Calculating Annual Compound Interest Rate

This problem asks us to find the annual rate of interest when a principal amount grows to a certain value over a period, with interest compounded annually.

Understanding Compound Interest

Compound interest means that the interest earned in each period is added to the principal amount for the next period. The formula for the amount (A) when interest is compounded annually is:

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

Where:

• $A$ = the future value of the investment/loan, including interest
• $P$ = the principal investment amount (the initial deposit or loan amount)
• $R$ = the annual interest rate (as a percentage)
• $n$ = the number of years the money is invested or borrowed for

Given Information

From the question, we have:

• Principal ($P$): ₹10,000
• Amount ($A$): ₹12,321
• Number of years ($n$): 2 years

Step-by-Step Calculation

1. Substitute the values into the formula:

   $12,321 = 10,000 \left(1 + \frac{R}{100}\right)^2$

2. Isolate the term with the rate ($R$): Divide both sides by the principal amount ($P = 10,000$):

   $\frac{12,321}{10,000} = \left(1 + \frac{R}{100}\right)^2$

   $1.2321 = \left(1 + \frac{R}{100}\right)^2$

3. Solve for the base term $\left(1 + \frac{R}{100}\right)$: Take the square root of both sides. Since the rate must be positive, we only consider the positive square root:

   $\sqrt{1.2321} = 1 + \frac{R}{100}$

   We know that $111 \times 111 = 12321$, so $\sqrt{1.2321} = 1.11$.

   $1.11 = 1 + \frac{R}{100}$

4. Solve for $\frac{R}{100}$: Subtract 1 from both sides:

   $1.11 - 1 = \frac{R}{100}$

   $0.11 = \frac{R}{100}$

5. Calculate the rate ($R$): Multiply both sides by 100:

   $R = 0.11 \times 100$

   $R = 11$

Conclusion

The calculated annual rate of interest is 11%.