Understanding the Compound Interest Problem

The question asks us to calculate the final amount obtained from an initial investment (principal) after a certain period, considering compound interest that is calculated quarterly.

Here are the details given:

• Principal Amount (\(P\)): INR 15000
• Annual Interest Rate (\(R\)): 10%
• Compounding Frequency: Quarterly
• Time Period (\(T\)): 2 years

Compound Interest Formula Explained

The formula to calculate the future value (\(A\)) of an investment with compound interest is:

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

Where:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (the initial amount 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 number of years the money is invested or borrowed for.

Step-by-Step Calculation

First, let's convert the annual interest rate to its decimal form and determine the values for \(n\) and \(nt\).

• Annual interest rate (\(R\)) = 10% = 0.10
• Since interest is compounded quarterly, the number of compounding periods per year (\(n\)) = 4.
• The time period (\(t\)) = 2 years.

Now, let's calculate the interest rate per quarter (\(\frac{r}{n}\)) and the total number of compounding periods (\(nt\)):

• Interest rate per quarter = \(\frac{r}{n} = \frac{0.10}{4} = 0.025\)
• Total number of compounding periods = \(nt = 4 \times 2 = 8\)

Next, we plug these values into the compound interest formula:

\(A = 15000 \left(1 + 0.025\right)^{8}\)

\(A = 15000 \left(1.025\right)^{8}\)

Now, we need to calculate \((1.025)^{8}\):

\((1.025)^{8} \approx 1.2184029\)

Finally, multiply this by the principal amount:

\(A = 15000 \times 1.2184029\)

\(A \approx 18276.0435\)

Determining the Final Yield

The calculated amount after 2 years, compounded quarterly, is approximately INR 18276.04.

Comparing this result with the given options:

• Option 1: INR 18432
• Option 2: INR 18276
• Option 3: INR 18654
• Option 4: INR 18225

The calculated value is closest to Option 2.