Question

Find the difference in the compound interest on Rs. 10000 for 1yr at 10% per annum, compounded half-yearly and when rate is compounded annually?​

Answer 1

$Principal = 10,000$

$Rate = 10\%$

$Time = 1 \: year$

$SI = \: \frac{ P×R×T}{100}$

$SI = \frac{10,000 \times 10 \times 1}{100}$

$SI = 1,000$

Answer 2

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

Case :- 1

Principal, P = Rs 10000

Rate of interest, r = 10 % 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

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

So, on substituting the values, we get

$\rm\:CI \: = \: 10000\: {\bigg[1 + \dfrac{10}{100} \bigg]}^{1} - 10000 \: \:$

$\rm\:CI \: = \: 10000\: {\bigg[1 + \dfrac{1}{10} \bigg]} - 10000 \: \:$

$\rm\:CI \: = \: 10000\: {\bigg[\dfrac{10 + 1}{10} \bigg]} - 10000 \: \:$

$\rm\:CI \: = \: 10000\: {\bigg[\dfrac{11}{10} \bigg]} - 10000 \: \:$

$\rm\:CI \: = \: 11000\: - \: 10000 \: \:$

$\rm\implies \:CI = Rs \: 1000$

Case :- 2

Principal, P = Rs 10000

Rate of interest, r = 10 % per annum compounded half yearly.

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 half yearly for n years is given by

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

So, on substituting the values, we get

$\rm\:CI \: = \: 10000\: {\bigg[1 + \dfrac{10}{200} \bigg]}^{2} - 10000 \: \:$

$\rm\:CI \: = \: 10000\: {\bigg[1 + \dfrac{1}{20} \bigg]}^{2} - 10000 \: \:$

$\rm\:CI \: = \: 10000\: {\bigg[ \dfrac{20 + 1}{20} \bigg]}^{2} - 10000 \: \:$

$\rm\:CI \: = \: 10000\: {\bigg[ \dfrac{21}{20} \bigg]}^{2} - 10000 \: \:$

$\rm\:CI \: = \: 10000\: \bigg(\dfrac{441}{400} \bigg) - 10000 \: \:$

$\rm\:CI \: = \: 11025 - 10000 \: \:$

$\rm\implies \:CI \: = \: Rs \: 1025$

Hence, the difference in Compound interest = 1025 - 1000 = Rs 25.

$\rule{190pt}{2pt}$

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

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

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

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

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

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

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

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