Problem Analysis:

The question asks to calculate the compound interest on a principal amount for a period that includes a fractional year, with interest compounded annually.

• Principal (P) = ₹ 24500
• Annual Interest Rate (R) = 10%
• Time (T) = \(2\frac{2}{5}\) years = 2.4 years
• Compounding Frequency = Yearly

Compound Interest Calculation Steps

Since the interest is compounded yearly and the time period is \(2.4\) years, we calculate the amount after 2 full years and then find the simple interest for the remaining \(0.4\) years on that amount.

Step 1: Calculate Amount after 2 Years

Using the compound interest formula for the full years:

Amount \(A_n = P \left(1 + \frac{R}{100}\right)^n\)

Here, \(n=2\) years.

\(A_2 = 24500 \left(1 + \frac{10}{100}\right)^2\)

\(A_2 = 24500 \left(1 + 0.1\right)^2\)

\(A_2 = 24500 \times (1.1)^2\)

\(A_2 = 24500 \times 1.21\)

\(A_2 = 29645\)

The amount after 2 years is ₹ 29645.

Step 2: Calculate Interest for the Fractional Year

The principal for the fractional part of the year is the amount calculated after 2 years, which is ₹ 29645.

The time period for the fractional part is \(0.4\) years.

The rate remains 10% per annum.

Calculate the Simple Interest (SI) for \(0.4\) years:

\(SI = \frac{P \times R \times T}{100}\)

\(SI_{0.4} = \frac{29645 \times 10 \times 0.4}{100}\)

\(SI_{0.4} = \frac{29645 \times 4}{100}\)

\(SI_{0.4} = 296.45 \times 4\)

\(SI_{0.4} = 1185.80\)

The simple interest for the remaining 0.4 years is ₹ 1185.80.

Step 3: Calculate Total Compound Interest

The total compound interest is the sum of the interest earned in the first 2 years and the simple interest earned in the fractional part of the year.

Interest for 2 years = \(A_2 - P = 29645 - 24500 = 5145\)

Total Compound Interest (CI) = Interest for 2 years + SI for 0.4 years

CI = \(5145 + 1185.80\)

CI = \(6330.80\)

Final Answer

The compound interest is ₹ 6330.80.