Understanding Simple Interest and the Problem

This question deals with simple interest earned on an investment. Simple interest is calculated only on the principal amount. The formula for simple interest is:

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

Where:

• \(P\) is the Principal amount (the initial sum invested).
• \(R\) is the Rate of interest per annum.
• \(T\) is the Time period in years.
• \(\text{SI}\) is the Simple Interest earned.

The problem provides us with the interest rate, two different time periods for investment, and the difference in the simple interest earned between these two periods. We need to find the initial sum invested, which is the Principal (\(P\)).

Setting up the Simple Interest Calculation

Let the initial sum invested be \(P\) (in Rupees).

The given Rate of Interest is \(R = 7.5\%\) per annum.

Case 1: Investment for 4 Years

Time period, \(T_1 = 4\) years.

Simple Interest earned in 4 years, \(\text{SI}_1\):

\( \text{SI}_1 = \frac{P \times R \times T_1}{100} = \frac{P \times 7.5 \times 4}{100} \)

\( \text{SI}_1 = \frac{30P}{100} \)

Case 2: Investment for 5 Years

Time period, \(T_2 = 5\) years.

Simple Interest earned in 5 years, \(\text{SI}_2\):

\( \text{SI}_2 = \frac{P \times R \times T_2}{100} = \frac{P \times 7.5 \times 5}{100} \)

\( \text{SI}_2 = \frac{37.5P}{100} \)

Calculating the Initial Sum Invested

The problem states that if the investment were for 5 years instead of 4 years, the interest earned would have been Rs. 375 more. This means the difference between the simple interest earned in 5 years (\(\text{SI}_2\)) and the simple interest earned in 4 years (\(\text{SI}_1\)) is Rs. 375.

\( \text{SI}_2 - \text{SI}_1 = 375 \)

Substitute the expressions for \(\text{SI}_1\) and \(\text{SI}_2\):

\( \frac{37.5P}{100} - \frac{30P}{100} = 375 \)

Combine the terms on the left side:

\( \frac{(37.5 - 30)P}{100} = 375 \)

\( \frac{7.5P}{100} = 375 \)

Now, solve for \(P\). Multiply both sides by 100:

\( 7.5P = 375 \times 100 \)

\( 7.5P = 37500 \)

Divide both sides by 7.5:

\( P = \frac{37500}{7.5} \)

To make the division easier, we can multiply the numerator and denominator by 10 to remove the decimal:

\( P = \frac{375000}{75} \)

Now, perform the division:

\( P = 5000 \)

So, the initial sum invested was Rs. 5,000.

Let's verify the difference:

• \( \text{SI}_1 = \frac{5000 \times 7.5 \times 4}{100} = \frac{5000 \times 30}{100} = \frac{150000}{100} = 1500 \)
• \( \text{SI}_2 = \frac{5000 \times 7.5 \times 5}{100} = \frac{5000 \times 37.5}{100} = \frac{187500}{100} = 1875 \)
• Difference = \( \text{SI}_2 - \text{SI}_1 = 1875 - 1500 = 375 \)

The calculated difference matches the value given in the problem (Rs. 375), confirming our calculated principal amount is correct.

Summary of Calculation Steps

Revision Table: Simple Interest Concepts

Additional Information: Simple vs. Compound Interest

It's important to distinguish between simple interest and compound interest, although this problem specifically uses simple interest.

• Simple Interest: As seen in this problem, simple interest is calculated only on the original principal amount. The interest earned in each period is constant if the principal and rate remain the same.
• Compound Interest: In compound interest, the interest earned in each period is added to the principal for the next period's calculation. This means interest is earned on both the original principal and the accumulated interest from previous periods. The formula for compound interest is \( A = P(1 + \frac{R}{100})^T \), where \(A\) is the amount after \(T\) years. Compound interest grows faster than simple interest over time.

Understanding which type of interest is being used is crucial for solving problems correctly.