A man borrows Rs. 820 and undertakes to pay back with compound interest @ 5%…

Concept: When a loan is cleared through equal yearly instalments under compound interest, apply the equation of value: pick one common reference date, grow (or discount) the loan and every instalment to that same date, and set the two totals equal. Any instalment size satisfying that single equation clears the loan exactly over its full term.

Application:

1. Loan P = Rs. 820, rate r = 5% per annum, term n = 2 years, equal yearly instalment = x, paid at the end of year 1 and the end of year 2.

2. Take the end of year 2 (the date of the last instalment) as the common reference date. Grown to that date: the loan becomes 820(1 + 5/100)2; the year-1 instalment grows for one further year to x(1 + 5/100); the year-2 instalment needs no further growth, so it stays x.

3. Equation of value: 820(1 + 5/100)2 = x(1 + 5/100) + x.

4. Simplify the powers: (1 + 5/100)2 = (21/20)2 = 441/400, so the left side is 820 × 441/400.

5. Simplify the right side: x(21/20) + x = x(21/20 + 20/20) = x(41/20).

6. Solve: 820 × 441/400 = x × 41/20 ⇒ x = 820 × 441/400 × 20/41. Since 820 ÷ 41 = 20, this reduces to x = 20 × 441 × 20/400 = 441.

Cross-check: Check by discounting each instalment back to today at 5%: the year-1 instalment's present value is 441 ÷ 1.05 = 420, and the year-2 instalment's present value is 441 ÷ (1.05)2 = 400; the two present values add to 420 + 400 = 820, exactly the amount borrowed — confirming the instalment amount is consistent.