Compound-Interest - 03

Explanation:-

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

Explanation:-

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

Explanation:-

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

Explanation:-

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

Explanation:-

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

Explanation:-

$$\eqalign{ & {\text{Principal}} \cr & {\text{ = Rs}}{\text{.}}\left( {\frac{{100 \times 1000}}{{5 \times 4}}} \right) \cr & = {\text{Rs}}{\text{.5}}000 \cr & {\text{Now, P = Rs}}{\text{.}}\,10000, \cr & {\text{T = 2 years,}} \cr & {\text{R = 5% }} \cr & {\text{Amount}} \cr & {\text{ = Rs}}{\text{.}}\left[ {10000 \times {{\left( {1 + \frac{5}{{100}}} \right)}^2}} \right] \cr & = {\text{Rs}}{\text{.}}\left( {10000 \times \frac{{21}}{{20}} \times \frac{{21}}{{20}}} \right) \cr & = {\text{Rs}}.11025 \cr & \therefore {\text{C}}{\text{.I}}{\text{. = }}\left( {11025 - 10000} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}.1025 \cr}$$

Explanation:-

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

Explanation:-

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

Explanation:-

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

Explanation:-

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