Compound Interest Difference: Half-Yearly vs. Quarterly Calculation

This solution explains how to find the difference between compound interest calculated at different compounding frequencies (half-yearly and quarterly) for a given principal amount, interest rate, and time period.

Understanding Compound Interest Calculations

Compound interest is calculated on the initial principal amount, plus the accumulated interest from previous periods. The formula for the amount (A) after time (t) is:

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

Where:

• P is the Principal amount (the initial sum of money)
• r is the annual interest rate (in decimal form)
• n is the number of times that interest is compounded per year
• t is the time the money is invested or borrowed for, in years

The Compound Interest (CI) is calculated as: CI = A - P.

Step 1: Calculate Interest Compounded Half-Yearly

Here, the interest is compounded twice a year.

• Principal (P): ₹40,000
• Annual Interest Rate (R): 20% or 0.20
• Time (t): 1 year
• Compounding Frequency (n): 2 (since it's half-yearly)

The interest rate per compounding period is $\frac{r}{n} = \frac{0.20}{2} = 0.10$.

The total number of compounding periods is $n \times t = 2 \times 1 = 2$.

Calculating the Amount (A1):

$$ A_1 = 40000 \left(1 + \frac{0.20}{2}\right)^{2 \times 1} $$ $$ A_1 = 40000 (1 + 0.10)^{2} $$ $$ A_1 = 40000 (1.10)^{2} $$ $$ A_1 = 40000 \times 1.21 $$ $$ A_1 = 48,400 $$

Calculating the Compound Interest (CI1):

$$ CI_1 = A_1 - P $$ $$ CI_1 = 48,400 - 40,000 $$ $$ CI_1 = 8,400 $$

So, the compound interest when compounded half-yearly is ₹8,400.

Step 2: Calculate Interest Compounded Quarterly

Here, the interest is compounded four times a year.

• Principal (P): ₹40,000
• Annual Interest Rate (R): 20% or 0.20
• Time (t): 1 year
• Compounding Frequency (n): 4 (since it's quarterly)

The interest rate per compounding period is $\frac{r}{n} = \frac{0.20}{4} = 0.05$.

The total number of compounding periods is $n \times t = 4 \times 1 = 4$.

Calculating the Amount (A2):

$$ A_2 = 40000 \left(1 + \frac{0.20}{4}\right)^{4 \times 1} $$ $$ A_2 = 40000 (1 + 0.05)^{4} $$ $$ A_2 = 40000 (1.05)^{4} $$ $$ A_2 = 40000 \times 1.21550625 $$ $$ A_2 = 48,620.25 $$

Calculating the Compound Interest (CI2):

$$ CI_2 = A_2 - P $$ $$ CI_2 = 48,620.25 - 40,000 $$ $$ CI_2 = 8,620.25 $$

So, the compound interest when compounded quarterly is ₹8,620.25.

Step 3: Find the Difference Between the Two Interests

To find the difference, subtract the interest calculated half-yearly from the interest calculated quarterly.

Difference = CI2 - CI1

Difference = ₹8,620.25 - ₹8,400

Difference = ₹220.25

Conclusion on Interest Difference

The difference between the compound interest on ₹40,000 at 20% per annum compounded half-yearly and compounded quarterly for one year is ₹220.25.