Calculating Compound Interest Compounded Half-Yearly

Let's break down how to calculate compound interest, especially when the interest is compounded more than once a year, like in this question where it's compounded half-yearly.

The problem asks us to find the compound interest on a principal amount of ₹8,000 at an annual interest rate of 20% for a period of 1 year, with the condition that the interest is compounded half-yearly.

Understanding Half-Yearly Compounding

When interest is compounded half-yearly, it means that the interest is calculated and added to the principal twice a year (every six months). This affects the rate and the time period used in the compound interest formula.

• The annual interest rate is divided by the number of compounding periods per year. Since it's half-yearly, there are 2 periods in a year.
• The total time period in years is multiplied by the number of compounding periods per year to get the total number of compounding periods.

Given:

• Principal amount (P) = ₹8,000
• Annual interest rate (R) = 20% per annum
• Time period (T) = 1 year
• Compounding frequency = Half-yearly (2 times a year)

Adjusting Rate and Time for Half-Yearly Calculation

For half-yearly compounding:

• Rate per compounding period (r) = Annual Rate / Number of periods per year
• r = 20% / 2 = 10% per half-year
• In decimal form, r = 10 / 100 = 0.10
• Number of compounding periods (n) = Time in years × Number of periods per year
• n = 1 year × 2 periods/year = 2 periods

Applying the Compound Interest Formula

The formula for the amount (A) after compounding is:

\(A = P(1 + r)^n\)

Where:

• \(P\) is the principal amount
• \(r\) is the interest rate per compounding period
• \(n\) is the total number of compounding periods

Let's plug in the values:

\(A = 8000(1 + 0.10)^2\)

\(A = 8000(1.10)^2\)

\(A = 8000 \times 1.21\)

\(A = 9680\)

So, the amount after 1 year compounded half-yearly is ₹9,680.

Calculating the Compound Interest

The compound interest (CI) is the difference between the total amount and the principal amount.

\(CI = A - P\)

\(CI = 9680 - 8000\)

\(CI = 1680\)

Therefore, the compound interest is ₹1,680.

Revision Table: Compound Interest Formulas

Understanding the different compounding periods is key. Here's a quick look at how the rate and number of periods change:

Additional Information: Simple vs. Compound Interest

It's helpful to compare compound interest to simple interest to see the difference.

• Simple Interest: Calculated only on the principal amount. The interest earned does not get added back to the principal to earn further interest.
• Compound Interest: Calculated on the principal amount and also on the accumulated interest of previous periods. This is often referred to as "interest on interest" and leads to faster growth of the investment or loan amount over time compared to simple interest.

In this problem, if the interest was simple interest:

Simple Interest = \(P \times R \times T / 100\)

Simple Interest = \(8000 \times 20 \times 1 / 100\)

Simple Interest = ₹1,600

Comparing this to the compound interest of ₹1,680, we can see that the compound interest is higher because the interest from the first half-year also earned interest in the second half-year.