Question

The difference between compound interest and simple interest on a sum for 3 years at 10% per annum, when the interest is compounded annually is 20. If the interest were compounded half yearly, what would be the difference in two interests ? ​

Answer 1

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

Given that,

The difference between compound interest and simple interest on a sum for 3 years at 10% per annum, when the interest is compounded annually is 20.

Let assume that Principal be P

Case :- 1

Compound interest

Principal = P

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

Time,n = 3 years

We know,

Compound interest 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

$\rm \: CI = P {\bigg[1 + \dfrac{10}{100} \bigg]}^{3} - P$

$\rm \: CI = P {\bigg[1 + \dfrac{1}{10} \bigg]}^{3} - P$

$\rm \: CI = P {\bigg[\dfrac{10 + 1}{10} \bigg]}^{3} - P$

$\rm \: CI = P {\bigg[\dfrac{11}{10} \bigg]}^{3} - P$

$\rm \: CI =\dfrac{1331P}{1000} - P$

$\rm \: CI =\dfrac{1331P - 1000P}{1000}$

$\rm\implies \:\rm \: CI =\dfrac{331P}{1000} - - - - (1)$

Simple Interest on a certain sum of money of Rs P invested at the rate of r % per annum for n years is given by

$\boxed{\tt{ \: \: SI\: = \: \frac{P \times r \times n}{100} \: }}$

So, on substituting the values, we get

$\rm \: SI = \dfrac{P \times 10 \times 3}{100}$

$\rm\implies \:SI = \dfrac{3P}{10} - - - (2)$

According to statement,

$\rm \: CI - SI = 20$

So, on substituting the values from equation (1) and (2), we get

$\rm \: \dfrac{331P}{1000} - \dfrac{3P}{10} = 20$

$\rm \: \dfrac{331P - 300P}{1000} = 20$

$\rm \: \dfrac{31P}{1000} = 20$

$\rm\implies \:P = \dfrac{20000}{31}$

Now, We have to find difference between compound interest and simple interest when rate of interest is compounded half-yearly.

So,

$\rm \: CI - SI$

$\rm \: = \: P {\bigg[1 + \dfrac{r}{200} \bigg]}^{2n} - P \: - \dfrac{P \times r \times n}{100} \:$

$\rm \: = \: P \bigg({\bigg[1 + \dfrac{r}{200} \bigg]}^{2n} - 1 \: - \dfrac{1 \times r \times n}{100}\bigg) \:$

So, on substituting the values, we get

$\rm \: = \: \dfrac{20000}{31} \bigg({\bigg[1 + \dfrac{10}{200} \bigg]}^{6} - 1 \: - \dfrac{10\times 3}{100}\bigg) \:$

$\rm \: = \: \dfrac{20000}{31} \bigg({\bigg[1 + \dfrac{1}{20} \bigg]}^{6} - 1 \: - \dfrac{1\times 3}{10}\bigg) \:$

$\rm \: = \: \dfrac{20000}{31} \bigg({\bigg[\dfrac{20 + 1}{20} \bigg]}^{6} - \dfrac{10 + 3}{10}\bigg) \:$

$\rm \: = \: \dfrac{20000}{31} \bigg({\bigg[\dfrac{21}{20} \bigg]}^{6} - \dfrac{10 + 3}{10}\bigg) \:$

$\rm \: = \: \dfrac{20000}{31} \bigg(1.34 - 1.3\bigg) \:$

$\rm \: = \: \dfrac{20000}{31} \times 0.04 \:$

$\rm \: = \: 25.81 \: \: \: \{approx \}$

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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 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

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

Answer 2

Refer the given Attachments.