Debt Discharge Calculation: Simple Interest Installment
This solution explains how to determine the equal annual installment required to settle a debt under a simple interest arrangement.
Problem Breakdown: Debt Amount, Time, and Interest Rate
We are asked to find the amount of an annual installment that will completely pay off a debt. Here's the information given:
- Principal Debt Amount ($P$): Rs. 3,658
- Number of Years ($n$): 4 years
- Annual Simple Interest Rate ($R$): 12% or 0.12
- Goal: Calculate the equal annual installment amount ($x$).
The challenge is to find a regular payment amount that covers both the principal and the simple interest accrued over the repayment period.
Applying the Simple Interest Installment Formula
For discharging a debt of principal amount $P$ due in $n$ years at a simple interest rate $R$ per annum, by making $n$ equal annual installments of $x$, the standard formula used is derived by considering the future value of each installment at the time the debt is due. The sum of the future values of these installments must equal the original debt amount.
The formula is:
$$ x \left[ n + R \frac{n(n-1)}{2} \right] = P $$
Where:
- $x$ is the amount of the annual installment.
- $n$ is the number of years (or installments).
- $R$ is the annual rate of simple interest (in decimal form).
- $P$ is the principal debt amount.
Calculating the Annual Installment Amount
Now, let's substitute the given values into the formula:
- $P = 3,658$
- $n = 4$
- $R = 0.12$
Substituting these values into the formula:
$$ x \left[ 4 + 0.12 \times \frac{4(4-1)}{2} \right] = 3,658 $$
First, calculate the term $\frac{n(n-1)}{2}$:
$$ \frac{4(4-1)}{2} = \frac{4 \times 3}{2} = \frac{12}{2} = 6 $$
Now, substitute this back into the main equation:
$$ x [4 + 0.12 \times 6] = 3,658 $$
Calculate the interest component within the bracket:
$$ 0.12 \times 6 = 0.72 $$
The equation becomes:
$$ x [4 + 0.72] = 3,658 $$
$$ x [4.72] = 3,658 $$
To find the installment amount $x$, divide the principal debt by 4.72:
$$ x = \frac{3,658}{4.72} $$
$$ x = 775 $$
Final Answer: Annual Installment
The calculation shows that an annual installment of Rs. 775 is required to discharge the debt of Rs. 3,658 in four years at a 12% simple interest rate per annum.
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