What equal annual instalment will discharge a debt of Rs. 1,092 due in 3 years at 12% simple interest?

Introduction / Context:
This question asks for an equal annual instalment that will clear a debt due at a future time when interest is calculated on a simple interest basis. It is a classic application of time value of money ideas with instalments. Instead of paying a lump sum at the end, the borrower pays yearly amounts that, together with accumulated interest, are equivalent to the required amount at the due date.

Given Data / Assumptions:

Debt due at the end of 3 years = Rs. 1,092.
Simple interest rate = 12% per annum.
Three equal annual instalments are paid, one at the end of each year.
We need to find the size of each annual instalment.

Concept / Approach:
The idea is to bring all instalments to the same point in time, typically the end of 3 years, by adding simple interest to earlier payments for the remaining years. The sum of the future values of all instalments must equal the amount of the debt due at that point, i.e., 1,092. Each instalment is some value x. The first instalment earns interest for 2 years, the second for 1 year, and the third for zero extra years, since it is paid at the due date. This gives an equation in x to solve.

Step-by-Step Solution:
Let each equal annual instalment be x rupees. The first instalment is paid at the end of year 1 and will earn interest for 2 more years at 12% simple interest. Future value of first instalment at end of year 3 = x * [1 + (12 * 2) / 100] = x * (1 + 24 / 100) = 1.24x. The second instalment is paid at the end of year 2 and will earn interest for 1 more year. Future value of second instalment at end of year 3 = x * [1 + (12 * 1) / 100] = x * 1.12. The third instalment is paid at the end of year 3, so it earns no additional interest; its future value is simply x. Total of all future values at the end of year 3 = 1.24x + 1.12x + x = 3.36x. This total must equal the debt due, 1,092 rupees, so 3.36x = 1,092. Therefore x = 1,092 / 3.36 = 325 rupees.

Verification / Alternative check:
Check the calculation by computing each future value explicitly. First instalment: 325 * 1.24 = 403.00. Second instalment: 325 * 1.12 = 364.00. Third instalment: 325. Sum = 403 + 364 + 325 = 1,092. This matches the required amount exactly, confirming that an instalment of Rs. 325 is correct.

Why Other Options Are Wrong:
Option Rs.545 or Rs.560 would produce future value totals much larger than 1,092 when multiplied by 3.36, meaning the borrower would overpay the debt.
Option Rs.550 similarly gives 3.36 * 550 = 1,848, which is far greater than 1,092 and does not match the equivalent value of the debt.
Only Rs.325 balances the future values of the instalments with the amount due at the given interest rate.

Common Pitfalls:
Some students forget that the instalments are paid at different times and simply divide the total amount by 3, ignoring the effect of interest.
Others treat the interest as compound rather than simple, which changes the future value factors and leads to incorrect instalment values.
A frequent error is to apply interest for 3 years to all instalments, ignoring the different time spans for which each instalment can earn interest.

Final Answer:
The equal annual instalment required to discharge the debt is Rs. 325 per year.