question_answer 1)

                Calculate the amount and compound interest on (a) Rs 10,800 for 3 years at \[12\frac{1}{2}\]% per annum compounded annually. (b) Rs 18,000 for \[2\frac{1}{2}\] years at 10% per annum compounded annually. (c) Rs 62,500 for \[1\frac{1}{2}\] years at 8% per annum compounded half yearly. (d) Rs 8,000 for 1 year at 9% per annum compounded half yearly. (You could use the year by year calculation using SI formula to verify). (e) Rs 10,000 for 1 year at 8% per annum compounded half yearly.

Answer:

Sol.        (a) By using year by year calculation SI on Rs 10,800 at \[12\frac{1}{2}%\] per annum for 1 year \[=10,800\times \frac{25}{2}\times \frac{1}{100}\] = Rs 1, 350 \[\therefore \] Amount at the end of 1st year = Rs 10,800 + Rs 1,350 (A = P + SI) = Rs 12,150 = Principal for 2nd year. SI on Rs 12,150 at \[12\frac{1}{2}%\] per annum for 1 year\[=12,150\times \frac{25}{2}\times \frac{1}{100}\] = Rs 1,518.75 \[\therefore \] Amount at the end of 2nd year = Rs 12,150 + Rs 1,518.75 = Rs 13,668.75 = Principal for 3rd year SI on Rs 13,668.75 at \[12\frac{1}{2}%\] per annum for 1 year \[=13,668.75\times \frac{25}{2}\times \frac{1}{100}\] = Rs 1,708.59 \[\therefore \] Amount at the end of 3rd year = Rs 13,668.75 + Rs 1,708.59 = Rs 15,377.34 this is the required amount. Now, CI = Rs 15,377.34 ? Rs 10,800 = Rs 4,577.34 OR CI = Rs 1,350 + Rs 1,518.75 + Rs 1,708.59 = Rs 4,577.34 By using compound interest formula P = Rs 10,800 R =  \[12\frac{1}{2}%\] per annum \[=\frac{25}{2}%\] per annum n = 3 years \[\therefore \]  \[A=P{{\left( 1+\frac{R}{100} \right)}^{n}}\] \[=10,800\,{{\left( 1+\frac{25}{2\times 100} \right)}^{3}}\]           \[=10,800{{\left( 1+\frac{1}{8} \right)}^{3}}\] \[=10,\,800\,{{\left( \frac{9}{8} \right)}^{3}}\] \[=10,800\times \frac{9}{8}\times \frac{9}{8}\times \frac{9}{8}\] = Rs 15, 377.34 \[\therefore \] CI =  A ? P = Rs 15,377.34 ? Rs 10,800    = Rs 4,577.34               (b) By using year by year calculation SI on Rs 18,000 at 10% p.a. for 1 year\[=\frac{18,000\times 10\times 1}{100}\,=\text{Rs}\,1,800\] \[\therefore \] Amount at the end of 1st year = Rs 18,000 + Rs 1,800 = Rs 19,800 = Principal for 2nd year SI on Rs 19,800 at 10% p.a. for 1 year \[=\frac{19,800\times 10\times 1}{100}\] = Rs 1,980 \[\therefore \] Amount at the end of 2nd year = Rs 19,800 + Rs 1,980 = Rs 21,780 = Principal for 3rd year SI on Rs 21,780 at 10% p.a. for \[\frac{1}{2}\] year \[=\frac{21,\,780\times 10\times 1}{2\times 100}\] = Rs 1,089 \[\therefore \] Amount at the end of \[2\frac{1}{2}\] years = Rs 21,780 + Rs 1,089 = Rs 22,869 This is the required amount. Now, CI = Rs 22,869 ? Rs 18,000 = Rs 4,869 By Using Compound Interest formula P = Rs 18,000 R = 10% p.a. n = 2 years \[\therefore \]  \[A=P{{\left( 1+\frac{R}{100} \right)}^{n}}\] \[=18,000{{\left( 1+\frac{10}{100} \right)}^{2}}\] \[=18,000{{\left( 1+\frac{1}{10} \right)}^{2}}\] \[=18,000{{\left( \frac{11}{10} \right)}^{2}}\] \[=18.000\times \frac{11}{10}\times \frac{11}{10}\] = Rs 21,780 SI on Rs 21,780 at 10% p.a. for \[\frac{1}{2}\] year \[=\frac{21,780\times 10\times 1}{2\times 100}\] = Rs 1,089 \[\therefore \] Amount at the end of \[2\frac{1}{2}\] years = Rs 21,780 + Rs 1,089 = Rs 22,869 CI = A ? P = Rs 22,869 ? Rs 18,000 = Rs 4,869. (c) By using half year by half year calculation SI on Rs 62,500 at 8% p.a. for half year \[\frac{62,500\times 8\times 1}{2\times 100}\] = Rs 2,500 \[\therefore \] Amount at the end of 1st half year = Rs 62,500 + Rs 2,500 = Rs 65,000 = Principal for 2nd half year SI on Rs 65,000 at 8% p.a. for half year \[=\frac{65,000\times 8\times 1}{2\times 100}\] = Rs 2,600 \[\therefore \] Amount at the end of 2nd half year = Rs 65,000 + Rs 2,600 = Rs 67,600 = Principal for 3rd half year  SI on Rs 67600 at 8% p.a. for 1 half year \[=\frac{67,600\times 8\times 1}{2\times 100}\] = Rs 2,704 \[\therefore \] Amount at the end of 3rd half year = Rs 67,600 + Rs 2,704 = Rs 70,304 this is the required amount. Now, CI = Rs 70,304 ? Rs 62,500 = Rs 7,804 By using compound interest formula P = Rs 62,500 R = 8% p.a. \[=\frac{1}{2}\times 8%\] per half year = 4% per half year \[n=1\frac{1}{2}\] year \[=1\frac{1}{2}\times 2\] half years = 3 half years \[\therefore \]  \[A=P{{\left( 1+\frac{R}{100} \right)}^{n}}\] \[=62,500{{\left( 1+\frac{4}{100} \right)}^{3}}\] \[=62,500{{\left( 1+\frac{1}{25} \right)}^{3}}\] \[=62,500{{\left( \frac{26}{25} \right)}^{3}}\] \[=62,500\times \frac{26}{25}\times \frac{26}{25}\times \frac{26}{25}\] = Rs 70, 304 \[\therefore \] CI = A ? P = Rs 70,304 ? Rs 62,500 = Rs 7804. (d) By using half-year by half-year calculation SI on Rs 8,000 at 9% p.a. for 1st half year \[=\frac{8,000\times 9\times 1}{2\times 100}=\text{Rs}\,360\] \[\therefore \] Amount at the end of 1st half year = Rs 8,000 + Rs 360 = Rs 8,360 = Principal for the 2nd half year SI on Rs 8,360 at 9% p.a. for 2nd half year \[=\frac{8,360\times 9\times 1}{2\times 100}\] = Rs 376.20 \[\therefore \] Amount at the end of 2nd half year = Rs 8,360 + Rs 376.20 = Rs 8,736.20 This is the required amount.           Now, CI = Rs 8,736.20 ? Rs 8,000 = Rs 736.20 By using compound interest formula P = Rs 8, 000 R = 9% p.a. \[=\frac{9}{2}%\] per half year \[n=1\] year \[=1\times 2\] half years = 2 half years  \[\therefore \]  \[A=P{{\left( 1+\frac{R}{100} \right)}^{n}}\] \[=8,000{{\left( 1+\frac{9}{2\times 100} \right)}^{2}}\]   \[=8,000{{\left( 1+\frac{9}{200} \right)}^{2}}\] \[=8,000{{\left( \frac{209}{200} \right)}^{2}}\] \[=8,000\times \,\frac{209}{200}\times \frac{209}{200}\] = Rs 8, 736.20 \[\therefore \]  CI = A ? P = Rs 8,736.20 - Rs 8,000 = Rs 736.20. (e) By using half-year by half-year calculation SI on Rs 10,000 at 8% per annum for 1st half year \[=\frac{10,000\times 8\times 1}{2\times 100}=\text{Rs}\,400\] \[\therefore \] Amount at the end of 1st half year = Rs 10,000 + Rs 400 = Rs 10,400 = Principal for the 2nd half year SI on Rs 10,400 at 8% per annum for 2nd half year \[=\frac{10,400\times 8\times 1}{2\times 100}=\text{Rs}\,416\] \[\therefore \] Amount at the end of 2nd half year = Rs 10,400 + Rs 416 = Rs 10,816 This is the required amount Now, CI = Rs 10,816 ? Rs 10,000 = Rs 816. By using compound interest formula P = Rs 10, 000 R = 8% per annum \[=\frac{8}{2}%\] per half year = 4% per half year \[n=1\] year \[=1\times 2\] half years = 2 half years \[\therefore \]  \[A=P{{\left( 1+\frac{R}{100} \right)}^{n}}\] \[=10,000\,{{\left( 1+\frac{4}{100} \right)}^{2}}\] \[=10,000{{\left( 1+\frac{1}{25} \right)}^{2}}\] \[=10,000\times \frac{26}{25}\times \frac{26}{25}\] = Rs 10, 816 \[\therefore \] \[CI=A-P\] = Rs 10, 816 ? Rs 10, 000 = Rs 816.