Understanding the Problem: SI vs CI Difference

The question requires us to calculate the original amount of money (the principal) that was invested. We are provided with the difference between the simple interest (SI) and the compound interest (CI), compounded annually, over a period of 2 years. The interest rate is given as 17% per annum, and the difference between CI and SI is ₹967.

Key Concepts: Simple Interest and Compound Interest

Simple Interest (SI) is calculated solely on the initial principal amount. The formula used is:

$$ \text{SI} = \frac{P \times R \times n}{100} $$

Where:

• $P$ represents the Principal sum
• $R$ represents the Rate of interest per annum
• $n$ represents the Time period in years

Compound Interest (CI) is calculated on the principal amount plus the accumulated interest from previous periods. The formula for CI is:

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

Formula for Difference between CI and SI for 2 Years

For investments held for exactly 2 years, the difference between compound interest (compounded annually) and simple interest can be calculated using a simplified formula:

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

This formula provides a direct method to find the principal when the difference, rate, and time (2 years) are known.

Applying the Formula to Find the Principal Sum

We are given the following information:

• The difference between CI and SI = ₹967
• The annual rate of interest ($R$) = 17%
• The time period = 2 years

Substituting the given values into the difference formula:

$$ 967 = P \left( \frac{17}{100} \right)^2 $$

To find the principal sum ($P$), we need to rearrange the equation:

$$ 967 = P \left( \frac{289}{10000} \right) $$

Now, isolate $P$:

$$ P = \frac{967 \times 10000}{289} $$

Calculation of the Principal Sum

Let's perform the calculation:

$$ P = \frac{9670000}{289} $$

$$ P \approx 33460.2076 $$

Rounding Off the Result

The question requires the principal sum to be rounded to the nearest integer. Rounding ₹33460.2076 gives:

$$ P = ₹33,460 $$

Conclusion

The calculated principal sum is ₹33,460. This value represents the initial amount invested, based on the provided difference between simple and compound interest over two years at a 17% annual interest rate.