Understanding the Difference Between Compound Interest and Simple Interest

This problem asks us to find the annual rate of interest when we are given the principal amount, the time period, and the difference between the compound interest (compounding annually) and the simple interest earned over that period.

We are given:

• Principal amount (P) = ₹500
• Time period (T) = 2 years
• Difference between Compound Interest and Simple Interest (CI - SI) = ₹1.25

Our goal is to find the Rate of Interest (R) per annum.

Formulas for Simple Interest and Compound Interest

The formula for Simple Interest (SI) is:

\( \text{SI} = \frac{P \times R \times T}{100} \)

The formula for the Amount (A) under Compound Interest is:

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

The formula for Compound Interest (CI) is:

\( \text{CI} = A - P = P \left(1 + \frac{R}{100}\right)^T - P \)

Setting up the Equation for the Difference

The difference between Compound Interest and Simple Interest is given as CI - SI = ₹1.25. We can write this as:

\( \text{CI} - \text{SI} = \left[ P \left(1 + \frac{R}{100}\right)^T - P \right] - \frac{P \times R \times T}{100} \)

Substitute the given values into this equation:

\( 1.25 = \left[ 500 \left(1 + \frac{R}{100}\right)^2 - 500 \right] - \frac{500 \times R \times 2}{100} \)

Solving for the Rate of Interest (R)

Let's simplify the equation step-by-step:

\( 1.25 = 500 \left(1 + \frac{R}{100}\right)^2 - 500 - \frac{1000R}{100} \)

\( 1.25 = 500 \left(1^2 + 2 \times 1 \times \frac{R}{100} + \left(\frac{R}{100}\right)^2\right) - 500 - 10R \)

\( 1.25 = 500 \left(1 + \frac{2R}{100} + \frac{R^2}{10000}\right) - 500 - 10R \)

\( 1.25 = 500 \times 1 + 500 \times \frac{2R}{100} + 500 \times \frac{R^2}{10000} - 500 - 10R \)

\( 1.25 = 500 + \frac{1000R}{100} + \frac{500R^2}{10000} - 500 - 10R \)

\( 1.25 = 500 + 10R + \frac{R^2}{20} - 500 - 10R \)

Notice that the \(500\) and \(-500\) terms cancel out, and the \(10R\) and \(-10R\) terms cancel out.

\( 1.25 = \frac{R^2}{20} \)

Now, solve for \(R^2\):

\( R^2 = 1.25 \times 20 \)

\( R^2 = 25 \)

Take the square root of both sides to find R:

\( R = \sqrt{25} \)

\( R = 5 \)

So, the rate of interest is 5% per annum.

Alternative Method using the Direct Formula for CI - SI for 2 Years

For a period of 2 years, the difference between Compound Interest and Simple Interest can be directly calculated using the formula:

\( \text{CI} - \text{SI} = P \left( \frac{R}{100} \right)^2 \)

Substitute the given values into this formula:

\( 1.25 = 500 \left( \frac{R}{100} \right)^2 \)

\( 1.25 = 500 \times \frac{R^2}{10000} \)

\( 1.25 = \frac{500R^2}{10000} \)

\( 1.25 = \frac{R^2}{20} \)

Solving for \(R^2\):

\( R^2 = 1.25 \times 20 \)

\( R^2 = 25 \)

Solving for R:

\( R = \sqrt{25} \)

\( R = 5 \)

Both methods yield the same result. The rate of interest is 5% per annum.

The final answer is 5%.

Revision Table: Key Concepts in Compound vs Simple Interest

Additional Information: Importance of Interest Rate in Financial Calculations

The interest rate is a crucial component in both simple and compound interest calculations. It represents the cost of borrowing money or the return on an investment, expressed as a percentage of the principal amount per period (usually per year).

• Simple Interest: The interest is calculated only on the initial principal amount. The interest earned does not contribute to the principal for subsequent periods.
• Compound Interest: The interest earned in each period is added to the principal for the next period. This means that interest is earned on the initial principal as well as on the accumulated interest from previous periods. This effect is known as compounding and leads to faster growth of the investment or debt compared to simple interest over time, especially for longer periods and higher interest rates.
• Rate Fluctuation: In real-world scenarios, interest rates can be fixed or variable. Fixed rates remain constant over the loan or investment term, while variable rates can change based on market conditions.
• Effective Annual Rate (EAR): When compounding happens more frequently than annually (e.g., semi-annually, quarterly, monthly), the Effective Annual Rate (EAR) or Annual Percentage Yield (APY) gives a true picture of the return, as it accounts for the effect of compounding within the year. The formula we used assumes annual compounding, so the rate R is the annual rate.

Understanding the difference between how interest is calculated and the impact of the interest rate is fundamental for making informed financial decisions, whether it's saving, investing, or borrowing.