Compound interest | Part-2

Eight Standard >> Compound interest | Part-2

Half-yearly, quarterly and monthly compound interest

Illustration: Suppose we want to calculate the compound interest, when principal sum is Rs 5000, rate of interest is 10% per annual for 3 years compounded annually.

Ans:

\(A = P(1 + \frac{r}{n})^{(nt)}\)

Where: A = the final amount after time t (including the principal and compound interest) P = the principal amount (Rs 5000 in this case)
r = the yearly interest rate (e.g., 10% expressed as 0.10 in decimal form).
n = the times the interest is compounded within a year (if compounded yearly, then n = 1).
t = the total time period in years. (3 years in this case)

Now, let's plug in the values and calculate:

A =\( 5000(1 + \frac{0.10}{1})^{(1*3)}\)

A = \(5000(1 + 0.10)^3\)

A = \(5000(1.10)^3\)

(A = \(5000 * 1.331\)

A = 6655

Therefore, the compound interest accrued on Rs 5,000 over 3 years at an annual interest rate of 10%, with yearly compounding, amounts to Rs 1,655 (i.e., Rs 6,655 − Rs 5,000).

When the interest is compounded more frequently, such as half-yearly, quarterly, or monthly, the formula for compound interest changes slightly to take into account the compounding periods.

1. Interest Compounded Half-Yearly: The formula for compound interest when interest is compounded half-yearly is:

   A = \(P * (1 + \frac{r}{2})^{(2*t)}\)

   Where: A represents the total amount after time t, including both the initial principal and the compound interest. P is the initial investment or principal. r stands for the yearly interest rate, written in decimal form. t stands for the number of years.

2. Interest Compounded Quarterly: The formula for compound interest when interest is compounded quarterly is:

   A = \(P * (1 + \frac{r}{4})^{(4*t)}\)

   A represents the total accumulated amount after time t, which includes both the principal and the compound interest.
   P stands for the initial principal amount.
   r is the annual interest rate expressed as a decimal.
   t refers to the number of years the money is invested or borrowed for.

3. Interest Compounded Monthly: The formula for compound interest when interest is compounded monthly is:

   A = \(P * (1 + \frac{r}{12})^{(12*t)}\)

   Where: A = the final amount after time t (including the principal and compound interest) P = the principal amount r = the annual interest rate (expressed as a decimal) t = the number of years

These formulas take into account the more frequent compounding periods, allowing for more precise calculations of compound interest when interest is applied more frequently throughout the year.