Calculating Compound Interest for Fractional Years

This problem asks us to calculate the compound interest on a principal amount of Rs. 2,000 for a period of \(2\frac{1}{2}\) years at an annual interest rate of 10%, compounded annually. Compound interest for a fractional period requires calculating the interest for the whole number of years and then the simple interest on the accumulated amount for the remaining fractional part.

Understanding the Given Information

• Principal amount (P) = Rs. 2,000
• Time (n) = \(2\frac{1}{2}\) years
• Rate of interest (R) = 10% per annum
• Compounding frequency: Annually

Formula for Amount with Fractional Time Period

When the time period is in the form of \(a\frac{b}{c}\) years, where 'a' is the whole number of years and \(\frac{b}{c}\) is the fractional part, and the interest is compounded annually, the amount (A) can be calculated using the formula:

\[A = P \left(1 + \frac{R}{100}\right)^a \left(1 + \frac{\frac{b}{c}R}{100}\right)\]

Where:

• P = Principal
• R = Annual interest rate
• a = Whole number of years
• \(\frac{b}{c}\) = Fractional part of a year

Step-by-Step Compound Interest Calculation

Here, P = 2000, a = 2 years, \(\frac{b}{c} = \frac{1}{2}\) year, and R = 10%.

Step 1: Calculate the Amount after the Whole Number of Years (2 Years)

We can first calculate the amount after 2 full years. This is the standard compound interest calculation for a whole number of years.

Amount after 2 years \(A_2 = P \left(1 + \frac{R}{100}\right)^2\)

\[A_2 = 2000 \left(1 + \frac{10}{100}\right)^2\]

\[A_2 = 2000 \left(1 + 0.1\right)^2\]

\[A_2 = 2000 \left(1.1\right)^2\]

\[A_2 = 2000 \times 1.21\]

\[A_2 = 2420\]

So, after 2 full years, the amount is Rs. 2,420.

Step 2: Calculate Interest for the Fractional Part (Remaining \(\frac{1}{2}\) Year)

Now, we need to calculate the interest for the remaining \(\frac{1}{2}\) year. This interest is calculated on the amount accumulated at the end of the whole number of years (Rs. 2,420) using the simple interest method. The rate for this period is the annual rate multiplied by the fraction of the year.

Rate for \(\frac{1}{2}\) year = \(R \times \frac{b}{c} = 10\% \times \frac{1}{2} = 5\%\)

Interest for \(\frac{1}{2}\) year = Simple Interest on Rs. 2,420 at 5% per annum for 1 year (or at 10% p.a. for \(\frac{1}{2}\) year).

Using the formula for Simple Interest \(SI = \frac{P \times R \times T}{100}\), where P is the amount at the start of the period, R is the annual rate, and T is the time in years:

\[\text{Interest for } \frac{1}{2} \text{ year} = \frac{2420 \times 10 \times \frac{1}{2}}{100}\]

\[\text{Interest for } \frac{1}{2} \text{ year} = \frac{2420 \times 5}{100}\]

\[\text{Interest for } \frac{1}{2} \text{ year} = \frac{12100}{100}\]

\[\text{Interest for } \frac{1}{2} \text{ year} = 121\]

The interest for the remaining half year is Rs. 121.

Step 3: Calculate the Total Amount after \(2\frac{1}{2}\) Years

The total amount at the end of \(2\frac{1}{2}\) years is the sum of the amount after 2 years and the interest for the next \(\frac{1}{2}\) year.

Total Amount \(A = A_2 + \text{Interest for } \frac{1}{2} \text{ year}\)

\[A = 2420 + 121\]

\[A = 2541\]

The total amount after \(2\frac{1}{2}\) years is Rs. 2,541.

Step 4: Calculate the Compound Interest

Compound Interest (CI) is the total amount minus the original principal.

\[CI = \text{Total Amount} - \text{Principal}\]

\[CI = 2541 - 2000\]

\[CI = 541\]

The compound interest is Rs. 541.

Summary Table

Final Answer

The compound interest on Rs. 2,000 for \(2\frac{1}{2}\) years at 10% per annum, compounded annually, is Rs. 541.

Revision Table: Compound Interest Key Concepts

Additional Information: Handling Fractional Time in CI

When calculating compound interest for a period that includes a fraction of a year, and compounding is annual:

• The full number of years are compounded normally.
• For the remaining fractional part of the year, the interest is calculated as simple interest on the amount accumulated at the end of the last full year.
• The rate for the simple interest calculation is the annual rate multiplied by the fraction of the year. For example, for \(\frac{1}{2}\) year at 10% p.a., the rate used for simple interest calculation on the accumulated amount is \(10\% \times \frac{1}{2} = 5\%\).

This method ensures that interest earned during the full periods is compounded, while the interest for the partial period is treated as a simple accrual on the amount at the beginning of that partial period.