Question

the compound interest on a sum of money in 1st and 2nd years are Rs.200 and Rs.232 respectively, Find the rate of compound interest compounded annually and the sum.​

Answer 1

Given that :-

• Compound interest on a sum of money in 1st is Rs.200.

• Compound interest on a sum of money for second year is Rs.232.

To Find :-

Rate of compound interest and sum invested.

Solution :-

Let assume that,

☆ Sum invested be Rs P

☆ Rate of interest be R % per annum compounded annually .

We know that,

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

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

According to statement,

Case :- 1

• Sum invested = Rs P

• Rate of interest = R % per annum

• Time, n = 1 year

• Compound interest, CI = Rs 200

On substituting all these values in above formula, we get

$\tt{\implies 200=P\bigg(1+\dfrac{R}{100}\bigg)^{1}-P}$

$\rm :\longmapsto\:200 = \cancel{P} + \dfrac{PR}{100} - \cancel P$

$\bf\implies \:PR = 20000 - - - (1)$

Case :- 2

• Sum invested = Rs P

• Rate of interest = R % per annum

• Time, n = 2 years

• Compound interest in 2 years = 232 + 200 = 432

So,

On substituting all these values in above formula, we get

$\tt{\longmapsto 432=P\bigg(1+\dfrac{R}{100}\bigg)^{2}-P}$

$\rm :\longmapsto\:432 = P\bigg(1 + \dfrac{ {R}^{2} }{10000} + \dfrac{2R}{100} \bigg) - P$

$\rm :\longmapsto\:432 = \cancel P + \dfrac{P {R}^{2} }{10000} + \dfrac{PR}{50} - \cancel P$

$\rm :\longmapsto\:432 = \dfrac{PR \times R}{10000} + \dfrac{20000}{50}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \: PR = 20000\bigg \}}$

$\rm :\longmapsto\:432 = \dfrac{20000R}{10000} + 400$

$\rm :\longmapsto\:432 - 400 = 2R$

$\rm :\longmapsto\:2R = 32$

$\bf\implies \:R = 16\% \: per \: annum$

On substituting the value of R = 16 in equation (1), we get

$\rm :\longmapsto\:P \times 16 = 20000$

$\bf\implies \:P = 1250$

Hence,

Sum invested = Rs 1250

Rate of interest = 16 % per annum compounded annually.

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

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

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

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

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

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