Calculating Compound Interest for Fractional Years

This problem asks us to calculate the compound interest on a principal amount over a period that includes a fraction of a year, with the interest compounded annually.

Understanding the Problem

We are given:

• Principal amount (P) = Rs. 30000
• Annual interest rate (R) = 15%
• Time period (T) = \(2\frac{2}{3}\) years
• Compounding frequency: Yearly

We need to find the total compound interest paid after \(2\frac{2}{3}\) years.

Formula for Compound Interest with Fractional Time

When interest is compounded yearly and the time period is \(n\frac{p}{q}\) years, where \(n\) is a whole number and \(\frac{p}{q}\) is a fraction, the total amount (A) is calculated using the formula:

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

In this specific case, \(n = 2\) and \(\frac{p}{q} = \frac{2}{3}\).

Step-by-Step Calculation of Compound Interest

Let's apply the formula with the given values:

Principal (P) = 30000

Rate (R) = 15%

Time (T) = \(2\frac{2}{3}\) years

Here, the whole number of years is \(n = 2\), and the fractional part is \(\frac{p}{q} = \frac{2}{3}\).

First, calculate the rate for the fractional part of the year:

Rate for \(\frac{2}{3}\) year = \(\frac{2}{3} \times 15\% = 10\%\)

Now, substitute the values into the compound interest formula for fractional time:

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

\(A = 30000 \left(1 + \frac{15}{100}\right)^2 \left(1 + \frac{10}{100}\right)\)

\(A = 30000 \left(1 + 0.15\right)^2 \left(1 + 0.10\right)\)

\(A = 30000 (1.15)^2 (1.10)\)

Calculate \((1.15)^2\):

\((1.15)^2 = 1.15 \times 1.15 = 1.3225\)

Now, substitute this value back into the equation for A:

\(A = 30000 \times 1.3225 \times 1.10\)

Calculate \(1.3225 \times 1.10\):

\(1.3225 \times 1.10 = 1.45475\)

Finally, calculate the total amount A:

\(A = 30000 \times 1.45475\)

\(A = 43642.50\)

The total amount after \(2\frac{2}{3}\) years is Rs. 43642.50.

To find the compound interest (CI), subtract the principal from the total amount:

\(CI = A - P\)

\(CI = 43642.50 - 30000\)

\(CI = 13642.50\)

The interest to be paid is Rs. 13642.50.

Revision Table: Compound Interest Concepts

Additional Information on Compound Interest

Compound interest is often called "interest on interest". It grows faster than simple interest because the interest earned in each period is added to the principal, and subsequent interest is calculated on this new, larger principal.

• Simple Interest: Interest is calculated only on the initial principal amount for the entire duration. Formula: \(SI = \frac{P \times R \times T}{100}\).
• Compounding Frequency: How often the interest is added to the principal. It can be yearly, half-yearly, quarterly, monthly, etc. The more frequent the compounding, the higher the interest earned (or paid) for the same annual rate.
• Impact of Time: Compound interest has a much greater impact over longer time periods compared to simple interest.
• Applications: Compound interest is used in calculating returns on investments (like savings accounts, fixed deposits) and the cost of loans (like mortgages, credit cards).

For periods involving fractions of a year, as seen in this problem, the compound interest is calculated by applying the compound interest formula for the whole number of years and then applying simple interest for the fractional part on the amount accumulated at the end of the last whole year. The formula \(A = P \left(1 + \frac{R}{100}\right)^n \left(1 + \frac{\frac{p}{q} R}{100}\right)\) effectively combines these two steps to give the total amount directly.