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.

For 2 years:

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

For 3 years:

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

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:

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

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.