Calculating Simple Interest Loan Installments

This question asks us to find the equal annual installment amount required to repay a simple interest loan.

We are given:

• Principal amount (P) = ₹2,000
• Annual simple interest rate (R) = 5%
• Number of equal annual installments (n) = 3

Let the equal annual installment amount be \(x\).

Understanding Simple Interest Installments

In simple interest installment problems, the total amount due at the end of the loan period (if no payments were made) is equal to the sum of the future values of all the installments paid, calculated using the simple interest formula.

The simple interest future value formula is: \(FV = P(1 + \frac{RT}{100})\), where \(P\) is the present value (the installment amount in this case), \(R\) is the rate, and \(T\) is the time from the payment date to the end of the loan term.

Step-by-Step Calculation of the Installment Amount

Step 1: Calculate the total amount due at the end of 3 years without any installments.

Total Simple Interest = \(\frac{P \times R \times T}{100}\)

Total Simple Interest = \(\frac{2000 \times 5 \times 3}{100} = \frac{30000}{100} = 300\)

Total Amount Due = Principal + Total Simple Interest

Total Amount Due = \(2000 + 300 = 2300\)

So, if the loan were repaid as a single sum at the end of 3 years, the total amount payable would be ₹2,300.

Step 2: Calculate the future value of each installment at the end of the loan term (after 3 years).

• The first installment is paid at the end of Year 1. It earns simple interest for the remaining 2 years (from end of Year 1 to end of Year 3).
• Future Value of 1st Installment = \(x(1 + \frac{5 \times 2}{100}) = x(1 + 0.10) = 1.10x\)
• The second installment is paid at the end of Year 2. It earns simple interest for the remaining 1 year (from end of Year 2 to end of Year 3).
• Future Value of 2nd Installment = \(x(1 + \frac{5 \times 1}{100}) = x(1 + 0.05) = 1.05x\)
• The third installment is paid at the end of Year 3. It earns simple interest for 0 years.
• Future Value of 3rd Installment = \(x(1 + \frac{5 \times 0}{100}) = x(1 + 0) = x\)

Step 3: Equate the total amount due to the sum of the future values of the installments.

Sum of Future Values of Installments = \(1.10x + 1.05x + x = 3.15x\)

Total Amount Due = Sum of Future Values of Installments

\(2300 = 3.15x\)

Step 4: Solve for \(x\).

\(x = \frac{2300}{3.15}\)

To remove the decimal, multiply the numerator and denominator by 100:

\(x = \frac{2300 \times 100}{3.15 \times 100} = \frac{230000}{315}\)

Now, simplify the fraction by dividing the numerator and denominator by their greatest common divisor. Both are divisible by 5:

\(230000 \div 5 = 46000\)

\(315 \div 5 = 63\)

So, \(x = \frac{46000}{63}\)

Step 5: Convert the fraction to a mixed number to match the options.

Divide 46000 by 63:

The quotient is 730, and the remainder is 10. So, the mixed number is \(730 \frac{10}{63}\).

Thus, the annual installment amount is ₹\(730 \frac{10}{63}\).

Summary of the Simple Interest Installment Calculation

The annual installment amount is ₹\(730 \frac{10}{63}\).

Let's verify the answer by converting the option back to an improper fraction:

\(730 \frac{10}{63} = \frac{(730 \times 63) + 10}{63} = \frac{45990 + 10}{63} = \frac{46000}{63}\).

This matches our calculated value for \(x\).

Revision Table: Simple Interest Key Concepts

Additional Information: Simple Interest vs. Compound Interest Installments

It's important to distinguish between simple interest and compound interest when dealing with loan installments.

• Simple Interest Installments: Interest is typically calculated based on the initial principal amount for the full term or on the reducing balance *before* the installment is applied for that year. The method used in the solution (equating future value of total amount due to sum of future values of installments) is a common approach for simple interest installment problems.
• Compound Interest Installments: Interest is calculated on the principal amount plus any accumulated interest from previous periods. The calculation involves concepts like present value or future value of annuities, and the formula \(EMI = P \times \frac{r(1+r)^n}{(1+r)^n - 1}\) (for Equated Monthly Installments) or similar annual installment formulas based on compounding. Compound interest installments are more common in real-world lending scenarios.

This question specifically mentioned "simple interest", guiding us to use the simple interest method described in the solution.