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. About the author Iris
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 [tex]\tt{\longmapsto CI=P\bigg(1+\dfrac{R}{100}\bigg)^{n}-P}[/tex] 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 [tex]\tt{\implies 200=P\bigg(1+\dfrac{R}{100}\bigg)^{1}-P}[/tex] [tex]\rm :\longmapsto\:200 = \cancel{P} + \dfrac{PR}{100} – \cancel P [/tex] [tex]\bf\implies \:PR = 20000 – – – (1)[/tex] 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 [tex]\tt{\longmapsto 432=P\bigg(1+\dfrac{R}{100}\bigg)^{2}-P}[/tex] [tex]\rm :\longmapsto\:432 = P\bigg(1 + \dfrac{ {R}^{2} }{10000} + \dfrac{2R}{100} \bigg) – P[/tex] [tex]\rm :\longmapsto\:432 = \cancel P + \dfrac{P {R}^{2} }{10000} + \dfrac{PR}{50} – \cancel P[/tex] [tex]\rm :\longmapsto\:432 = \dfrac{PR \times R}{10000} + \dfrac{20000}{50} [/tex] [tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \: PR = 20000\bigg \}}[/tex] [tex]\rm :\longmapsto\:432 = \dfrac{20000R}{10000} + 400 [/tex] [tex]\rm :\longmapsto\:432 – 400 = 2R[/tex] [tex]\rm :\longmapsto\:2R = 32[/tex] [tex]\bf\implies \:R = 16\% \: per \: annum[/tex] On substituting the value of R = 16 in equation (1), we get [tex]\rm :\longmapsto\:P \times 16 = 20000[/tex] [tex]\bf\implies \:P = 1250[/tex] 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 [tex]\tt{\implies A=P\bigg(1+\dfrac{r}{100}\bigg)^{n}}[/tex] Amount on a certain sum of money Rs P invested at the rate of r % per annum compounded semi- annually for n years is [tex]\tt{\implies A=P\bigg(1+\dfrac{r}{200}\bigg)^{2n}}[/tex] Amount on a certain sum of money Rs P invested at the rate of r % per annum compounded quarterly for n years is [tex]\tt{\implies A=P\bigg(1+\dfrac{r}{400}\bigg)^{4n}}[/tex] Reply
Given that :-
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
[tex]\tt{\longmapsto CI=P\bigg(1+\dfrac{R}{100}\bigg)^{n}-P}[/tex]
According to statement,
Case :- 1
On substituting all these values in above formula, we get
[tex]\tt{\implies 200=P\bigg(1+\dfrac{R}{100}\bigg)^{1}-P}[/tex]
[tex]\rm :\longmapsto\:200 = \cancel{P} + \dfrac{PR}{100} – \cancel P [/tex]
[tex]\bf\implies \:PR = 20000 – – – (1)[/tex]
Case :- 2
So,
On substituting all these values in above formula, we get
[tex]\tt{\longmapsto 432=P\bigg(1+\dfrac{R}{100}\bigg)^{2}-P}[/tex]
[tex]\rm :\longmapsto\:432 = P\bigg(1 + \dfrac{ {R}^{2} }{10000} + \dfrac{2R}{100} \bigg) – P[/tex]
[tex]\rm :\longmapsto\:432 = \cancel P + \dfrac{P {R}^{2} }{10000} + \dfrac{PR}{50} – \cancel P[/tex]
[tex]\rm :\longmapsto\:432 = \dfrac{PR \times R}{10000} + \dfrac{20000}{50} [/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \: PR = 20000\bigg \}}[/tex]
[tex]\rm :\longmapsto\:432 = \dfrac{20000R}{10000} + 400 [/tex]
[tex]\rm :\longmapsto\:432 – 400 = 2R[/tex]
[tex]\rm :\longmapsto\:2R = 32[/tex]
[tex]\bf\implies \:R = 16\% \: per \: annum[/tex]
On substituting the value of R = 16 in equation (1), we get
[tex]\rm :\longmapsto\:P \times 16 = 20000[/tex]
[tex]\bf\implies \:P = 1250[/tex]
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
[tex]\tt{\implies A=P\bigg(1+\dfrac{r}{100}\bigg)^{n}}[/tex]
Amount on a certain sum of money Rs P invested at the rate of r % per annum compounded semi- annually for n years is
[tex]\tt{\implies A=P\bigg(1+\dfrac{r}{200}\bigg)^{2n}}[/tex]
Amount on a certain sum of money Rs P invested at the rate of r % per annum compounded quarterly for n years is
[tex]\tt{\implies A=P\bigg(1+\dfrac{r}{400}\bigg)^{4n}}[/tex]