Understanding the Compound Interest Problem

The question requires us to first determine the original amount of money (the principal) that was invested. We are told that this investment grew to ₹ 3,267 over a period of 2 years, earning an interest rate of 10% per year, compounded annually.

Calculating the Principal Sum

To find the principal sum ($P$), we can use the formula for the final amount ($A$) when interest is compounded annually:

$$ A = P \left(1 + \frac{R}{100}\right)^T $$

In this formula:

• $A$ represents the final amount, which is ₹ 3,267.
• $P$ represents the principal sum, which we need to find.
• $R$ represents the annual interest rate, given as 10%.
• $T$ represents the time period in years, given as 2 years.

Let's substitute the known values into the formula:

$$ 3267 = P \left(1 + \frac{10}{100}\right)^2 $$

First, simplify the term inside the parentheses:

$$ 1 + \frac{10}{100} = 1 + 0.1 = 1.1 $$

Now substitute this back into the equation:

$$ 3267 = P (1.1)^2 $$

Calculate the value of $(1.1)^2$:

$$ (1.1)^2 = 1.21 $$

The equation becomes:

$$ 3267 = P \times 1.21 $$

To find the principal ($P$), we rearrange the equation:

$$ P = \frac{3267}{1.21} $$

Performing the division gives us:

$$ P = 2700 $$

Therefore, the original sum invested was ₹ 2,700.

Calculating the Simple Interest

The question now asks for the simple interest earned on this same principal sum (₹ 2,700) at the same interest rate (10% per year) but over a different time period: $2\frac{1}{2}$ years.

The formula for calculating simple interest ($SI$) is:

$$ SI = \frac{P \times R \times T}{100} $$

Where:

• $SI$ is the simple interest we want to calculate.
• $P$ is the principal sum, which is ₹ 2,700.
• $R$ is the annual interest rate, which is 10%.
• $T$ is the time period in years.

First, let's express the time period $T$ as a decimal:

$$ T = 2\frac{1}{2} \text{ years} = 2.5 \text{ years} $$

Now, substitute the values of $P$, $R$, and $T$ into the simple interest formula:

$$ SI = \frac{2700 \times 10 \times 2.5}{100} $$

Calculate the product in the numerator:

$$ SI = \frac{27000 \times 2.5}{100} $$

$$ SI = \frac{67500}{100} $$

Finally, perform the division to find the simple interest:

$$ SI = 675 $$

Final Answer Summary

The simple interest calculated on the principal sum of ₹ 2,700 at an annual rate of 10% for $2\frac{1}{2}$ years is ₹ 675.