An automobile financier claims to lend money at simple interest, but he effectively adds the interest to the principal every six months for calculating interest for the next period. If he charges 10% interest per annum in this way, what is his effective annual rate of interest?

Introduction / Context:
This question explores the concept of effective interest rate when interest is added to principal more than once per year. The financier claims to use simple interest, but by adding interest to the principal every six months, he is effectively compounding the interest half yearly. The given nominal annual rate is 10%, and you must find the effective annual rate when compounding happens twice a year.

Given Data / Assumptions:

Nominal rate of interest per annum = 10%.
Interest is added to principal every six months, so compounding is half yearly.
Number of compounding periods per year n = 2.
Effective annual rate is the percentage increase in the principal after one full year.
We use the compound interest factor (1 + i / n)^n to find the effective rate.

Concept / Approach:
If the nominal annual rate is R% and interest is compounded n times per year, the effective annual factor is (1 + R / (100n))^n. The effective annual rate Reff is then (this factor - 1) * 100%. In this question, R = 10 and n = 2, so each half year the rate is 10 / 2 = 5%. We apply the factor (1 + 0.05)^2 and find the effective rate.

Step-by-Step Solution:
Step 1: Nominal annual rate R = 10%. Step 2: Number of compounding periods per year n = 2 (every 6 months). Step 3: Rate per half year = R / n = 10 / 2 = 5% = 0.05 in decimal form. Step 4: Annual multiplication factor under half-yearly compounding is (1 + 0.05)^2. Step 5: Compute (1 + 0.05)^2 = 1.05^2 = 1.1025. Step 6: Effective annual rate Reff is given by (factor - 1) * 100% = (1.1025 - 1) * 100%. Step 7: Reff = 0.1025 * 100% = 10.25%. Step 8: Thus, the financier effectively charges an annual rate of 10.25%.

Verification / Alternative check:
Think in terms of a Rs 100 principal. After the first 6 months at 5%, it becomes 100 * 1.05 = 105. After the next 6 months, interest is again 5%, now on 105, so amount = 105 * 1.05 = 110.25. The increase from 100 to 110.25 is Rs 10.25 on Rs 100, corresponding to 10.25%, confirming the effective rate.

Why Other Options Are Wrong:
10% would be correct only if there were no compounding. 10.5% arises if one incorrectly multiplies the nominal rate by 1.05 or uses an approximate method. 9.75% is less than the nominal rate and does not account for compounding. Therefore, none of these match the correct effective rate of 10.25%, so 10.25% is the right answer and option "None of these" is not needed here.

Common Pitfalls:
Students sometimes confuse nominal and effective rates or forget that compounding more frequently than annually increases the effective rate. Another common error is to multiply 10% by the number of compounding periods, which is incorrect. Always use the compounding factor (1 + R / (100n))^n to find the effective annual rate.

Final Answer:
The effective annual rate of interest charged by the financier is 10.25%.