Question

ruchi deposited certain sum of money at 8% per annum for 1 year if the difference between the compound interest compounded semi annualy and annualy is 88 rupees. ruchi deposited how much money

Answer 1

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

Let assume that Ruchi deposited Rs P in the bank.

Case :- 1 Compounded annually

We have,

Principal = Rs P

Rate of interest, r = 8 % per annum compounded annually

Time, n = 1 year

We know,

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

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

So, on substituting the values, we get

$\rm \: CI _1 = P {\bigg[1 + \dfrac{8}{100} \bigg]}^{1} - P$

$\rm \: CI _1 = P {\bigg[1 + \dfrac{2}{25} \bigg]} - P$

$\rm \: CI _1 = P {\bigg[\dfrac{25 + 2}{25} \bigg]} - P$

$\rm \: CI _1 = P {\bigg[\dfrac{27}{25} \bigg]} - P$

$\rm \: CI _1 = \dfrac{27P - 25P}{25}$

$\rm\implies \:\rm \: CI _1 = \dfrac{2P}{25} - - - (1)$

Case :- 2 Compounded semi - annually

We have

Principal = Rs P

Rate of interest, r = 8 % per annum compounded semi - annually

Time, n = 1 year

We know,

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

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

So, on substituting the values, we get

$\rm \: CI _2 = P {\bigg[1 + \dfrac{8}{200} \bigg]}^{2} - P$

$\rm \: CI _2 = P {\bigg[1 + \dfrac{1}{25} \bigg]}^{2} - P$

$\rm \: CI _2 = P {\bigg[\dfrac{25 + 1}{25} \bigg]}^{2} - P$

$\rm \: CI _2 = P {\bigg[\dfrac{26}{25} \bigg]}^{2} - P$

$\rm \: CI _2 = \dfrac{676P}{625} - P$

$\rm \: CI _2 = \dfrac{676P - 625P}{625}$

$\rm\implies \:\rm \: CI _2 = \dfrac{51P}{625} - - - (2)$

Now, According to statement, It is given that the difference between the compound interest semi - annually and annually is Rs 88.

$\rm \: CI _2 - CI _1 = 88$

$\rm \: \dfrac{51P}{625} - \dfrac{2P}{25} = 88$

$\rm \: \dfrac{51P - 50P}{625} = 88$

$\rm \: \dfrac{P}{625} = 88$

$\rm\implies \:P = 55000$

It means, Ruchi deposited Rs 55, 000 in the bank.

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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 annually for n years is given by

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

2. 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

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

3. 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

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

4. 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

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

Answer 2

Answer:

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

Let assume that Ruchi deposited Rs P in the bank.

Case :- 1 Compounded annually

We have,

Principal = Rs P

Rate of interest, r = 8 % per annum compounded annually

Time, n = 1 year

We know,

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

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

So, on substituting the values, we get

$\rm \: CI _1 = P {\bigg[1 + \dfrac{8}{100} \bigg]}^{1} - P$

$\rm \: CI _1 = P {\bigg[1 + \dfrac{2}{25} \bigg]} - P$

$\rm \: CI _1 = P {\bigg[\dfrac{25 + 2}{25} \bigg]} - P$

$\rm \: CI _1 = P {\bigg[\dfrac{27}{25} \bigg]} - P$

$\rm \: CI _1 = \dfrac{27P - 25P}{25}$

$\rm\implies \:\rm \: CI _1 = \dfrac{2P}{25} - - - (1)$

Case :- 2 Compounded semi - annually

We have

Principal = Rs P

Rate of interest, r = 8 % per annum compounded semi - annually

Time, n = 1 year

We know,

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

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

So, on substituting the values, we get

$\rm \: CI _2 = P {\bigg[1 + \dfrac{8}{200} \bigg]}^{2} - P$

$\rm \: CI _2 = P {\bigg[1 + \dfrac{1}{25} \bigg]}^{2} - P$

$\rm \: CI _2 = P {\bigg[\dfrac{25 + 1}{25} \bigg]}^{2} - P$

$\rm \: CI _2 = P {\bigg[\dfrac{26}{25} \bigg]}^{2} - P$

$\rm \: CI _2 = \dfrac{676P}{625} - P$

$\rm \: CI _2 = \dfrac{676P - 625P}{625}$

$\rm\implies \:\rm \: CI _2 = \dfrac{51P}{625} - - - (2)$

Now, According to statement, It is given that the difference between the compound interest semi - annually and annually is Rs 88.

$\rm \: CI _2 - CI _1 = 88$

$\rm \: \dfrac{51P}{625} - \dfrac{2P}{25} = 88$

$\rm \: \dfrac{51P - 50P}{625} = 88$

$\rm \: \dfrac{P}{625} = 88$

$\rm\implies \:P = 55000$

It means, Ruchi deposited Rs 55, 000 in the bank.

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

Additional Information :-

1. 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

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

2. 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

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

3. 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

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

4. 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

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