Debt Discharge Calculation Using Simple Interest Installments
Understanding the Problem
This question requires us to determine the original amount of debt (the principal) that is paid off through a series of 5 equal monthly payments. We are given the value of each installment and the applicable simple interest rate per year.
Key Information Provided:
- Monthly Installment Amount (M): ₹1,845
- Number of Installments (n): 5 months
- Annual Simple Interest Rate (R): 48%
Calculating the Monthly Interest Rate
Since the installments are paid monthly, we need to convert the annual interest rate into a monthly interest rate.
The formula for the monthly interest rate (r) is:
$ r = \frac{R}{12} $
Substituting the given annual rate:
$ r = \frac{48\%}{12} = \frac{0.48}{12} = 0.04 $
Therefore, the monthly simple interest rate is 0.04 or 4%.
Calculating the Principal Debt
To find the principal debt discharged by the installments, we calculate the future value (FV) of each installment at the end of the loan term (the 5th month). This method accounts for the simple interest earned by each installment from the time it is paid until the end of the term.
The loan duration is 5 months.
- First Installment (paid at the end of month 1): This payment earns simple interest for the remaining 4 months (months 2, 3, 4, 5). FV_1 = $ M \times (1 + r \times 4) = 1845 \times (1 + 0.04 \times 4) = 1845 \times (1 + 0.16) = 1845 \times 1.16 $
- Second Installment (paid at the end of month 2): This payment earns simple interest for the remaining 3 months (months 3, 4, 5). FV_2 = $ M \times (1 + r \times 3) = 1845 \times (1 + 0.04 \times 3) = 1845 \times (1 + 0.12) = 1845 \times 1.12 $
- Third Installment (paid at the end of month 3): This payment earns simple interest for the remaining 2 months (months 4, 5). FV_3 = $ M \times (1 + r \times 2) = 1845 \times (1 + 0.04 \times 2) = 1845 \times (1 + 0.08) = 1845 \times 1.08 $
- Fourth Installment (paid at the end of month 4): This payment earns simple interest for the remaining 1 month (month 5). FV_4 = $ M \times (1 + r \times 1) = 1845 \times (1 + 0.04 \times 1) = 1845 \times (1 + 0.04) = 1845 \times 1.04 $
- Fifth Installment (paid at the end of month 5): This payment is made on the final day and earns no interest. FV_5 = $ M = 1845 $
Summing the Future Values to Find the Principal
The total amount of the debt discharged is the sum of the future values of all the installments.
Principal Debt (P) = $ FV_1 + FV_2 + FV_3 + FV_4 + FV_5 $
$ P = (1845 \times 1.16) + (1845 \times 1.12) + (1845 \times 1.08) + (1845 \times 1.04) + 1845 $
We can factor out the installment amount:
$ P = 1845 \times (1.16 + 1.12 + 1.08 + 1.04 + 1.00) $
First, sum the multipliers:
$ 1.16 + 1.12 + 1.08 + 1.04 + 1.00 = 5.40 $
Now, calculate the principal debt:
$ P = 1845 \times 5.40 $
$ P = 9963 $
Conclusion
The amount of debt discharged by the 5 equal monthly installments of ₹1,845 at a 48% simple interest rate is ₹9,963.
Similar Questions
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Important Questions from Installments
A computer is available for Rs. 39,000 on cash payment or Rs. 19,000 as cash payment followed by five monthly instalments of Rs. 4,200 each. What is the rate of interest per annum under the instalment plan?
What is the amount (in Rs.) of debt that will be discharged in 6 equal instalments of Rs. 800 each, if the debt is due in 6 years at 5% per annum?
A sum of Rs. P was borrowed and paid back in two equal yearly instalments, each of Rs. 35,280. If the rate of interest was 5% per annum and interest is compounding annually, then the value of P is ________.
A loan of Rs. 1,50,000 is availed with compound interest rate of 10% per annum for two years compounded annually. It is to be paid in equal yearly installments, and the installment is to be paid at the end of each year. The value of the equal yearly installment is : (Rounded off to two places of decimal)
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