A loan of ₹10,200 is to be paid back in two equal half-yearly instalments. If…

Concept: When a loan is repaid through equal instalments at fixed intervals, the loan amount equals the sum of the present values of all the instalments, each discounted at the periodic rate of interest for the number of periods until it is paid. If the interest is compounded m times a year, first convert the annual rate to the equivalent periodic rate (annual rate ÷ m) before writing this present-value equation. The total interest charged equals the total amount actually repaid minus the amount originally borrowed.

Method:

1. Half-yearly rate = 8% ÷ 2 = 4% = 0.04 (converting the annual compounding rate to the half-yearly period rate).

2. Let each instalment be R. Since there are two half-yearly instalments, the present-value equation is: 10,200 = R/1.04 + R/1.042 (discounting instalment 1 by one period and instalment 2 by two periods).

3. Combine the discount factors: 1/1.04 + 1/1.042 = (1.04 + 1)/1.042 = 2.04/1.0816.

4. So 10,200 = R × 2.04/1.0816, giving R = 10,200 × 1.0816/2.04 = ₹5,408.

5. Total amount actually repaid = 2 × R = 2 × 5,408 = ₹10,816.

6. Total interest charged = Amount repaid − Amount borrowed = 10,816 − 10,200 = ₹616.

Cross-check: Substitute R = ₹5,408 back into the present-value equation — 5,408/1.04 = ₹5,200 and 5,408/1.042 = ₹5,000; their sum is ₹5,200 + ₹5,000 = ₹10,200, exactly the amount borrowed, confirming the instalment (and hence the interest of ₹616) is correct.

Answer: Total interest charged = ₹616.