A man borrows some money at compound interest and agrees to repay the loan in two yearly instalments: Rs. 3150 at the end of the 1st year and Rs. 4410 at the end of the 2nd year. If the rate of compound interest is 5% per annum, what is the original sum borrowed?

Introduction / Context:
This is a loan repayment problem under compound interest. The borrower repays the loan by making two instalments at the end of successive years. Each instalment includes interest and partial repayment of principal. To find the original amount borrowed, we treat each instalment as a payment whose present value (discounted back to the date of borrowing) must sum to the initial loan amount.

Given Data / Assumptions:

Loan is repaid in two instalments:
  • Rs. 3150 at the end of year 1
  • Rs. 4410 at the end of year 2
Rate of compound interest r = 5% per annum
Compounding yearly
Let the principal borrowed be P

Concept / Approach:
At 5% per annum, the present value (today's value) of a payment made after n years is given by:
PV = Payment / (1.05)^n. Since the man borrows P today and repays with two future payments, the sum of the present values of those payments must equal P. Hence:
P = 3150 / 1.05 + 4410 / (1.05)^2. We compute this expression to find the original loan amount.

Step-by-Step Solution:
Step 1: Discount the first instalment back 1 year. PV1 = 3150 / 1.05. PV1 = 3000 (since 3150 ÷ 1.05 = 3000). Step 2: Discount the second instalment back 2 years. PV2 = 4410 / (1.05)^2. (1.05)^2 = 1.1025. PV2 = 4410 / 1.1025 = 4000. Step 3: Add the present values to find the loan. P = PV1 + PV2 = 3000 + 4000 = Rs. 7000.

Verification / Alternative check:
We can check by forward calculation. If P = 7000 at 5%, amount after 1 year is 7000 * 1.05 = 7350. After paying 3150 at the end of year 1, the remaining balance is 7350 − 3150 = 4200. This 4200 grows at 5% for another year to 4200 * 1.05 = 4410, which is exactly the second instalment. Therefore, a loan of Rs. 7000 is completely cleared by the given instalments.

Why Other Options Are Wrong:
If the loan were Rs. 5000 or Rs. 6500, the future values and remaining balances would not match the instalment schedule and interest rate. Likewise, Rs. 9200 would require significantly larger repayments to settle the loan in just two years at 5%. Only Rs. 7000 leads to a consistent repayment pattern.

Common Pitfalls:
A common error is to simply add the two instalments and subtract some estimated interest, instead of properly discounting each instalment back to today. Another mistake is treating the second instalment as if it were discounted only one year instead of two. Always remember that each payment must be discounted for the full time between the borrowing date and the payment date.

Final Answer:
The original sum borrowed was Rs. 7000.