Finding the Principal Amount with Varying Interest Rates

This problem involves understanding how a change in the simple interest rate affects the total interest earned over a fixed period. We are given that a 5% higher rate of interest would have resulted in Rs. 200 more interest over 2 years for a certain sum of money. We need to find the original sum of money (the principal).

Understanding Simple Interest

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

\( \text{Simple Interest (SI)} = \frac{\text{Principal (P)} \times \text{Rate of Interest (R)} \times \text{Time (T)}}{100} \)

Where:

• P is the principal amount (the sum of money invested).
• R is the annual rate of interest (in percent).
• T is the time period (in years).

Setting Up the Problem

Let the principal amount be \( P \) rupees.

Let the original annual rate of interest be \( r\% \).

The time period is \( T = 2 \) years.

Scenario 1: Original Interest Rate

The simple interest earned at the original rate \( r\% \) is:

\( SI_1 = \frac{P \times r \times 2}{100} \)

Scenario 2: Higher Interest Rate

The new annual rate of interest is \( (r+5)\% \).

The simple interest earned at the higher rate \( (r+5)\% \) is:

\( SI_2 = \frac{P \times (r+5) \times 2}{100} \)

Calculating the Difference in Interest

According to the problem, the interest gained with the higher rate is Rs. 200 more than the interest gained with the original rate. Therefore:

\( SI_2 - SI_1 = 200 \)

Substitute the expressions for \( SI_1 \) and \( SI_2 \):

\( \frac{P \times (r+5) \times 2}{100} - \frac{P \times r \times 2}{100} = 200 \)

\( \frac{2P(r+5)}{100} - \frac{2Pr}{100} = 200 \)

\( \frac{2Pr + 10P - 2Pr}{100} = 200 \)

\( \frac{10P}{100} = 200 \)

Solving for the Principal Amount

Now, we can solve the equation for \( P \):

\( \frac{P}{10} = 200 \)

\( P = 200 \times 10 \)

\( P = 2000 \)

So, the principal amount is Rs. 2000.

Verifying the Solution

Let's check if a principal of Rs. 2000 results in a Rs. 200 difference for a 5% rate increase over 2 years. The difference in interest is simply due to the difference in the rate (5%) applied to the principal for the given time (2 years).

Extra Interest = \( \frac{\text{Principal} \times \text{Extra Rate} \times \text{Time}}{100} \)

Extra Interest = \( \frac{2000 \times 5 \times 2}{100} \)

Extra Interest = \( \frac{20000}{100} \)

Extra Interest = \( 200 \)

This matches the given information that the additional interest is Rs. 200. Thus, the calculated principal amount is correct.

Conclusion

The sum of money invested, which is the principal amount, is Rs. 2000.

Comparing this result with the given options:

• A. Rs. 3000
• B. Rs. 2600
• C. Rs. 5000
• D. Rs. 2000

The calculated principal amount matches option D.

Revision Table: Simple Interest Concepts

Additional Information on Simple Interest Problems

When tackling simple interest word problems, it's crucial to correctly identify the principal, rate, and time. Problems involving changes in rate or time often require setting up equations based on the difference in interest earned or the final amount. Remember that in simple interest, the interest earned each year is constant for a fixed principal and rate.

Key points to remember:

• Ensure the rate and time are in consistent units (e.g., rate per year, time in years).
• The difference in simple interest due to a change in rate or time is proportional to the change.
• The difference in interest can be calculated directly using the difference in rate/time in the simple interest formula. For example, a 5% difference in rate over 2 years on principal P results in a simple interest difference of \( \frac{P \times 5 \times 2}{100} \).