Calculating Compound Interest with Half-Yearly Compounding

To calculate the compound interest on a principal amount, we need to consider the rate of interest, the time period, and the compounding frequency. In this case, we have an initial principal amount of Rs. 1000, an interest rate of 20% per annum, and the interest is compounded half-yearly for a period of 18 months.

Formula for Compound Interest
The formula to calculate compound interest is:
A = P(1 + r/n)^(nt) - P

Where:
A = the final amount including interest
P = the principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times that interest is compounded per year
t = time period in years

Step 1: Convert Annual Interest Rate and Time Period
Since the interest is compounded half-yearly, we need to adjust the annual interest rate and time period accordingly.

Annual Interest Rate:
The annual interest rate is 20%, which can be expressed as 0.20 in decimal form.

Time Period:
The given time period is 18 months. To convert it into years, we divide it by 12 (months in a year) resulting in 1.5 years.

Step 2: Calculate Compound Interest Using the Formula
Now, let's substitute the values into the compound interest formula:

A = P(1 + r/n)^(nt) - P
A = 1000(1 + 0.20/2)^(2*1.5) - 1000

Step 3: Simplify the Calculation
Simplifying the equation:

A = 1000(1 + 0.10)^3 - 1000
A = 1000(1.10)^3 - 1000
A = 1000(1.331) - 1000
A = 1331 - 1000
A = 331

Step 4: Calculate the Compound Interest
To find the compound interest, subtract the principal amount from the final amount:

Compound Interest = A - P
Compound Interest = 331 - 1000
Compound Interest = -669

Interpretation of the Result
The compound interest on Rs. 1000 at a rate of 20% per annum for 18 months, compounded half-yearly, is -Rs. 669. This negative value suggests that the investment has decreased over time. It is important to note that compound interest can be negative when the final amount is less than the principal amount.