Calculating Annual Installment for Debt Discharge with Simple Interest

This problem asks us to find the fixed annual payment, or installment, required to pay off a specific debt amount over a set number of years, considering simple interest.

Understanding the Problem: Simple Interest Installments

When a debt is repaid through equal annual installments under simple interest, there's a specific method to calculate the installment amount. The core idea is that the total amount paid back (sum of installments) plus the total simple interest charged on the varying principal balances over the years must equal the original debt plus the total simple interest on the original debt for the entire period if no payments were made. A common formula used for simple interest installment problems simplifies this relationship.

Let's identify the given information:

• Total Debt Amount (Principal), $P = \text{₹}5,664$
• Time Period, $n = 4$ years
• Simple Interest Rate, $R = 12\%$ per annum
• Annual Installment Amount, $x$ (which we need to find)

Formula for Simple Interest Installments

For a debt of principal $P$ to be discharged in $n$ equal annual installments of $x$ at a simple interest rate of $R\%$ per annum, the relationship is often given by the formula:

$\text{Debt} = nx + \frac{R x}{100} [ (n-1) + (n-2) + \dots + 1 + 0 ]$

The sum of the series $(n-1) + (n-2) + \dots + 1 + 0$ is the sum of the first $(n-1)$ natural numbers, which is $\frac{(n-1)n}{2}$.

So, the formula becomes:

$P = nx + \frac{R x}{100} \frac{n(n-1)}{2}$

Step-by-Step Calculation of the Annual Installment

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

• $P = 5664$
• $n = 4$
• $R = 12$
• $x = ?$

$5664 = 4x + \frac{12x}{100} \frac{4(4-1)}{2}$

$5664 = 4x + \frac{12x}{100} \frac{4 \times 3}{2}$

$5664 = 4x + \frac{12x}{100} \times \frac{12}{2}$

$5664 = 4x + \frac{12x}{100} \times 6$

$5664 = 4x + \frac{72x}{100}$

$5664 = 4x + 0.72x$

$5664 = (4 + 0.72)x$

$5664 = 4.72x$

To find $x$, we need to divide the debt amount by $4.72$:

$x = \frac{5664}{4.72}$

Let's perform the division:

$x = 1200$

So, the annual installment required is ₹1,200.

Verification with the Formula Logic

According to the formula's underlying logic, the total amount paid back through installments (which is $4x$) plus the simple interest calculated on the installment amount 'x' for the respective remaining periods (represented by the sum $3+2+1+0$) should equal the original principal. Let's check:

• Total Installments = $4 \times \text{₹}1200 = \text{₹}4800$
• Total Interest part calculated in the formula = $\frac{12 \times 1200}{100} \times 6 = \frac{14400}{100} \times 6 = 144 \times 6 = \text{₹}864$
• Sum = $\text{₹}4800 + \text{₹}864 = \text{₹}5664$

This sum matches the original debt amount, ₹5,664, confirming our calculation based on this specific simple interest installment model.

Comparing with Options

The calculated annual installment is ₹1,200. Let's compare this with the given options:

• Option 1: ₹1,230
• Option 2: ₹1,210
• Option 3: ₹1,200
• Option 4: ₹1,220

Our calculated value matches Option 3.

Conclusion

The annual installment that will discharge a debt of ₹5,664 in 4 years at 12% simple interest is ₹1,200.

Revision Table: Key Concepts in Simple Interest Installments

Additional Information: Simple vs. Compound Interest Installments

It is important to distinguish between debt discharge under simple interest and compound interest.

• Simple Interest Installments: As seen in this problem, a specific formula is often used which relates the original principal to the sum of installments and a calculated simple interest component. The interest calculation is often simplified and might not directly reflect interest on the true reducing balance. The formula $P = nx + \frac{Rx}{100} \frac{n(n-1)}{2}$ is typical for this type of problem.
• Compound Interest Installments (EMI): For compound interest, installments (often called EMIs - Equated Monthly Installments) are calculated using annuity formulas. The interest for each period is calculated on the outstanding principal at the beginning of that period. The formula for the present value of an annuity is typically used: $P = \frac{x}{r} [1 - (1+r)^{-n}]$, where r is the per-period interest rate and n is the number of periods. This is a more realistic model for loans like mortgages or car loans.

This question specifically mentions "simple interest", so we use the method appropriate for simple interest installment problems.