Question

The difference between simple and compound
Interests compounded annually on a certain
sum of money for 2 years at 4% per annum is rs. 1 . find the sum​

Answer 1

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

Let assume that the sum be Rs x.

Case :- 1 Compound interest

Principal, P = Rs x

Time, n = 2 years

Rate, r = 4 % per annum compounded annually.

We know,

Compound interest on a certain sum of money Rs P invested at the rate of r % per annum compounded annually for n years is

$\rm :\longmapsto\:\bf{ CI=P\bigg(1+\dfrac{r}{100}\bigg)^{n}-P}$

So, on substituting the values, we get

$\tt{\implies CI=x\bigg(1+\dfrac{4}{100}\bigg)^{2}-x}$

$\tt{\implies CI=x\bigg(1+\dfrac{1}{25}\bigg)^{2}-x}$

$\tt{\implies CI=x\bigg(\dfrac{25 + 1}{25}\bigg)^{2}-x}$

$\tt{\implies CI=x\bigg(\dfrac{26}{25}\bigg)^{2}-x}$

$\tt{\implies CI=x\bigg(\dfrac{676}{625}\bigg)-x}$

$\tt{\implies CI=\dfrac{676x - 625x}{625}}$

$\bf{\implies CI=\dfrac{51x}{625}} - - - (1)$

Case :- 2 Simple Interest

Principal, P = Rs x

Time, n = 2 years

Rate, r = 4 % per annum.

We know,

Simple interest on a certain sum of money Rs p invested at the rate of r % per annum compounded annually for n years is

$\rm :\longmapsto\:Simple \: Interest, \: SI = \dfrac{p \times r \times n}{100}$

So, on substituting the values, we get

$\rm :\longmapsto\:SI = \dfrac{x \times 4 \times 2}{100}$

$\bf :\longmapsto\:SI = \dfrac{2x}{25} - - - (2)$

Now,

According to statement,

$\rm :\longmapsto\:CI - SI = 1$

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

$\rm :\longmapsto\:\dfrac{51x}{625} - \dfrac{2x}{25} = 1$

$\rm :\longmapsto\:\dfrac{51x - 50x}{625} = 1$

$\rm :\longmapsto\:\dfrac{x}{625} = 1$

$\bf\implies \:x = 625$

So,

• Sum = Rs 625

Additional Information :-

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

$\rm :\longmapsto\:\bf{Amount =P\bigg(1+\dfrac{r}{100}\bigg)^{n}}$

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

$\rm :\longmapsto\:\bf{Amount =P\bigg(1+\dfrac{r}{200}\bigg)^{2n}}$

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

$\rm :\longmapsto\:\bf{Amount =P\bigg(1+\dfrac{r}{400}\bigg)^{4n}}$