Simple & Compound Interest

Principal: The amount of money borrowed or invested.

Interest: The extra amount paid to the lender or investor.

Interest Rate (%): The extra amount paid to the lender or investor in percentage. It is usually calculated yearly (per annum) and sometimes monthly (per month).

\[Interest\;Rate\;per\;Annum = Interest\;Rate\;per\;Month \times 12 \]

There are 2 types of interest calculation:

• Simple (Flat) Interest: The interest is calculated based on the original principal amount only.
• Compound Interest: The interest is calculated based on the accumulated prinicpal amount.

Simple Interest

$$\begin{align}
Interest & = Principal \times Interest\;Rate \\
& = \frac{pr}{100} \\
\end{align}$$

$$\begin{align}
Total\;Interest & = Principal \times Interest\;Rate \times Duration \\
& = \frac{prt}{100} \\
\end{align}$$

$$\begin{align}
Total\;Amount & = Principal + Total\;Interest \\
& = Principal + \left( Principal \times Interest\;Rate \times Duration \right) \\
& = Principal \left( 1 + Interest\;Rate \times Duration \right) \\
& = p \left( 1 + \frac{rt}{100} \right) \\
\end{align}$$


Compound Interest

$$\begin{align}
Total\;Amount\;in\;1st\;year, A_1 & = Principal \left( 1 + Interest\;Rate\;per\;annum \right) \\
Total\;Amount\;in\;2nd\;year, A_2 & = A_1 \left( 1 + Interest\;Rate\;per\;annum \right) \\
& = Principal \left( 1 + Interest\;Rate\;per\;annum \right)^2 \\
Total\;Amount\;in\;3rd\;year, A_3 & = A_2 \left( 1 + Interest\;Rate\;per\;annum \right)\\
& = Principal \left( 1 + Interest\;Rate\;per\;annum \right)^3 \\
& \vdots \\
Total\;Amount\;in\;nth\;year, A_n & = A_{n-1} \left( 1 + Interest\;Rate\;per\;annum \right) \\
& = Principal \left( 1 + Interest\;Rate\;per\;annum \right)^n \\
& = p \left( 1 + \frac{r}{100} \right)^n \\
\end{align}$$

\[Total\;Interest = Total\;Amount - Principal\]

Note: Compound interest is like calculating simple interest every year/month with a new (accumulated) principal.