Introduction / Context:
This is a present value problem under compound interest. The borrower makes two future payments, and the question asks you to compute the equivalent single sum at the time of borrowing. Essentially, you discount each future payment back to the present using the given compound interest rate and add them to find the initial loan amount.
Given Data / Assumptions:
- Payment at the end of 1st year = Rs 3150.
- Payment at the end of 2nd year = Rs 4410.
- Rate of interest r = 5% per annum.
- Interest is compounded annually.
- Original loan amount (principal) P is unknown.
Concept / Approach:
The present value of a future payment is found by dividing that payment by (1 + r)^n, where n is the number of years until payment. Here, we treat both yearly payments as clearing the loan and find the sum of their present values. Therefore, principal P equals the present value of Rs 3150 due in 1 year plus the present value of Rs 4410 due in 2 years, both discounted at 5% per annum.
Step-by-Step Solution:
Step 1: Present value of the 1st payment:
PV1 = 3150 / (1.05).
Step 2: Present value of the 2nd payment:
PV2 = 4410 / (1.05^2).
Step 3: Compute the numerical values:
PV1 = 3150 / 1.05 = 3000.
Step 4: Compute PV2:
1.05^2 = 1.1025, so PV2 = 4410 / 1.1025 = 4000.
Step 5: Add the present values to get the principal:
P = PV1 + PV2 = 3000 + 4000 = 7000.
Verification / Alternative check:
If the principal is Rs 7000 at 5% per annum, after 1 year it becomes 7000 * 1.05 = 7350. After paying 3150, the balance is 4200. After the 2nd year, this 4200 grows to 4200 * 1.05 = 4410, which matches the second payment exactly.
Why Other Options Are Wrong:
Rs 5000 or Rs 6500 are too small; if you compound these and subtract the given payments, either some amount remains unpaid or the schedule does not balance.
Rs 9200 is too large and would lead to overpayment when compared to the scheduled payments and interest.
Common Pitfalls:
Some students mistakenly add payments without discounting or use simple interest formulas. Another error is to discount both payments by the same factor, ignoring the different time periods.
Final Answer:
The original sum borrowed is Rs 7000.