Finding the Principal Sum Using Simple Interest

The question asks us to find the original sum of money (the principal) that generated a specific amount of simple interest over a given period at a certain rate. We are provided with the simple interest earned, the annual interest rate, and the time duration.

Understanding Simple Interest

Simple interest is calculated only on the initial principal amount. It is the easiest way to calculate interest on a principal amount. The formula for simple interest is:

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

Where:

• \(\text{SI}\) is the simple interest earned.
• \(\text{P}\) is the principal amount (the initial sum).
• \(\text{R}\) is the annual rate of interest (in percent).
• \(\text{T}\) is the time the money is invested or borrowed for (in years).

Given Information

From the question, we have the following values:

• Simple Interest (\(\text{SI}\)) = Rs. 3,040
• Rate of Interest (\(\text{R}\)) = 8% p.a.
• Time (\(\text{T}\)) = 5 years

We need to find the Principal (\(\text{P}\)).

Calculating the Principal Amount

We can rearrange the simple interest formula to solve for the Principal (\(\text{P}\)):

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

Now, let's substitute the given values into this formula:

\( \text{P} = \frac{3040 \times 100}{8 \times 5} \)

First, calculate the product of the rate and time in the denominator:

\( 8 \times 5 = 40 \)

Now, substitute this back into the formula for P:

\( \text{P} = \frac{3040 \times 100}{40} \)

\( \text{P} = \frac{304000}{40} \)

To simplify, we can cancel out a zero from the numerator and the denominator:

\( \text{P} = \frac{30400}{4} \)

Finally, perform the division:

\( \text{P} = 7600 \)

So, the principal sum is Rs. 7,600.

Verification

Let's verify our answer by calculating the simple interest on Rs. 7,600 at 8% p.a. for 5 years:

\( \text{SI} = \frac{7600 \times 8 \times 5}{100} \)

\( \text{SI} = \frac{7600 \times 40}{100} \)

\( \text{SI} = \frac{304000}{100} \)

\( \text{SI} = 3040 \)

The calculated simple interest (Rs. 3,040) matches the simple interest given in the question, confirming that our calculated principal sum of Rs. 7,600 is correct.

Summary of Calculation Steps

1. Identify the known values: Simple Interest (SI), Rate (R), and Time (T).
2. Recall or derive the formula for Principal (P) using the simple interest formula: \( \text{P} = \frac{\text{SI} \times 100}{\text{R} \times \text{T}} \).
3. Substitute the known values into the formula.
4. Perform the necessary calculations (multiplication and division).
5. State the final principal amount.

Revision Table: Simple Interest Concepts

Additional Information on Simple Interest Calculations

Simple interest problems often involve finding one of the four variables: Principal, Rate, Time, or Simple Interest, when the other three are known. The single simple interest formula \(\text{SI} = \frac{\text{PRT}}{100}\) is sufficient to solve all such variations by rearranging it appropriately.

• To find Rate (R): \( \text{R} = \frac{\text{SI} \times 100}{\text{P} \times \text{T}} \)
• To find Time (T): \( \text{T} = \frac{\text{SI} \times 100}{\text{P} \times \text{R}} \)

Remember that the Rate (\(\text{R}\)) and Time (\(\text{T}\)) must be in consistent units. If the rate is per annum, the time must be in years. If time is given in months or days, convert it to years before using the formula.

For example, if time is 6 months, \( \text{T} = \frac{6}{12} = 0.5 \) years. If time is 73 days, \( \text{T} = \frac{73}{365} = 0.2 \) years (assuming a non-leap year).