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.