\text{Amount = 2P}} \cr & {\text{Case (I) 2P = P}}{\left( {1 + \frac{R}{{100}}} \right)^4} \cr & 2 = {\left( {1 + \frac{R}{{100}}} \right)^4}.....(i) \cr & {\text{Case (II) Let after t years it will be 8 times}} \cr & {\text{8P = P}}{\left( {1 + \frac{R}{{100}}} \right)^t} \cr & {\left( 2 \right)^3} = {\left( {1 + \frac{R}{{100}}} \right)^t}.....(ii) \cr & {\text{By using equation (i) & equation (ii)}} \cr & {\left( {1 + \frac{R}{{100}}} \right)^{12}} = {\left( {1 + \frac{R}{{100}}} \right)^t} \cr & {\text{By comparing both sides,}} \cr & {\text{t = 12 years}} \cr} $$

Let the sum be Rs. x $$\eqalign{ & \therefore 1352 = x{\left( {1 + \frac{4}{{100}}} \right)^2} \cr & \Rightarrow 1352 = x{\left( {1 + \frac{1}{{25}}} \right)^2} \cr & \Rightarrow 1352 = x{\left( {\frac{{26}}{{25}}} \right)^2} \cr & \Rightarrow x = \frac{{1352 \times 25 \times 25}}{{26 \times 26}} \cr & \Rightarrow x = {\text{Rs}}{\text{.}}\,1250 \cr} $$

$$\eqalign{ & {\text{Given,}}\,{\text{Principal}},\,P = Rs.\,2800 \cr & {\text{Compound}}\,{\text{rate}},\,R = 10\% \,{\text{per}}\,{\text{annum}} \cr & = \frac{{10}}{2} = 5\% \,{\text{half - yearly}} \cr & {\text{Amount}} \cr & = {\text{Rs}}{\text{.}}\left[ {2800 \times \left( {1 + \frac{{10}}{{100}}} \right)\left( {1 + \frac{{5}}{{100}}} \right)} \right] \cr & = {\text{Rs.}}\left( {2800 \times \frac{{11}}{{10}} \times \frac{{21}}{{20}}} \right) \cr & = {\text{Rs. }}3234 \cr & \therefore {\text{C}}{\text{.I}}{\text{. = Rs.}}\left( {3234 - 2800} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs. }}434 \cr} $$

$$\eqalign{ & {\text{Rate = 5% }} \cr & {\text{Time = 3 years}} \cr & {\text{Compound Interest Rs. 252}}{\text{.20}} \cr & {\text{Effective rate% of CI for 3 years}} \cr & {\text{ = 15}}{\text{.7625% }} \cr & {\text{Effective rate% of SI for 3 years}} \cr & {\text{ = 5}} \times {\text{3 = 15% }} \cr & {\text{Required SI }} \cr & {\text{ = }}\frac{{252.20}}{{15.7625}} \times 15 \cr & = 240 \cr} $$

$$\eqalign{ & {\text{CI for 2 years}}\,{\text{ = Rs. 282}}{\text{.15}} \cr & {\text{SI for 2 year}}\,{\text{ = Rs. 270}} \cr & {\text{SI for 1 year = }}\frac{{270}}{2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs.}}\,135 \cr & {\text{Difference between CI and SI}} \cr & {\text{ = }}\left( {282.15 - 270} \right) \cr & {\text{ = Rs. 12}}{\text{.15}} \cr & {\text{Required rate % }} \cr & {\text{ = }}\frac{{12.15}}{{135}} \times 100 = 9\% \cr} $$ Note : Always remainder for first year CI and SI will be same.

$$\eqalign{ & {\text{ = Rs}}.10000\left[ {\left( {1 + \frac{4}{{100}}} \right)\left( {1 + \frac{5}{{100}}} \right)\left( {1 + \frac{6}{{100}}} \right)} \right] \cr & = {\text{Rs}}.\left( {10000 \times \frac{{26}}{{25}} \times \frac{{21}}{{20}} \times \frac{{53}}{{50}}} \right) \cr & = {\text{Rs}}.\left( {\frac{{57876}}{5}} \right) = {\text{Rs}}.11575.20 \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left( {11575.20 - 10000} \right) \cr & \,\,\,\,\,\,\,\,\, = {\text{Rs}}.1575.20 \cr} $$

$$\eqalign{ & {\text{P = Rs}}.10000, \cr & {\text{R}} = 20\% \,p.a. \cr & \,\,\,\,\,\,\, = 10\% \,{\text{per}}\,{\text{half year}} \cr & T = 2\,{\text{years}} = 4\,{\text{half}}\,{\text{years}} \cr & {\text{Amount}} \cr & {\text{ = Rs}}.\left[ {10000 \times {{\left( {1 + \frac{{10}}{{100}}} \right)}^4}} \right] \cr & = {\text{Rs}}.\left( {10000 \times \frac{{11}}{{10}} \times \frac{{11}}{{10}} \times \frac{{11}}{{10}} \times \frac{{11}}{{10}}} \right) \cr & = {\text{Rs}}.14641 \cr & \therefore {\text{C}}{\text{.I}}{\text{. = Rs}}.\left( {14641 - 10000} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}.\,4641 \cr} $$

Interest on 650 for one year = 676 - 650 = 26 $$\eqalign{ & 26 = \frac{{650 \times r \times 1}}{{100}} \cr & r = 4\% \cr & 650 = P\left[ {1 + \frac{4}{{100}}} \right] \cr & \Rightarrow 650 = P \times \frac{{26}}{{25}} \cr & \Rightarrow p = \frac{{650 \times 25}}{{26}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,625 \cr} $$

$$\eqalign{ & 4\% = \frac{1}{{25}} \cr & \,\,\,\,\,\,\,\,\, = \frac{{26 \to {\text{Amount}}}}{{25 \to {\text{Principal}}}} \cr & {\text{Time = 2 years}} \cr & {\text{Principal}}\,\,\,\,\,{\text{Amount}} \cr & \,\,\,\,\,\,\,\,\,{\text{25}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{26}} \cr & \,\,\,\,\,\,\,\,\,{\text{25}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{26}} \cr & \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \cr & \,\,\,\,\,\,\,\,625\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,676 \cr & \,\,\,\, \downarrow \times 0.4\,\,\,\,\,\, \downarrow \times 0.4 \cr & \,\,\,\,\,\,\,\,\,250\,\,\,\,\,\,\,\,\,\,\,\,270.40 \cr & {\text{Hence required principal}} \cr & {\text{ = Rs.250}} \cr} $$

$$\eqalign{ & {\text{Amount}} \cr & {\text{ = Rs}}.\left[ {20000{{\left( {1 + \frac{{10}}{{100}}} \right)}^2}\left( {1 + \frac{{20}}{{100}}} \right)} \right] \cr & = {\text{Rs}}.\left( {20000 \times \frac{{11}}{{10}} \times \frac{{11}}{{10}} \times \frac{6}{5}} \right) \cr & = {\text{Rs}}.29040 \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}.\left( {29040 - 20000} \right) \cr & \,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}. 9040 \cr} $$

Difference in CI and SI for 2 years $$\eqalign{ & = \left( {40.80 - 40} \right) \cr & = {\text{Rs 0}}{\text{.80}} \cr & {\text{SI for first year }} \cr & {\text{ = }}\frac{{40}}{2} = {\text{Rs}}{\text{.}}\,20 \cr & {\text{Required Rate }}\% \cr & {\text{ = }}\frac{{0.80}}{{20}} \times 100 = 4\% \cr} $$

Let the sum be Rs. P and rate of interest be R% per annum. Then, $$\eqalign{ & P{\left( {1 + \frac{R}{{100}}} \right)^2} = 9680\,.....\,\left( 1 \right) \cr & P{\left( {1 + \frac{R}{{100}}} \right)^3} = 10648\,.....\,\left( 2 \right) \cr} $$ On dividing equation (2) by (1) : $$\eqalign{ & 1 + \frac{R}{{100}} = \frac{{10648}}{{9680}} \cr & \Rightarrow \frac{R}{{100}} = \frac{{10648}}{{9680}} - 1 \cr & \Rightarrow \frac{R}{{100}} = \frac{{10648 - 9680}}{{9680}} \cr & \Rightarrow \frac{R}{{100}} = \frac{{968}}{{9680}} \cr & \Rightarrow \frac{R}{{100}} = \frac{1}{{10}} \cr & \Rightarrow R = \frac{1}{{10}} \times 100 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 10\% \cr} $$

The interest is compounded quarterly, $$\therefore R = \frac{{20}}{4} = 5\% $$ Time = 3 quarters $$\eqalign{ & \therefore C.I. = P\left[ {{{\left( {1 + \frac{R}{{100}}} \right)}^T} - 1} \right] \cr & = 16000\left[ {{{\left( {1 + \frac{5}{{100}}} \right)}^3} - 1} \right] \cr & = 16000\left[ {{{\left( {\frac{{21}}{{20}}} \right)}^3} - 1} \right] \cr & = 16000\left( {\frac{{9261 - 8000}}{{8000}}} \right) \cr & = 16000 \times \frac{{1261}}{{8000}} \cr & = {\text{Rs}}{\text{.}}\,\,2522 \cr} $$

$$\eqalign{ & {\text{Principal}} \cr & {\text{ = Rs}}{\text{.}}\left( {\frac{{100 \times 1000}}{{5 \times 4}}} \right) \cr & = {\text{Rs}}{\text{. 5}}000 \cr & {\text{Now, P = Rs}}{\text{.}}\,10000, \cr & {\text{T = 2 years,}} \cr & {\text{R = 5% }} \cr & {\text{Amount}} \cr & {\text{ = Rs}}{\text{.}}\left[ {10000 \times {{\left( {1 + \frac{5}{{100}}} \right)}^2}} \right] \cr & = {\text{Rs}}{\text{.}}\left( {10000 \times \frac{{21}}{{20}} \times \frac{{21}}{{20}}} \right) \cr & = {\text{Rs}}. 11025 \cr & \therefore {\text{C}}{\text{.I}}{\text{. = }}\left( {11025 - 10000} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}. 1025 \cr} $$ Details Explanation Understanding Simple Interest: Simple interest is calculated only on the principal amount (the initial amount of money). The formula for simple interest is: Simple Interest = (Principal × Rate × Time) / 100 Solving for the Principal: In this problem, we know the simple interest (Rs. 1000), the rate (5% per year), and the time (4 years). We can use the simple interest formula to find the principal: 1000 = (Principal × 5 × 4) / 100 Solving for Principal, we get: Principal = Rs. 5000 Understanding Compound Interest: Compound interest is calculated on the principal amount plus accumulated interest from previous periods. It grows faster than simple interest. The formula for compound interest is a bit more complex, but we can break it down step-by-step. Calculating Compound Interest: The question asks for the compound interest on twice the principal (2 × Rs. 5000 = Rs. 10000) for 2 years at 5% interest. Year 1: Interest = (10000 × 5 × 1) / 100 = Rs. 500 Amount at the end of Year 1 = 10000 + 500 = Rs. 10500 Year 2: Interest = (10500 × 5 × 1) / 100 = Rs. 525 Total compound interest over 2 years = 500 + 525 = Rs. 1025 Therefore, the correct answer is B: Rs. 1025 (Note: There's a discrepancy in the provided options. The calculation leads to Rs. 1025, which is not listed.)

$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left( {\frac{{P \times R \times 2}}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\left( {\frac{{2PR}}{{100}}} \right) \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left[ {P \times {{\left( {1 + \frac{R}{{100}}} \right)}^2} - P} \right] \cr & \,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\left[ {\frac{{P{R^2}}}{{{{\left( {100} \right)}^2}}} + \frac{{2PR}}{{100}}} \right] \cr & \therefore {\text{Difference}} \cr & {\text{ = Rs}}{\text{.}}\left[ {\left\{ {\frac{{P{R^2}}}{{{{\left( {100} \right)}^2}}} + \frac{{2PR}}{{100}}} \right\} - \frac{{2PR}}{{100}}} \right] \cr & = {\text{Rs}}{\text{.}}\left[ {\frac{{P{R^2}}}{{{{\left( {100} \right)}^2}}}} \right] \cr} $$

Time (t) = 2 years Rate % = 4% Effective rate of CI of 2 years $$\eqalign{ & {\text{ = 4 + 4 + }}\frac{{4 \times 4}}{{100}} \cr & = 8.16\% \cr} $$ Effective Rate of SI for 2 years = 8% According to the question $$\eqalign{ & {\text{8}}{\text{.16% of sum}} \cr & {\text{ = Rs. 2448}} \cr & {\text{1% of sum}} \cr & {\text{ = Rs. }}\frac{{2448}}{{8.16}} \cr & {\text{8% of sum}} \cr & {\text{ = }}\frac{{2448}}{{8.16}} \times {\text{8}} \cr & {\text{ = Rs. 2400 }} \cr} $$

$$\eqalign{ & {\text{Principal (}}{{\text{P}}_1}{\text{) = Rs. 6000}} \cr & {\text{Time (t) = 2 years}} \cr & {\text{Rate % = 5% }} \cr & {\text{Simple interest}} \cr & {\text{ = }}\frac{{6000 \times 5 \times 2}}{{100}}{\text{ = Rs. 600}} \cr & {\text{Principal (}}{{\text{P}}_2}{\text{) = Rs. 5000}} \cr & {\text{Time (t) = 2 years}} \cr & {\text{Rate % = 8% }} \cr} $$ 2 year effective rate for Compound interest $$\eqalign{ & = 8 + 8 + \frac{{8 \times 8}}{{100}} = 16.64\% \cr & {\text{Compound}}\,{\text{Interest}} \cr & {\text{ = 5000}} \times \frac{{16.64}}{{100}} = {\text{Rs}}{\text{. 832}} \cr & {\text{Difference}} \cr & {\text{ = Rs}}{\text{. }}\left( {832 - 600} \right) \cr & = {\text{Rs}}{\text{.}}\,232 \cr & {\text{ }} \cr} $$

$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left( {\frac{{19200 \times 12 \times 3}}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.6912}} \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left[ {19200 \times {{\left( {1 + \frac{{12}}{{100}}} \right)}^3} - 19200} \right] \cr & = {\text{Rs}}{\text{.}}\left[ {\left( {19200 \times \frac{{28}}{{25}} \times \frac{{28}}{{25}} \times \frac{{28}}{{25}}} \right) - 19200} \right] \cr & = {\text{Rs}}{\text{. }}\left( {\frac{{16859136}}{{625}} - 19200} \right) \cr & = {\text{Rs}}{\text{. }}\left( {26974.6176 - 19200} \right) \cr & = {\text{Rs}}{\text{. 7774}}{\text{.6176}} \cr & \therefore {\text{Difference }} \cr & {\text{ = Rs}}{\text{.}}\left( {7774.6176 - 6912} \right) \cr & = {\text{Rs}}{\text{. 862}}{\text{.6176}} \cr} $$

$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. = Rs}}{\text{. }}\left( {15000 \times \frac{{25}}{2} \times 2 \times \frac{1}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}3750 \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}{\text{. }}\left[ {15000{{\left( {1 + \frac{{25}}{{2 \times 100}}} \right)}^2} - 15000} \right] \cr & = {\text{Rs}}{\text{. }}\left( {15000 \times \frac{9}{8} \times \frac{9}{8} - 15000} \right) \cr & = {\text{Rs}}{\text{. }}\left( {18948.375 - 15000} \right) \cr & = {\text{Rs}}{\text{. }}3984.375 \cr & {\text{Difference }} \cr & {\text{ = Rs}}{\text{. }}\left( {3984.375 - 3750} \right) \cr & = {\text{Rs}}{\text{. }}234.375 \cr} $$

Let the principal be x Rs. Now, $$\eqalign{ & C.I. = P\left[ {{{\left( {1 + \frac{R}{{100}}} \right)}^T} - 1} \right] \cr & \Rightarrow 1261 = x\left[ {{{\left( {1 + \frac{5}{{100}}} \right)}^3} - 1} \right] \cr & \Rightarrow 1261 = x\left( {\frac{{9261}}{{8000}} - 1} \right) \cr & \Rightarrow 1261 = x\left( {\frac{{9261 - 8000}}{{8000}}} \right) \cr & \Rightarrow 1261 = \frac{{1261x}}{{8000}} \cr & \Rightarrow x = \frac{{1261 \times 8000}}{{1261}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,8000 \cr} $$

Time (t) = 1 years Rate % = 4% Amount = Rs. 7803 When interest is compounded half yearly New Rate = $$\frac{4}{2}$$ = 2% Time = 1 × 2 = 2 years Required rate% for 2 years CI $${\text{ = 2}} + {\text{2}} + \frac{{2 \times 2}}{{100}} = 4.04\% $$ According to question, (100 + 4.04)% of sum = Rs. 7803 $$\eqalign{ & \therefore {\text{Sum = }}\frac{{7803}}{{104.04}} \times 100 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}7500 \cr} $$

$$\eqalign{ & {\text{Amount after }}{{\text{1}}^{{\text{st}}}}{\text{ year}} \cr & {\text{ = Rs}}{\text{. }}\left[ {4000\left( {1 + \frac{{15}}{{100}}} \right) - 1500} \right] \cr & = {\text{Rs}}{\text{. }}\left[ {\left( {4000 \times \frac{{23}}{{20}}} \right) - 1500} \right] \cr & = {\text{Rs}}{\text{. }}\left( {4600 - 1500} \right) \cr & = {\text{Rs}}{\text{. }}3100 \cr & {\text{Amount after }}{{\text{2}}^{{\text{nd}}}}{\text{ year}} \cr & {\text{ = Rs}}{\text{. }}\left[ {3100\left( {1 + \frac{{15}}{{100}}} \right) - 1500} \right] \cr & = {\text{Rs}}{\text{. }}\left[ {\left( {3100 \times \frac{{23}}{{20}}} \right) - 1500} \right] \cr & = {\text{Rs}}{\text{. }}\left( {3565 - 1500} \right) \cr & = {\text{Rs}}{\text{. }}2065 \cr & {\text{Amount after }}{{\text{3}}^{{\text{rd}}}}{\text{ year}} \cr & {\text{ = Rs}}{\text{. }}\left[ {2065\left( {1 + \frac{{15}}{{100}}} \right) - 1500} \right] \cr & = {\text{Rs}}{\text{. }}\left[ {\left( {2065 \times \frac{{23}}{{20}}} \right) - 1500} \right] \cr & = {\text{Rs}}{\text{. }}\left( {2374.75 - 1500} \right) \cr & = {\text{Rs}}{\text{. }}874.75 \cr} $$

$${\text{Rate 8% = }}\frac{2}{{25}}$$ Principal Amount 25 27 25 27 625 729 ↓ × 8 ↓ × 8 5000 5832 Hence required sum = Rs. 5000