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Answer :
(a) Given, P=₹ 10,800,n=3 years.
We have, R=\(12\frac{1}2\)% = \(\frac{25}2\)%p.a.
So, we know that \(Amount=P(1+\frac{R}{100})^n\)
\(=10,800(1+\frac{25}{100})^3\)
\(=10,800(\frac{9}{8})^3\)
\(=10,800\times(\frac{9}{8})\times(\frac{9}{8})\times(\frac{9}{8})\)
\(=\frac{4,92,075}{32}=₹ \; 15,377.34\)
Thus, we have Compound interest=A - P
=₹ 15,377.34-₹ 10,800 = ₹ 4,577.35
Hence we have, amount = ₹ 15,377.34 and CI = ₹ 4,577.34
(b)We have: P = ₹ 18,000, n = \(2\frac { 1 }{ 2 }\) years and R = 10% p.a.
The amount for \(2\frac { 1 }{ 2 }\) years, can be calculated by first calculating the amount to 2 years using CI formula and then calculating the simple interest by using SI formula.
The amount for 2 years:
Amount = \(18000(1+\frac{10}{100})^2\)
= \(18000\times(\frac{110}{100})^2\)
= \(18000\times(\frac{11}{10})\times(\frac{11}{10})=₹ \; 21,780\)
∴ Interest after 2 years = Amount - P
=21,780-18,000=₹ 3,780
So, now taking principal amount as ₹ 21,780, the SI for the next \(\frac{1}2\)years will be as:
SI=\(\frac{P\times R\times n}{100}\)
=\(\frac{21,780\times10\times1}{100\times2}\)
=\(₹\; 1,089\)
Therefore,Total CI = ₹ 3780 +₹ 1,089= ₹ 4,869
Amount = P + I = ₹ 21,780 + ₹ 1,089 =₹ 22,869
Hence, the amount = ₹ 22,869 and CI = ₹ 4,869
(c)Given, P=₹ 62,500,n=\(1\frac{1}2=\frac{3}2\) years per annum compounded half yearly i.e., \(\frac{3}2\times 2\) years=3 half years.
We have, R=\(8\)%=\(\frac{8}2\)% = \(4\)%half yearly
So, we know that \(Amount=P(1+\frac{R}{100})^n\)
\(=62,500(1+\frac{4}{100})^3\)
\(=62,500(\frac{26}{25})^3\)
\(=62,500\times(\frac{26}{25})\times(\frac{26}{25})\times(\frac{26}{25})\)
\(=4 \times 26\times 26\times 26=₹ \; 70,304\)
Thus, we have Compound interest=A - P
=₹ 70,304-₹ 62,500 = ₹ 7,804
Hence we have, amount = ₹ 70,304 and CI = ₹ 7804
(d)Given, P=₹ 8,000,n=1 years and R = 9% per annum compounded half yearly
Since, the interest is compounded half yearly n =\(1 \times 2 = 2\) half years.So, R=\(\frac{9}2\)% per half years.
So, we know that \(Amount=P(1+\frac{R}{100})^n\)
\(=8,000(1+\frac{9}{2\times100})^2\)
\(=8,000(\frac{209}{200})^2\)
\(=8,000\times(\frac{209}{200})\times(\frac{209}{200})\)
\(=₹ \;(\frac{8,7362}{10})\)
\(=₹ \; 8,736.20\)
Thus, we have Compound interest=A - P
=₹ 8,736.20-₹ 8,000 = ₹ 736.20
Hence we have, amount = ₹ 8736.20 and CI = ₹ 736.20
(e)Given, P=₹ 10,000,n=1 years and R = 8% per annum compounded half yearly
Since, the interest is compounded half yearly n =\(1 \times 2 = 2\) half years.So, R=\(\frac{8}2=4\)% per half years.
So, we know that \(Amount=P(1+\frac{R}{100})^n\)
\(=10,000(1+\frac{4}{100})^2\)
\(=10,000(\frac{26}{25})^2\)
\(=10,000\times(\frac{26}{25})\times(\frac{26}{25})\)
\(=16\times 26\times 26\)
\(=₹ \; 10,816\)
Thus, we have Compound interest=A - P
=₹ 10,816 -₹ 10,000 = ₹ 816
Hence we have, amount = ₹ 10,816 and CI = ₹ 816