A loan was repaid in two annual instalments of Rs.121 each. If the rate of…

Concept: When a loan is repaid through equal annual instalments under compound interest, the sum borrowed equals the sum of the present values of all future instalments -- each instalment discounted back to today by the compound-interest factor (1 + r) raised to the number of years before it falls due, i.e. Sum = C/(1+r) + C/(1+r)2 + ... for instalments due at years 1, 2, ...

Application to this problem:

1. Let the rate r = 10% = 0.1 per annum and each instalment C = Rs.121, due at the end of year 1 and the end of year 2.

2. Present value of the first instalment (due after 1 year): PV1 = 121 / 1.1 = Rs.110.

3. Present value of the second instalment (due after 2 years): PV2 = 121 / (1.1)2 = 121 / 1.21 = Rs.100.

4. Sum borrowed = PV1 + PV2 = 110 + 100 = Rs.210.

Cross-check (forward balance method): start the loan at Rs.210. In year 1, interest at 10% adds Rs.21, taking the balance to Rs.231; paying the first Rs.121 instalment leaves a balance of Rs.110. In year 2, interest at 10% on Rs.110 adds Rs.11, taking the balance to Rs.121; paying the second Rs.121 instalment clears the loan exactly -- confirming that Rs.210 was indeed the sum borrowed.