Understanding the Scenario: Interest Difference

The question asks us to determine the initial amount of money invested, known as the principal. We are given that the difference between the compound interest (calculated annually) and the simple interest earned over a period of 3 years is ₹228. The annual interest rate is 4%.

Key Formulas for Interest Calculation

To solve this, we need the formulas for Simple Interest (SI) and Compound Interest (CI).

Let:

• \( P \) = Principal amount
• \( R \) = Annual interest rate (in percent)
• \( T \) = Time period (in years)

The formula for Simple Interest is:

$$ SI = \frac{P \times R \times T}{100} $$

The formula for Compound Interest compounded annually is:

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

Difference Between CI and SI for 3 Years

For calculations involving the difference between compound interest and simple interest over 3 years, a specific formula can be used, which simplifies the process:

$$ \text{Difference} = P \left( \frac{R}{100} \right)^2 \left( 3 + \frac{R}{100} \right) $$

This formula directly relates the principal, rate, and the interest difference for a 3-year period.

Applying the Formula to the Problem

We are provided with the following information:

• Difference = ₹228
• Annual Interest Rate \( R = 4\% \)

We need to find the Principal \( P \). Substitute the given values into the difference formula:

$$ 228 = P \left( \frac{4}{100} \right)^2 \left( 3 + \frac{4}{100} \right) $$

Step-by-Step Calculation

Let's break down the calculation step-by-step:

1. Simplify the rate fraction: \( \frac{4}{100} = \frac{1}{25} \).
2. Substitute this simplified value back into the equation:

   $$ 228 = P \left( \frac{1}{25} \right)^2 \left( 3 + \frac{1}{25} \right) $$

3. Calculate the squared term and the term in the parenthesis:

   $$ \left( \frac{1}{25} \right)^2 = \frac{1}{625} $$

   $$ 3 + \frac{1}{25} = \frac{75}{25} + \frac{1}{25} = \frac{76}{25} $$

4. Substitute these results back into the equation:

   $$ 228 = P \left( \frac{1}{625} \right) \left( \frac{76}{25} \right) $$

5. Multiply the fractions:

   $$ 228 = P \left( \frac{1 \times 76}{625 \times 25} \right) $$

   $$ 228 = P \left( \frac{76}{15625} \right) $$

6. Now, solve for \( P \) by rearranging the equation:

   $$ P = \frac{228 \times 15625}{76} $$

7. To simplify, notice that \( 228 \) is exactly \( 3 \) times \( 76 \) (since \( 3 \times 76 = 228 \)):

   $$ P = \frac{(3 \times 76) \times 15625}{76} $$

8. Cancel out the common factor of \( 76 \) in the numerator and the denominator:

   $$ P = 3 \times 15625 $$

9. Perform the final multiplication:

   $$ P = 46875 $$

Final Answer Determination

The calculation shows that the principal amount is ₹46,875. This value corresponds to option 4.