An automobile financier claims to lend money at simple interest, but in practice he adds the interest calculated every six months to the principal and then charges interest again, effectively using half-yearly compounding. If his stated (nominal) rate is 8% per annum, what is the effective annual rate of interest actually charged by him?

Introduction / Context: This problem focuses on the difference between simple interest and compound interest, and on how a financier can effectively charge a higher rate than the one declared. Instead of truly using simple interest, he compounds the interest every six months. The question asks us to convert the nominal rate of 8% per annum with semi-annual compounding into an equivalent effective annual rate. Given Data / Assumptions:

• Stated (nominal) annual rate R = 8% per annum. • Interest is added every six months, so there are 2 compounding periods in a year. • Each half-yearly rate r_half = 8% / 2 = 4% per half-year. • We want the effective annual rate, meaning the single rate that produces the same increase over one full year.

Concept / Approach: For compound interest with n compounding periods per year, the effective annual rate is calculated using the formula: Effective rate = (1 + r_half / 100)^n - 1, expressed as a percentage. Here, n = 2 because interest is compounded twice a year. We compare this effective rate with the nominal 8% to see how much extra the borrower is actually paying. Step-by-Step Solution: Half-yearly rate r_half = 8 / 2 = 4%. Number of compounding periods in a year n = 2. Effective annual multiplier = (1 + r_half / 100)^n. Substitute values: (1 + 4 / 100)^2 = (1 + 0.04)^2. Compute: 1.04 * 1.04 = 1.0816. Effective annual rate = (1.0816 - 1) * 100%. Effective annual rate = 0.0816 * 100% = 8.16% per annum. Verification / Alternative check: Consider Rs. 100 as principal. In one year, at 8% simple interest, interest is Rs. 8, so the amount is Rs. 108. Under the financier’s method with semi-annual compounding at 4% per half-year, amount after one year is 100 * 1.0816 = Rs. 108.16. The interest is Rs. 8.16, which corresponds to 8.16% on Rs. 100, confirming our effective annual rate. Why Other Options Are Wrong: 8% ignores the compounding effect and represents only the nominal rate. 8.5%, 9%, and 10.25% all overstate the growth; using them would give amounts significantly above Rs. 108.16 on Rs. 100 in one year, which does not match the computed compound amount at 4% per half-year. Common Pitfalls: Students often forget that semi-annual compounding requires using half the annual rate and squaring the growth factor. Another mistake is to simply add 8% and half of 8% or to treat the situation as if it were simple interest. Ignoring the compounding periods leads to incorrect effective rates. Final Answer: The effective annual rate of interest charged by the financier is 8.16% per annum.