Understanding the Compound Interest Problem

This problem involves calculating the annual interest rate for an investment where the interest is compounded annually. We are given the initial principal amount, the final amount after a certain period, and the time period in years.

Key Concepts: Compound Interest

Compound interest is interest calculated on the initial principal and also on the accumulated interest from previous periods. This means the interest earned also earns interest over time, leading to faster growth compared to simple interest.

The formula for compound interest when interest is compounded annually is:

\(A = P(1 + \frac{r}{100})^n\)

Where:

• \(A\) is the final amount after \(n\) years
• \(P\) is the principal amount (initial investment)
• \(r\) is the annual interest rate (in percent)
• \(n\) is the number of years

Analyzing the Given Information

From the question, we have the following values:

• Principal Amount (\(P\)) = Rs. 10,000
• Final Amount (\(A\)) = Rs. 11,449
• Time Period (\(n\)) = 2 years
• We need to find the annual interest rate (\(r\)).

Step-by-Step Solution to Find the Interest Rate

We will plug the given values into the compound interest formula and solve for \(r\).

\(A = P(1 + \frac{r}{100})^n\)

Substitute the values:

\(11449 = 10000(1 + \frac{r}{100})^2\)

Divide both sides by 10000:

\(\frac{11449}{10000} = (1 + \frac{r}{100})^2\)

\(1.1449 = (1 + \frac{r}{100})^2\)

To isolate the term \((1 + \frac{r}{100})\), take the square root of both sides:

\(\sqrt{1.1449} = \sqrt{(1 + \frac{r}{100})^2}\)

\(\sqrt{1.1449} = 1 + \frac{r}{100}\)

Calculate the square root of 1.1449:

\(1.07 = 1 + \frac{r}{100}\)

Now, subtract 1 from both sides to find the value of \(\frac{r}{100}\):

\(1.07 - 1 = \frac{r}{100}\)

\(0.07 = \frac{r}{100}\)

Multiply both sides by 100 to find \(r\):

\(r = 0.07 \times 100\)

\(r = 7\)

So, the annual interest rate is 7%.

Conclusion

The calculated annual interest rate is 7%. This matches option 3 provided in the question.

Revision Table: Compound Interest Terms

Additional Information: Compound vs. Simple Interest

It's important to distinguish compound interest from simple interest. Simple interest is calculated only on the principal amount.

• Simple Interest Formula: \(SI = \frac{P \times r \times n}{100}\)
• Total Amount with Simple Interest: \(A = P + SI = P(1 + \frac{rn}{100})\)

In simple interest, the interest earned in each period remains constant, while in compound interest, the interest earned increases with each period because it is calculated on a growing principal (original principal + accumulated interest).

For the same principal, rate, and time (greater than 1 year), compound interest will always yield a higher final amount than simple interest.