Fundamentals of Interest Calculation :

Interest is defined as the additional money paid by a borrower to a lender (such as a bank or patapedhi) after a stipulated time.

Simple Interest :

Simple interest is calculated using a fixed formula based on the original principal:

Formula: I = \(\frac{PNR}{100}\)

Variables:

• P (Principal): The original sum of money borrowed or invested.
• N (Number of years): The duration of the loan or investment.
• R (Rate): The interest rate per cent per annum (p.c.p.a.).

Compound Interest :

When interest is charged on the principal and also on the interest accrued in the previous year, the total interest payable at the end of the period is called the compound interest.

The Transition to Compound Interest :

Compound interest occurs when a borrower fails to pay the interest at the end of a year. In such cases, the bank adds the first year’s interest to the original principal to form a new principal for the second year.

• Interest on Interest: The second year’s interest is calculated on the "amount" (Principal + Interest) of the first year.
• Constant Ratio: For every year, the ratio of \(\frac{Amount}{Principle}\) remains constant.

The Compound Interest Formula :

To streamline calculations over multiple years, a standard formula is used to find the final amount (A) instead of calculating interest year-by-year.

If the interest rate is R, the amount of ₹1 after one year is \(1×(1+\frac{R}{100})\). To find the amount after N years, the principal is multiplied by this ratio N times.

Standard Formula: A = \(P(1+\frac{R}{100})^N\)

Finding Interest (I): Once the total amount is calculated, the compound interest is found by subtracting the principal from the amount:

Compound interest (I) = Amount – Principal = A - P

Compound Interest Variable :

Variations in Compounding Frequency :

While standard calculations are annual, financial institutions may use different intervals. The formula is adjusted accordingly:

• Six-monthly (Semi-annual): If interest is calculated every six months, the rate is taken as \(\frac{R}{2}\) and the duration is considered as 2N stages.
• Monthly: The rate is adjusted to \(\frac{R}{12}\) and the duration is taken as 12 x N stages.
• Daily: Many modern banks now calculate compound interest on a daily basis.

Application of formula for compound interest :

(i) The formula for finding the amount by compound interest can be used to solve examples involving population growth and the depreciation or decrease in the value of a machinery or property.

(ii) The formula A = \(P(1+\frac{R}{100})^N\) is used in those situations where increase on the previous increase and reduction on the previous reduction has to be calculated.

(iii) In case of reduction, the rate R is taken to be negative.

For example, if a city's population of 250,000 increases at a compounding rate of 8% per year, the population after two years is calculated as:

A = 250000 × \((1+\frac{8}{100})^2\) = 291,600

Depreciation :

Depreciation refers to the reduction in the value of an article (like a vehicle or machine) over time due to use.

• Negative Rate: Because the value decreases, the rate of depreciation (R) is taken as a negative value in the formula.
• Example: A scooter purchased for ₹60,000 with a depreciation rate of 20% will be valued at ₹38,400 after two years [600000 × \((1+\frac{-20}{100})^2\)  = 38,400]