Interest

The principal amount is P, rate of interest is R and time of loan is T

• Simple Interest = $$\dfrac{P*T*R}{100}$$
• Amount = Principal + Simple Interest
• Compound Interest = $$ P(1+\dfrac{R}{100})^{T}$$ - P
• For the same principal, positive rate of interest and time period, the compound interest on the loan is always greater than or equal to the simple interest.

When interest is not compounded annually but with a periodicity of "n", i.e. interest is incurred "n" times annually:

A = $$P\left(1+\frac{R}{n\cdot100}\right)^{n\cdot T}$$

For example, when the periodicity is 2, i.e. "n" = 2, the interest is compounded half-yearly:

A=$$P\left(1+\dfrac{R}{200}\right)^{2T}$$

and when the periodicity is 4, i.e. "n" = 4, the interest is compounded quarterly:

$$A = P\left(1+\dfrac{R}{400}\right)^{4T}$$

For 2 years:

$$CI - SI = P\left(\dfrac{R}{100}\right)^2$$

For 3 years:

$$CI - SI = P\left(\dfrac{R}{100}\right)^2\left(3+\dfrac{R}{100}\right)$$

If an amount 'P' is borrowed for 'n' years at r% per annum compounded annually, and x is the instalment that is paid at the end of each year, starting from the first year, then:

$$P\ =\ \dfrac{\ x}{1+\dfrac{r}{100}}+\ \dfrac{\ x}{\left(1+\dfrac{r}{100}\right)^{^2}}+...+\ \dfrac{\ x}{\left(1+\dfrac{r}{100}\right)^{^n}}$$

or 

$$P\ \left(1+\frac{r}{100}\right)^{n\ }=\ x\ \left(\left(1+\frac{r}{100}\right)^{n-1}+\left(1+\frac{r}{100}\right)^{n-2\ }...\ +1\right)$$

The same formula can be used when a man purchased an article costing Rs P and decided to pay the amount in yearly instalments of Rs. X with an interest rate of r% per annum compounded annually over a time period of 'n' years.