Calculating the Original Loan Amount with Varying Simple Interest

This problem involves calculating the original principal amount of a loan based on the total simple interest paid over four years, where the interest rate changes annually.

The key information provided is:

• The initial simple interest rate is 5% per annum.
• The rate increases by 0.5% each year starting from the second year.
• The loan duration is 4 years.
• The total simple interest paid over 4 years is Rs. 4,600.

We need to find the principal amount (the original loan amount).

Step 1: Determine the Simple Interest Rate for Each Year

The simple interest rate changes annually. Let's list the rate for each of the four years:

• Year 1 Rate: 5%
• Year 2 Rate: 5% + 0.5% = 5.5%
• Year 3 Rate: 5.5% + 0.5% = 6.0%
• Year 4 Rate: 6.0% + 0.5% = 6.5%

Step 2: Calculate the Simple Interest for Each Year

The formula for simple interest is:

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

In this case, the 'Time' for each calculation is 1 year, and the 'Principal' remains constant each year because it is simple interest. Let 'P' be the principal amount.

• Interest for Year 1 (SI₁): \( \text{SI}_1 = \frac{P \times 5 \times 1}{100} = \frac{5P}{100} \)
• Interest for Year 2 (SI₂): \( \text{SI}_2 = \frac{P \times 5.5 \times 1}{100} = \frac{5.5P}{100} \)
• Interest for Year 3 (SI₃): \( \text{SI}_3 = \frac{P \times 6 \times 1}{100} = \frac{6P}{100} \)
• Interest for Year 4 (SI₄): \( \text{SI}_4 = \frac{P \times 6.5 \times 1}{100} = \frac{6.5P}{100} \)

Step 3: Calculate the Total Simple Interest over 4 Years

The total interest paid is the sum of the simple interest from each year:

\( \text{Total Interest} = \text{SI}_1 + \text{SI}_2 + \text{SI}_3 + \text{SI}_4 \)

\( \text{Total Interest} = \frac{5P}{100} + \frac{5.5P}{100} + \frac{6P}{100} + \frac{6.5P}{100} \)

Combine the terms with the common denominator:

\( \text{Total Interest} = \frac{(5 + 5.5 + 6 + 6.5)P}{100} \)

Sum the percentages:

\( 5 + 5.5 + 6 + 6.5 = 23 \)

So, the total interest is:

\( \text{Total Interest} = \frac{23P}{100} \)

Step 4: Equate Total Interest to the Given Amount and Solve for Principal

We are given that the total interest paid is Rs. 4,600.

\( \frac{23P}{100} = 4600 \)

Now, solve for P:

\( 23P = 4600 \times 100 \)

\( 23P = 460000 \)

\( P = \frac{460000}{23} \)

\( P = 20000 \)

So, the original loan amount (principal) was Rs. 20,000.

The total simple interest paid corresponds to a total rate of 23% of the principal over the four years.

Conclusion

The calculated principal amount based on the total simple interest paid is Rs. 20,000.

Revision Table: Simple Interest Calculation

Additional Information: Understanding Simple Interest

Simple interest is a basic method of calculating interest on a loan or investment. Unlike compound interest, simple interest is only calculated on the initial principal amount. This means the interest earned or charged each period does not get added to the principal for the purpose of calculating interest in subsequent periods.

In this problem, even though the rate changes, the interest for each year is still calculated based on the original principal amount, not the principal plus accumulated interest (which would be the case in compound interest).

Steps to solve problems with varying simple interest rates:

1. Identify the interest rate for each specific time period (usually each year).
2. Calculate the simple interest for each period using the formula \( \frac{P \times R \times T}{100} \), where T is the duration of that specific period (often 1 year).
3. Sum up the interest calculated for all periods to find the total simple interest.
4. If the total interest is given, use the total interest equation to solve for the unknown variable, usually the principal or the rate.

This approach was used in the solution to find the original loan amount.