A person borrowed a sum of Rs 6000 at 10% p.a., interest compounded annually.…

Concept: For a loan repaid in equal annual installments under compound interest, the future value of the loan at the end of the repayment period must equal the combined future value of all the installments, since each installment paid earlier keeps earning interest for the remaining years until the loan is fully cleared.

Working:

1. Given: Principal P = Rs 6000, rate r = 10% per annum compounded annually, n = 3 years, and x = each equal installment paid at the end of every year.

2. Since the installments are paid at year-ends, compare all amounts at the end of year 3. The loan grows for the full 3 years: FV(loan) = 6000 × (1.10)3 = 6000 × 1.331 = Rs 7986.

3. The first installment (paid at the end of year 1) still earns interest for the remaining 2 years: its value at year 3 is x × (1.10)2 = 1.21x. The second installment (paid at the end of year 2) earns interest for 1 more year: x × 1.10 = 1.1x. The third installment (paid at the end of year 3) contributes just x.

4. Equate the loan's future value to the combined future value of the three installments: 7986 = x(1.21 + 1.1 + 1) = 3.31x.

5. Solve for x: x = 7986 / 3.31, which is approximately 2412.68, rounding to Rs 2,413.

Cross-check:

Discounting the installments back to the present (year 0) instead of compounding the loan forward gives the same equation: 6000 = x/1.10 + x/1.102 + x/1.103 = x(0.9091 + 0.8264 + 0.7513) = 2.4869x, so x = 6000 / 2.4869, approximately 2412.68 — consistent with the future-value method, confirming Rs 2,413 as each installment.