Understanding the Interest Calculation Problem

This problem asks us to find the difference between two ways of calculating interest on a specific amount of money: compound interest (calculated annually) and simple interest. We are given the principal amount, the time period, and the annual interest rate.

Key Information Provided:

• Principal Amount (P): ₹31,250
• Time Period (T or n): 2 years
• Annual Interest Rate (R): 8%

Calculating Simple Interest (SI)

Simple interest is calculated only on the initial principal amount. The formula for Simple Interest is:

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

Plugging in the values:

$$ SI = \frac{31250 \times 8 \times 2}{100} $$

$$ SI = \frac{31250 \times 16}{100} $$

$$ SI = 312.50 \times 16 $$

$$ SI = 5000 $$

So, the Simple Interest earned is ₹5,000.

Calculating Compound Interest (CI)

Compound interest is calculated on the principal amount plus the accumulated interest from previous periods. Since the interest is compounded annually, we calculate the total amount after 2 years.

The formula for the Amount (A) with compound interest is:

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

Where:

• P = Principal = ₹31,250
• R = Annual Rate = 8%
• n = Number of years = 2

Let's calculate the Amount (A):

$$ A = 31250 \left(1 + \frac{8}{100}\right)^2 $$

$$ A = 31250 \left(1 + 0.08\right)^2 $$

$$ A = 31250 \left(1.08\right)^2 $$

$$ A = 31250 \times 1.1664 $$

$$ A = 36450 $$

Now, we find the Compound Interest (CI) by subtracting the principal from the total amount:

$$ CI = A - P $$

$$ CI = 36450 - 31250 $$

$$ CI = 5200 $$

So, the Compound Interest earned is ₹5,200.

Finding the Difference Between Interests

The question asks for the difference between the compound interest and the simple interest.

Difference = CI - SI

Difference = ₹5,200 - ₹5,000

Difference = ₹200

Alternative Method: Difference Formula for 2 Years

For a period of 2 years, the difference between compound interest and simple interest can be directly calculated using the formula:

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

Using the given values:

$$ \text{Difference} = 31250 \left(\frac{8}{100}\right)^2 $$

$$ \text{Difference} = 31250 \times (0.08)^2 $$

$$ \text{Difference} = 31250 \times 0.0064 $$

$$ \text{Difference} = 200 $$

Both methods confirm that the difference between the compound interest and the simple interest is ₹200.

Conclusion

The difference between the interest compounded annually and the simple interest on a sum of ₹31,250 for 2 years at 8% per annum is ₹200.