Understanding Compound and Simple Interest: A Practical Example with Step-by-Step Solutions

Understanding the nuances between compound and simple interest is crucial for anyone dealing with finances. This article dives into a specific scenario where we compare compound interest over two years to simple interest over four years at half the rate. Through a series of detailed steps, we will illustrate how to calculate the principal amount and the simple interest on the same sum.

Scenario: Calculating Compound Interest

Let’s begin with a given scenario: The compound interest on a certain sum for 2 years at a rate of 10% per annum is Rs. 725. How do we find the principal amount and subsequently calculate the simple interest on the same sum for double the time at half the rate per annum?

Given:

Compound Interest (CI) Rs. 725 Rate (R) 10% Time (T) 2 years

Step 1: Calculate the Principal Amount

The formula for compound interest is:

CI P left(1 frac{R}{100} right)^T - P

Substitute the given values:

725 P left(1 frac{10}{100} right)^2 - P

Simplify the expression:

725 P (1.1^2 - 1)

725 P (1.21 - 1)

725 0.21P

P 725 / 0.21

[](P 3452.38, approximately Rs. 2500)

Thus, the principal amount (P) is approximately Rs. 2500.

Step 2: Calculate Simple Interest

Now, we need to calculate the simple interest on the same sum for 4 years at a rate of 5% per annum.

The formula for simple interest is:

SI (P * R * T) / 100

Substitute the values P 2500, R 5, and T 4:

SI (2500 * 5 * 4) / 100

SI 500

Hence, the simple interest is Rs. 500.

Alternate Method: Shortcut Approach

To verify and simplify the process, we can use a more straightforward approach:

Step 1: Calculate Principal Amount using CI Formula

CI Principal [1 (R/100)^T - 1]

525 Principal [1 (10/100)^2 - 1]

Simplify:

525 Principal [1.1^2 - 1]

525 Principal [1.21 - 1]

525 Principal [0.21]

Principal 525 / 0.21

Principal 2500

Step 2: Calculate Simple Interest

SI (Principal * 5 * 4) / 100

SI 2500 * 5 * 4 / 100

SI 500

Through both methods, we arrive at the same result: Rs. 500.

Summary and Conclusion

The key takeaway is that the principal amount influencing both compound interest and simple interest must be calculated accurately. In this example, the principal is Rs. 2500, and the simple interest for a period of 4 years at 5% per annum amounts to Rs. 500.

Understanding these calculations and being able to apply the formulas in practical scenarios can significantly enhance financial planning and decision-making. The given example and detailed step-by-step solutions provide a clear and comprehensive approach to handling compound and simple interest.

Keywords: compound interest, simple interest, financial math