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8. a sum of rupees 5000 is invested at a rate of 5% per annum four year find the amount if the interest is compounded annually. 9. A sum of 2,000 is invested for 1 year at the rate of 10% per annum. Find the amount, if the interest is compounded quarterly.​

Answer 1

$\large\underline{\sf{Solution-8}}$

Principal, P = Rs 5000

Rate of interest, r = 5 % per annum compounded annually

Time, n = 4 years

We know,

Amount received on a certain sum of money of Rs P invested at the rate of r % per annum compounded annually for n years is given by

$\boxed{ \rm{ \:Amount \: = \: P \: {\bigg[1 + \dfrac{r}{100} \bigg]}^{n} \: }}$

So, on substituting the values, we get

$\rm \: Amount \: = \: 5000\: {\bigg[1 + \dfrac{5}{100} \bigg]}^{4}$

$\rm \: Amount \: = \: 5000\: {\bigg[1 + \dfrac{1}{20} \bigg]}^{4}$

$\rm \: Amount \: = \: 5000\: {\bigg[\dfrac{20 + 1}{20} \bigg]}^{4}$

$\rm \: Amount \: = \: 5000\: {\bigg[\dfrac{21}{20} \bigg]}^{4}$

$\rm\implies \:Amount \: = \: Rs \: 6077.53 \: \{approx. \}$

$\large\underline{\sf{Solution-9}}$

Principal, P = Rs 2000

Rate of interest, r = 10 % per annum compounded quarterly

Time, n = 1 years

We know,

Amount received on a certain sum of money of Rs P invested at the rate of r % per annum compounded quarterly for n years is given by

$\boxed{ \rm{ \:Amount \: = \: P \: {\bigg[1 + \dfrac{r}{400} \bigg]}^{4n} \: }}$

So, on substituting the values, we get

$\rm \: Amount \: = \: 2000 \: {\bigg[1 + \dfrac{10}{400} \bigg]}^{4 \times 1}$

$\rm \: Amount \: = \: 2000 \: {\bigg[1 + \dfrac{1}{40} \bigg]}^{4}$

$\rm \: Amount \: = \: 2000 \: {\bigg[\dfrac{40 + 1}{40} \bigg]}^{4}$

$\rm \: Amount \: = \: 2000 \: {\bigg[\dfrac{41}{40} \bigg]}^{4}$

$\rm\implies \:Amount \: = \: Rs \: 2207.63 \: \{approx. \}$

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Additional Information :-

1. Amount received on a certain sum of money of Rs P invested at the rate of r % per annum compounded semi - annually for n years is given by

$\boxed{ \rm{ \:Amount \: = \: P \: {\bigg[1 + \dfrac{r}{200} \bigg]}^{2n} \: }}$

2. Amount received on a certain sum of money of Rs P invested at the rate of r % per annum compounded monthly for n years is given by

$\boxed{ \rm{ \:Amount \: = \: P \: {\bigg[1 + \dfrac{r}{1200} \bigg]}^{12n} \: }}$

3. Compound interest ( CI ) received on a certain sum of money of Rs P invested at the rate of r % per annum compounded annually for n years is given by

$\boxed{ \rm{ \:CI \: = \: P \: {\bigg[1 + \dfrac{r}{100} \bigg]}^{n} - P \: }}$

4. Compound interest ( CI ) received on a certain sum of money of Rs P invested at the rate of r % per annum compounded semi - annually for n years is given by

$\boxed{ \rm{ \:CI \: = \: P \: {\bigg[1 + \dfrac{r}{200} \bigg]}^{2n} - P \: }}$

Answer 2

Answer:

Question No 1 :-

• A sum of Rs 5000 is invested at a rate of 5% per annum for 4 years. Find the amount if the interest is compounded annually.

Given :-

• A sum of Rs 5000 is invested at the rate of 5% per annum for 4 years.

To Find :-

• What is the amount if the interest is compounded annually.

Formula Used :-

$\clubsuit$ Amount formula when the interest is compounded annually :

$\bigstar \: \: \sf\boxed{\bold{\pink{A =\: P\bigg(1 + \dfrac{r}{100}\bigg)^n}}}\: \: \: \bigstar$

where,

• A = Amount
• P = Principal
• r = Rate of Interest
• n = Time Period

Solution :-

Given :

• Principal = Rs 5000
• Rate of Interest = 5% per annum
• Time Period = 4 years

According to the question by using the formula we get,

$\implies \bf A =\: P\bigg(1 + \dfrac{r}{100}\bigg)^n$

$\implies \sf A =\: 5000\bigg(1 + \dfrac{5}{100}\bigg)^4$

$\implies \sf A =\: 5000\bigg(\dfrac{100 \times 1 + 5}{100}\bigg)^4$

$\implies \sf A =\: 5000\bigg(\dfrac{100 + 5}{100}\bigg)^4$

$\implies \sf A =\: 5000\bigg(\dfrac{105}{100}\bigg)^4$

$\implies \sf A =\: 5000\bigg(\dfrac{105}{100} \times \dfrac{105}{100} \times \dfrac{105}{100} \times \dfrac{105}{100}\bigg)$

$\implies \sf A =\: 5000\bigg(\dfrac{121550625}{100000000}\bigg)$

$\implies \sf A =\: 5000(1.21550625)$

$\implies \sf A =\: 5000 \times 1.21550625$

$\implies \sf\bold{\red{A =\: Rs\: 6077.53}}$

$\therefore$ The amount is Rs 6077.53 .

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Question No 2 :-

• A sum of Rs 2000 is invested for 1 year at the rate of 10% per annum. Find the amount, if the interest is compounded quarterly.

Given :

• A sum of Rs 2000 is invested for 1 year at the rate of 10% per annum.

To Find :-

• What is the amount, if the interest is compounded quarterly.

Formula Used :-

$\clubsuit$ Amount formula when the interest is compounded quarterly :

$\bigstar \: \: \sf\boxed{\bold{\pink{A =\: P\Bigg(1 + \dfrac{\dfrac{r}{4}}{100}\Bigg)^{4n}}}}\: \: \: \bigstar$

where,

• A = Amount
• P = Principal
• r = Rate of Interest
• n = Time Period

Solution :-

Given :

• Principal = Rs 2000
• Rate of Interest = 10% per annum
• Time Period = 1 year

According to the question by using the formula we get,

$\implies \bf A =\: P\Bigg(1 + \dfrac{\dfrac{r}{4}}{100}\Bigg)^{4n}$

$\implies \sf A =\: 2000\Bigg(1 + \dfrac{\dfrac{10}{4}}{100}\Bigg)^{(4 \times 1)}$

$\implies \sf A =\: 2000\bigg(1 + \dfrac{10}{4} \times \dfrac{1}{100}\bigg)^{4}$

$\implies \sf A =\: 2000\bigg(1 + \dfrac{10}{400}\bigg)^4$

$\implies \sf A =\: 2000\bigg(\dfrac{400 \times 1 + 10}{400}\bigg)^4$

$\implies \sf A =\: 2000\bigg(\dfrac{400 + 10}{400}\bigg)^4$

$\implies \sf A =\: 2000\bigg(\dfrac{410}{400}\bigg)^4$

$\implies \sf A =\: 2000\bigg(\dfrac{410}{400} \times \dfrac{410}{400} \times \dfrac{410}{400} \times \dfrac{410}{400}\bigg)$

$\implies \sf A =\: 2000\bigg(\dfrac{2825761\cancel{0000}}{2560000\cancel{0000}}\bigg)$

$\implies \sf A =\: 2{\cancel{000}} \times \dfrac{2825761}{2560\cancel{000}}$

$\implies \sf A =\: \dfrac{5651522}{2560}$

$\implies \sf\bold{\red{A =\: Rs\: 2207.63}}$

$\therefore$ The amount is Rs 2207.63 .

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