Understanding Compound Interest: Solving Semi-Compounded Interest Problems

Compound interest can be a powerful tool to increase one's wealth over time. This article will guide you through the process of calculating semi-compounded interest, particularly focusing on a specific sum of money over different periods. Using the provided problem, we will break down the calculations and provide a step-by-step solution to finding the semi-compounded interest over three years.

Solving Semi-Compounded Interest Problems

Suppose the semi-compounded interest on a sum of money over one year and two years is Rs 4200 and Rs 9282, respectively. We will determine the compound interest for the same sum over three years.

Step 1: Establishing the Equation

We can use the formula for simple interest in a differential form to represent the interest earned in the first and second year. Let P be the principal amount and r be the annual interest rate.

C.I. in 1st year P [1 r/100] - P Rs 4200 P(r/100) 4200 rarr; (1)

C.I. in 2nd year P [1 r/100]^2 - P Rs 9282 P (1 r/100)^2 - P 9282 rarr; (2)

To simplify (2), we subtract (1) from (2) to find the interest earned on the first year's interest:

9282 - 4200 (P [1 r/100]^2 - P) - P [1 r/100] - P 5082

5082 P [1 (r/100)^2] - P [1 r/100] - 4200

5082 P [1 (r/100)^2 - (1 r/100)] - 4200

5082 P [(r/100)^2 - r/100] - 4200

5082 4200 P [(r/100)^2 - r/100]

9282 P [(r/100)^2 - r/100]

Let u (r/100)^2, so the equation becomes:

9282 - 4200 5082 Pu - P rarr; 5082 Pu - P 5082 P[(r/100)^2 - r/100] Pu - P

By solving the quadratic equation and substituting the known values, we find:

(Pu - P) 5082 rarr; 5082 4200u - 4200 rarr; 9282 (r/100)^4 - (r/100)^2 (r/100)^2 121/100 rarr; r/100 11/10 r 11%

Now, substituting r 11% into (1) to find P:

P (11/100) 4200 rarr; P 20000 Thus, the principal amount, P 20000

C.I. for 3 Years

The compound interest for 3 years can be calculated as:

C.I. 20000 [1 (11/100)^3 - 1] 20000 [1.11^3 - 1] 20000 [0.221561] 15431.22

Therefore, the compound interest over three years is Rs 15431.22.

Conclusion

The detailed step-by-step process showcases the importance of leveraging compound interest to maximize returns. By using the given data, we were able to accurately calculate the interest and demonstrate the power of compound interest over time. This knowledge is valuable for financial planning and investment strategies.