Solving Simple Interest Problems: Finding the Principal Sum

This solution explains how to find the original sum of money (principal) when you know the total amount after certain periods at a simple interest rate. We'll break down the problem step-by-step.

Understanding Simple Interest

Simple interest is a basic way to calculate the interest charged on a loan or earned on an investment. The interest is calculated only on the initial amount borrowed or invested (the principal). The formula for Simple Interest (SI) is:

$ SI = \frac{P \times R \times T}{100} $

Where:

• $P$ = Principal sum (the initial amount of money)
• $R$ = Rate of interest per year (in percent)
• $T$ = Time period (in years)

The total Amount ($A$) after time $T$ is the sum of the Principal and the Simple Interest:

$ A = P + SI $

Problem Analysis

We are given two scenarios for the same principal sum ($P$) and the same simple interest rate ($R$):

• Scenario 1: The sum amounts to Rs. 724 in 4 years.
• Scenario 2: The sum amounts to Rs. 870 in 6 years.

We need to find the original sum ($P$).

Step-by-Step Solution

Step 1: Calculate the difference in amount and time.

The difference between the amounts in the two scenarios is solely due to the simple interest earned during the extra time period.

• Amount after 6 years = Rs. 870
• Amount after 4 years = Rs. 724
• Difference in Amount = Rs. 870 - Rs. 724 = Rs. 146

The difference in time is:

• Time difference = 6 years - 4 years = 2 years

This means that the simple interest earned in 2 years is Rs. 146.

Step 2: Calculate the Simple Interest per year.

If Rs. 146 is the interest for 2 years, we can find the interest for one year:

Simple Interest per year = ↔146 / 2 = Rs. 73

Step 3: Calculate the Simple Interest for the first period (4 years).

Now that we know the simple interest earned per year, we can calculate the total simple interest earned over the first 4 years:

Simple Interest for 4 years = Simple Interest per year × 4 years

Simple Interest for 4 years = Rs. 73 × 4 = Rs. 292

Step 4: Calculate the Principal Sum (P).

We know that the total amount after 4 years is Rs. 724, and this amount is the sum of the Principal ($P$) and the Simple Interest for 4 years (Rs. 292).

Amount = Principal + Simple Interest for 4 years

Rs. 724 = $P$ + Rs. 292

To find $P$, we subtract the interest from the amount:

$P$ = Rs. 724 - Rs. 292

$P$ = Rs. 432

Step 5: Verify the result using the second scenario.

Let's check if this principal sum works for the second scenario (Rs. 870 in 6 years).

Simple Interest for 6 years = Simple Interest per year × 6 years

Simple Interest for 6 years = Rs. 73 × 6 = Rs. 438

Amount after 6 years = Principal + Simple Interest for 6 years

Amount = Rs. 432 + Rs. 438 = Rs. 870

This matches the given information, confirming our calculated principal sum is correct.

Conclusion

The sum of money (principal) that will amount to Rs. 724 in 4 years and Rs. 870 in 6 years at simple interest is Rs. 432.