Interest is money paid to the *lender* by the *borrower *for using his money for a specified period of time. This article gives the basic fundamentals of Interest, specifically the Simple Interest and the Compound Interest.

The terms used in this topic and their general representation are:

Interest *(I)*, Principal *(P)*, Time *(n; n* is expressed in number of periods, which is normally one year), Rate of Interest *(r; r* is taken as the rate of interest on Re. 1 for one year, instead of the rate on Rs. 100 ), Amount *(A)***To find the interest (I) and amount (A) of a given sum (P) in a given time (n) at simple interest**

The interest of *P *for one year is* Pr*, and therefore for* n* years is *Pnr*;

i.e I = P*nr * ...(1)

Also, *A = P + I*

i.e *A = (1 + nr) * ....(2)

From (1) and (2), we see that if of the quantities—*P, r, I or P, n, r, and A—*any three be given the fourth may be found.

**To find the interest (I) and amount (A) of a given sum (P) in a given time (n) at compound interest**

The amount of P at the end of the first year is* PR*;

and since, this is the principal for the second year,

the amount at the end of the second year is *PR × R or PR ^{2}*

Similarly, the amount at the end of the third year is *PR ^{3}*, and so on;

hence the amount in n years is *PR ^{n}*

i.e., *A = PR ^{n}*

therefore *I = P(R ^{n}-1)*

**Note: **

- If
*r*denotes the interest on Re 1 for one year, we have:*R = 1 + r*

- In business transactions, when the time contains a fraction of a year, it is usual to allow
*simple interest*for the fraction of the year. Thus, the amount of Re. 1 in ½ years is reckoned ; and the amount of P is years at compound interest is

Similarly, the amount of P in years is

- If the interest is payable more than once a year there is a distinction between the
*nominal annual rate*of interest and that actually received, which may be called the*true annual rate*; thus if the interest is payable twice a year, and if r is the*nominal annual*rate of interest, the amount of Re. 1 in half a year is , and therefore in the whole year the amount of Re. 1 is ; so that the true annual rate of interest is

- If the interest is payable q times a year, i.e. compounding is done every q months, and if r is the nominal annual rate, the interest on Re. 1 for each interval is and therefore, the amount of P in n years is . In this case, the interest is said to be “converted into principal” q times a year.

- If the interest is convertible into principal every moment, then q becomes infinitely great. To find the value of the amount, put so that
*q - rx*; - Thus the amount =

- =
*Pe*(since, x is infinite when q is infinite)^{nr }

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himanshu pandey, 1 Year Ago

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