The problem asks us to find the original sum of money, also known as the principal, given the difference between compound interest (CI) and simple interest (SI) over a specific period and rate. This is a common type of question in financial mathematics and quantitative aptitude.

Understanding Interest Concepts: Simple vs. Compound Interest

Let's first understand the two types of interest involved in this question:

• Simple Interest (SI): Simple interest is calculated only on the principal amount, or on that portion of the principal amount that remains unpaid. It does not compound, meaning interest earned does not earn further interest.
• Compound Interest (CI): Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. In other words, interest earns interest. This is why compound interest typically yields a higher return than simple interest over time. When compounding is done annually, it means the interest is added to the principal once a year.

Formula for Interest Difference over 2 Years

For a principal amount (\(\text{P}\)), a rate of interest (\(\text{R}\)) per annum, and a time period of 2 years, there is a specific formula to find the difference between compound interest and simple interest.

• The formula for Simple Interest (SI) for 2 years is:

  \(\text{SI} = \frac{\text{P} \times \text{R} \times 2}{100}\)

• The formula for Compound Interest (CI) for 2 years (compounded annually) is:

  \(\text{CI} = \text{P} \left(1 + \frac{\text{R}}{100}\right)^2 - \text{P}\)

The difference between Compound Interest and Simple Interest (\(\text{CI} - \text{SI}\)) for 2 years can be directly calculated using the formula:

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

Applying the Data to Find the Sum of Money

We are given the following information:

• Difference between CI and SI = Rs. 2400
• Rate of interest (\(\text{R}\)) = 40% per annum
• Time period (\(\text{N}\)) = 2 years

We need to find the principal amount (\(\text{P}\)). Let's substitute the given values into the formula for the difference:

Step 1: Write down the formula for the difference between CI and SI for 2 years.

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

Step 2: Substitute the known values into the formula.

\(2400 = \text{P} \left(\frac{40}{100}\right)^2\)

Step 3: Simplify the term inside the parenthesis.

\(2400 = \text{P} \left(\frac{2}{5}\right)^2\)

Step 4: Square the fraction.

\(2400 = \text{P} \times \frac{4}{25}\)

Step 5: Solve for \(\text{P}\) by rearranging the equation.

\(\text{P} = \frac{2400 \times 25}{4}\)

Step 6: Perform the calculation.

\(\text{P} = 600 \times 25\)

\(\text{P} = 15000\)

So, the principal sum of money is Rs. 15000.

Final Amount (Principal)

The question asks "What is the amount?". In the context of the options provided, and the general usage in such problems, "amount" here refers to the initial sum of money (principal) on which the interest is calculated. We have successfully calculated this principal amount.

The calculated principal amount is Rs. 15000.

Therefore, the sum of money is Rs. 15000.