Compound Interest Rate Calculation

This problem involves calculating the annual rate of interest when the amount after different periods under compound interest is known. We are given the amount a sum of money becomes after 2 years and 3 years, with interest compounded annually.

Understanding Compound Interest

Compound interest means that the interest earned in each period is added to the principal amount for calculating the interest for the next period. The formula for the amount (A) after 'n' years when a principal amount (P) is invested at a rate of interest (R%) per annum, compounded annually, is:

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

Applying the Formula to the Problem

Let P be the principal amount and R be the annual rate of interest.

According to the question:

• The amount after 2 years (\(A_2\)) is ₹5,290. \(5290 = P \left(1 + \frac{R}{100}\right)^2\) (Equation 1)
• The amount after 3 years (\(A_3\)) is ₹6,083.50. \(6083.50 = P \left(1 + \frac{R}{100}\right)^3\) (Equation 2)

Calculating the Rate of Interest

To find the rate of interest (R), we can divide Equation 2 by Equation 1. This eliminates the principal amount (P) and helps us find the factor \((1 + R/100)\).

\( \frac{6083.50}{5290} = \frac{P \left(1 + \frac{R}{100}\right)^3}{P \left(1 + \frac{R}{100}\right)^2} \)

\( \frac{6083.50}{5290} = \left(1 + \frac{R}{100}\right)^{(3-2)} \)

\( \frac{6083.50}{5290} = \left(1 + \frac{R}{100}\right) \)

Now, let's perform the division:

\( 1 + \frac{R}{100} = 1.15 \)

To find the rate R, we rearrange the equation:

\( \frac{R}{100} = 1.15 - 1 \)

\( \frac{R}{100} = 0.15 \)

\( R = 0.15 \times 100 \)

\( R = 15 \)

Therefore, the rate of interest per annum is 15%.

Conclusion

The rate of interest per annum, compounded annually, is 15%.