Question

3. A sum of money is lent at 8% per annum
compound interest. If the interest for the
second year exceeds that for the first year by
*96, find the sum of money.​

Answer 1

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

• Let the sum of money invested be Rs P.

Case :- 1

• Principal = Rs P

• Time, n = 1 year

• Rate, r = 8 % per annum

So,

Compound interest is given by

$\tt{\implies CI=P\bigg(1+\dfrac{r}{100}\bigg)^{n}-P}$

$\tt{\longmapsto CI=P\bigg(1+\dfrac{8}{100}\bigg)^{1}-P}$

$\tt{\longmapsto CI=P\bigg(\dfrac{100 + 8}{100}\bigg)-P}$

$\tt{\longmapsto CI=P\bigg(\dfrac{108}{100}\bigg)-P}$

$\tt{\longmapsto CI=P(1.08)-P}$

$\tt{\longmapsto CI=0.08P} - - (1)$

Case :- 2

• Principal = Rs P

• Time, n = 3 year

• Rate, r = 8 % per annum

So,

Compound interest is given by

$\tt{\implies CI=P\bigg(1+\dfrac{r}{100}\bigg)^{n}-P}$

$\tt{\longmapsto CI=P\bigg(1+\dfrac{8}{100}\bigg)^{2}-P}$

$\tt{\longmapsto CI=P\bigg(\dfrac{100 + 8}{100}\bigg)^{2}-P}$

$\tt{\longmapsto CI=P\bigg(\dfrac{108}{100}\bigg)^{2}-P}$

$\tt{\longmapsto CI=P\bigg(1.08\bigg)^{2}-P}$

$\tt{\longmapsto CI=P\bigg(1.1664\bigg)-P}$

$\tt\longmapsto CI=0.1664P - - - (2)$

Now,

According to statement,

Cimpound interest for the second year exceeds that for the first year by Rs 96.

$\tt\longmapsto \: (0.1664P \: - \: 0.08 P) - 0.08 P= 96$

$\tt\longmapsto \: 0.0064P \: = 96$

$\bf\implies \:P = 15000$

Hence,

• Sum invested is Rs 15000

Additional Information :-

Let us assume that certain sum of money Rs P is invested at the rate of r % per annum compounded annually for n years then Amount is given by

$\tt{\implies Amount=P\bigg(1+\dfrac{r}{100}\bigg)^{n}}$

Let us assume that certain sum of money Rs P is invested at the rate of r % per annum compounded half yearly for n years then Amount is given by

$\tt{\implies Amount=P\bigg(1+\dfrac{r}{200}\bigg)^{2n}}$

Let us assume that certain sum of money Rs P is invested at the rate of r % per annum compounded quarterly for n years then Amount is given by

$\tt{\implies Amount=P\bigg(1+\dfrac{r}{400}\bigg)^{4n}}$