Understanding Compound Interest Differences

The question asks for the difference in compound interest earned on a sum of Rs. 10,000 for 1 year at a rate of 10% per annum, calculated under two different compounding frequencies: yearly and half-yearly.

Let's break down the calculations for each scenario.

Scenario 1: Compound Interest Compounded Yearly

When interest is compounded yearly, the interest is calculated and added to the principal once a year.

• Principal (P) = Rs. 10,000
• Annual Interest Rate (R) = 10%
• Time (T) = 1 year
• Number of times interest is compounded per year (n) = 1 (for yearly compounding)

The formula for the Amount (A) when compounded yearly is:

\(A = P \left(1 + \frac{R}{100}\right)^T\)

Plugging in the values:

\(A_{yearly} = 10000 \left(1 + \frac{10}{100}\right)^1\)

\(A_{yearly} = 10000 \left(1 + 0.1\right)^1\)

\(A_{yearly} = 10000 \times (1.1)\)

\(A_{yearly} = 11000\)

The compound interest (\(CI_{yearly}\)) is the Amount minus the Principal:

\(CI_{yearly} = A_{yearly} - P\)

\(CI_{yearly} = 11000 - 10000\)

\(CI_{yearly} = 1000\)

So, the compound interest when compounded yearly is Rs. 1000.

Scenario 2: Compound Interest Compounded Half-Yearly

When interest is compounded half-yearly, the interest is calculated and added to the principal twice a year.

• Principal (P) = Rs. 10,000
• Annual Interest Rate (R) = 10%
• Time (T) = 1 year
• Number of times interest is compounded per year (n) = 2 (for half-yearly compounding)

When compounding half-yearly, the rate used per compounding period is half the annual rate, and the number of periods is double the number of years.

• Rate per half-year (\(r_{half}\)) = R/n = 10%/2 = 5%
• Number of periods (\(N\)) = T \(\times\) n = 1 year \(\times\) 2 periods/year = 2 periods

The formula for the Amount (A) when compounded n times a year is:

\(A = P \left(1 + \frac{R/n}{100}\right)^{Tn}\)

Using the adjusted rate and periods for half-yearly compounding:

\(A_{half-yearly} = 10000 \left(1 + \frac{5}{100}\right)^2\)

\(A_{half-yearly} = 10000 \left(1 + 0.05\right)^2\)

\(A_{half-yearly} = 10000 \times (1.05)^2\)

\(A_{half-yearly} = 10000 \times (1.05 \times 1.05)\)

\(A_{half-yearly} = 10000 \times 1.1025\)

\(A_{half-yearly} = 11025\)

The compound interest (\(CI_{half-yearly}\)) is the Amount minus the Principal:

\(CI_{half-yearly} = A_{half-yearly} - P\)

\(CI_{half-yearly} = 11025 - 10000\)

\(CI_{half-yearly} = 1025\)

So, the compound interest when compounded half-yearly is Rs. 1025.

Calculating the Difference

The difference between the compound interests is the compound interest compounded half-yearly minus the compound interest compounded yearly.

Difference = \(CI_{half-yearly} - CI_{yearly}\)

Difference = \(1025 - 1000\)

Difference = \(25\)

The difference between the compound interests is Rs. 25.

This shows that compounding more frequently (half-yearly instead of yearly) results in slightly more compound interest over the same time period, due to the interest earned in the first half-year also earning interest in the second half-year.

Difference in CI = Rs. 1025 - Rs. 1000 = Rs. 25.

Revision Table: Compound Interest Key Terms

Additional Information: Why Compounding Frequency Matters

The more frequently interest is compounded within a year, the higher the effective annual rate tends to be, even if the nominal annual rate is the same. This is because the interest earned in earlier parts of the year starts earning interest itself in the subsequent periods. For example, with half-yearly compounding, the interest earned after 6 months is added to the principal, and this new, larger principal earns interest for the remaining 6 months. With yearly compounding, the initial principal earns interest for the full year, but no interest is earned on the interest within that year.

This difference becomes more significant with higher interest rates and longer time periods.