Question

Difference between ci and si for 3 years formula deriviation

Answer 1

AnswEr :

• Let's Assume that Principal and Rate is same here with time 3 Years.

• Derivation of Formula Now :

$\leadsto\sf{Difference = Compound \: Interest - Simple \:Interest}$

$\leadsto\sf{Diff. = \bigg[P \bigg(1 + \dfrac{r}{100} \bigg)^{t} - 1\bigg]- \bigg[\dfrac{Prt}{100} }\bigg]$

$\leadsto\sf{Diff. = \bigg[P \bigg(1 + \dfrac{r}{100} \bigg)^{3} - 1 \bigg]- \bigg[\dfrac{Pr\times 3}{100} }\bigg]$

$\leadsto\sf{Diff. = \bigg[P \bigg(\dfrac{100 + r}{100} \bigg)^{3} - 1 \bigg]- \bigg[\dfrac{Pr \times 3}{100} }\bigg]$

$\leadsto\sf{Diff. = \bigg[P \bigg(\dfrac{(100 + r)^{3} }{(100)^{3} } - 1\bigg)\bigg]- \bigg[\dfrac{3Pr}{100} }\bigg]$

$\leadsto\sf{Diff. = \bigg[P \bigg(\dfrac{(100 + r)^{3} - (100)^{3}}{(100)^{3} }\bigg)\bigg]- \bigg[\dfrac{3Pr}{100} }\bigg]$

$\leadsto\sf{Diff. = \bigg[P \bigg(\dfrac{\cancel{(100)^{3}} + {r}^{3} + 300r(100 + r) - \cancel{(100)^{3}}}{(100)^{3} }\bigg)\bigg]- \bigg[\dfrac{3Pr}{100} }\bigg]$

$\leadsto\sf{Diff. = \bigg[P \bigg(\dfrac{ {r}^{3} + 300r(100 + r)}{(100)^{3} }\bigg)\bigg]- \bigg[\dfrac{3Pr}{100} }\bigg]$

$\leadsto\sf{Diff. = P\bigg[ \bigg(\dfrac{ {r}^{3} + 30000r + 300{r}^{2}}{(100)^{3} }\bigg)- \dfrac{3r}{100} }\bigg]$

$\leadsto\sf{Diff.= P\bigg( \dfrac{ {r}^{3} + 30000r + 300{r}^{2} - (3r \times{100}^{2})}{(100)^{3}}} \bigg)$

$\leadsto\sf{Diff.= P\bigg( \dfrac{ {r}^{3} + \cancel{30000r} + 300{r}^{2} - \cancel{30000R}}{(100)^{3}}} \bigg)$

$\leadsto\sf{Diff.= P\bigg( \dfrac{ {r}^{3} + 300{r}^{2}}{(100)^{3}}} \bigg)$

$\leadsto\boxed{\sf{Diff.= P{r}^{2}\bigg( \dfrac{r + 300}{(100)^{3}}} \bigg)}$

Answer 2

$\Large\underline\mathfrak{Question}$

• Difference between ci and si for 3 years formula deriviation.....

$\Large\bold\star\underline{\underline\textbf{Formula\:used}}$

• CI = A - P where A = $\tiny\red{\boxed{\sf A\:=\:P( 1 + \frac{r}{100})^{t}}}$
• SI = P×R×T/100

$\Large\underline{\underline{\sf{Solution}:}}$

_________________________

Let ,

D = CI for 3 years - SI for 3 year ,

$\textbf{ Putting values we get,,}$

$\green{\red \leadsto D = [P(1 + \frac{r}{100})^{3} - P] - ( \frac{P \times r \times 3}{100})} \\ \\ \green{ \red \leadsto D =P( \frac{(100 + r)^{3} }{ {100}^{3}} - 1) - \frac{3Pr}{100}} \\ \\ \green{\red \leadsto D = P[ \frac{(100 + r)^{3} - {100}^{3} }{ {100}^{3} } - \frac{3r}{100}]} \\ \\ using , \orange{\large\boxed{\bold{( {x}^{3} - {y}^{3} ) = (x - y)( {x}^{2} + {y}^{2} + xy)}}} \\ \\ \red \leadsto \pink{D =P[ \frac{(100 + r - 100)( {100}^{2} + {r}^{2} + {100}^{2} + {100}^{2} + 300r) }{ {100}^{3} } - \frac{3r}{100}]} \\ \\ \red \leadsto \pink{D =P[ \: \frac{r(3 \times {100}^{2} + 300r + {r}^{2}) }{ {100}^{3} } - \frac{3r}{100}]} \\ \\ \red \leadsto \blue{D =P[ \: \frac{3r {100}^{2} + 300 {r}^{2} + {r}^{3} - 3r {100}^{2} }{100^{3} } ]} \\ \\ \red \leadsto D =P[ \: \frac{ {r}^{3} + 300 {r}^{2} }{ {100}^{3} } ] \:$

$\textbf{Or, we can Learn it As,,}$

$\red{\large\boxed{\bold{D = P( \frac{r}{100} )^{2}( \frac{r}{100} + 3)}}}$

The above formula is applicable only in the following conditions::----

1. The principal in simple interest and compound interest must be same.

2. Rate of interest must be same in simple interest and compound interest.

3. In compound interest, interest has to be compounded annually.

$\large\underline\textbf{Hope it Helps You.}$