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# Interest Calculated on Varying Principal Amounts

## Introduction

The additional income on the money invested is termed as interest. In simple words, interest is charged on any loan or credit given or obtained. Interest charged is of two types: compound interest and simple interest. The interest charged depends on the principal loan amount, rate of interest and period for which the same is granted.

Let’s study the concepts in detail.

## Compound interest (C.I.)

The calculation of compound interest depends upon the amount given as loan or credit, I.e., principal which varies according to change in tenure or period. The rate of interest that is compounded in a specified tenure when added to the principal amount gives a new loan amount and this stands as the principal for the upcoming year. Thus, in the compound interest format, the principal amount rises from the first interest interval to another. In simple and easy words, the interest calculated on the principal amount plus the interest collected in past accounting years is known as the compound interest. The banking and the financial institutions or sectors utilise the concept for accuracy in calculations. Some of the applications for the same are:

Variations in population levels.

Bacterial terminology.

Variations in values of goods and services.

This is the prior concept of the routine schedule. Mostly the rates charged by banks on loans or credits etc. are compound ones as it goes on varying for the principal.

Simply, C.I. can be calculated by subtracting the original principal amount from the final amount which includes the interest for the respective year, i.e.,

Compound interest= Amount-Principal

Where Amount= Principal (1+ R/100) ^Time

If the interest is subject to annual calculations, then the above-mentioned are the base relations regarding the concept.

But there are different conditions of the interest rate under this concept. The compound interest is subject to a yearly, half-yearly, quarterly, monthly etc. calculating basis. The period changes the related calculations and finally the resultant computation.

The online availability of a compound interest calculator is a big help in calculating the accurate amount and interest.

## Derivation of the formula

To derive the respective formula, the relation between simple interest formula, principal amount, rate of interest and period are required to be discussed because simple interest, S.I., as for the first accounting year both types of interests are same.

If P is taken as principal, R is assumed to be a rate of interest and T refers to the time in years so, S.I. for the first year will be found by the relation of S.I.= (P*R*T)/100

Hence, Amount= P+S.I. (after the first year)

Now, in the case of Compound interest, the amount at the end of the first year becomes the principal amount for the next consecutive accounting year. Hence, the interest for the second year is calculated on the new amount and similarly, this goes on till n number of years. For an nth year, the relation to calculate compound interest is,

Amount=P (1+ Rate/100) ^n

C.I.= A-P= P [(1+ R/100) ^n -1]

## Computing the same half-yearly

Cuemath describes the topic in an efficient manner and even include the computation of the compound interest when the rate of interest is compounded half-yearly.

As the interest is to be calculated in half the year so the change in the principal amount is subject to variations every six months. Then the interest for the next (second) six months will be calculated on the new amount calculated after adding the interest to the original amount. This means that if, S.I.1= (P*R*1)/ (100*2) then, the Amount after six months will be=P+S.I. which is the principal for the next six months.

Now the interest will be calculated by taking the new amount as principal. Similarly, this will continue till the last final period.

Remember: While calculating the interest compounded half-yearly the rate is divided by 2 and time is multiplied by 2 before substituting the values in the general formulas.

The same is the procedure of calculating compound interest quarterly and periodically. Here, the rate is divided by 4 and time is multiplied by 4 if the same is calculated quarterly.

Total accumulated value, under the concept, is given by the relation

New Principal, P’=P [1+ (r/n)]^(n*t)

where P implies the principal amount

r implies a rate of interest

P’ implies the new principal amount

n implies a number of times principal is compounded

t  is time.

Here, C.I.= P’-P

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