Concept of Simple Interest

When interest is calculated for every period only on the principal, then the total amount of interest gained on all the periods is called simple interest.

Let principal amount = P
Rate of interest = r
Then the amount of interest after one year = Pr.

\[\begin{gathered} {\text{Amount = Principal + Interest}} \\ {\text{ = P}} + \Pr = {\text{P}}\left( {1 + {\text{r}}} \right) \\ \end{gathered} \]

Amount after two years

\[\begin{gathered} {\text{Amount = Principal + Interest}} \\ {\text{ = P}}\left( {1 + {\text{r}}} \right) + \Pr \\ {\text{ = P}}\left[ {{\text{1}} + {\text{r}} + {\text{r}}} \right] \\ {\text{ = P}}\left[ {{\text{1}} + 2{\text{r}}} \right] \\ \end{gathered} \]

Amount after three years
\[\begin{gathered} {\text{Amount = Principal + Interest}} \\ {\text{ = P}}\left( {1 + 2{\text{r}}} \right) + \Pr \\ {\text{ = P}}\left[ {{\text{1}} + 2{\text{r}} + {\text{r}}} \right] \\ {\text{ = P}}\left[ {{\text{1}} + 3{\text{r}}} \right] \\ \end{gathered} \]
And so on; the amount after n – years is
\[\begin{gathered} {\text{Amount = P}}\left( {1 + n{\text{r}}} \right) + \Pr \\ {\text{Amount = P}} + {\text{P}}n{\text{r}} \\ \end{gathered} \]

Hence
\[\begin{gathered} {\text{Amount = Principal + Amount of interest after n years}} \\ {\text{Amount of interest after n years = Pnr}} \\ \end{gathered} \]