The nominal rate of interest is 15% per annum, payable half-yearly (that is, interest is compounded twice a year). What is the corresponding effective annual rate of interest, expressed as a percentage per year?

Introduction / Context:
This problem asks us to convert a nominal annual interest rate with semi-annual compounding into an effective annual rate. The nominal rate is 15% per annum, but because interest is actually applied twice a year, the true annual growth is slightly higher than 15%. The task is to compute that effective annual rate.

Given Data / Assumptions:

• Nominal annual rate R_nominal = 15% per annum.
• Compounding is half-yearly, so there are 2 compounding periods per year.
• Each half-yearly rate r_half = 15 / 2 = 7.5%.
• We seek the effective annual rate R_effective that produces the same growth over one full year.

Concept / Approach:
If the interest is compounded twice a year at 7.5% each half-year, then after one year, the amount multiplies by (1 + 7.5 / 100)^2. The effective annual rate is the net percentage increase of this multiplier compared to the original principal. Mathematically, R_effective = [(1 + r_half / 100)^2 - 1] * 100%.

Step-by-Step Solution:
Compute half-yearly rate: r_half = 15 / 2 = 7.5%. Growth factor per half-year = 1 + 7.5 / 100 = 1.075. Effective annual growth factor = (1.075)^2. Calculate (1.075)^2 = 1.075 * 1.075 = 1.155625. Effective annual rate R_effective = (1.155625 - 1) * 100%. R_effective = 0.155625 * 100% = 15.5625%. Rounded suitably, this is approximately 15.56 percent per annum.
Verification / Alternative check:
Consider a principal of Rs. 100. At 15% simple interest, after one year it becomes Rs. 115. At 7.5% compounded half-yearly, after the first half-year it becomes 100 * 1.075 = 107.50. After the second half-year it grows to 107.50 * 1.075 = 115.5625. The net increase is 15.5625%, which matches the earlier calculation of about 15.56%.

Why Other Options Are Wrong:
15 percent ignores the extra growth due to compounding. 15.75 percent and 31.13 percent significantly overstate the effect of semi-annual compounding. 30 percent would correspond to simply doubling the nominal rate, which is incorrect for this context. Only 15.56 percent correctly reflects the true annual growth factor from two periods at 7.5% each.

Common Pitfalls:
Students often either forget to divide the nominal rate by the number of compounding periods or fail to square the growth factor for two periods. Others may incorrectly add 15% and half of 15% instead of compounding. Always remember that compounding uses multiplication of growth factors, not simple addition of percentages.

Final Answer:
The effective annual rate of interest is approximately 15.56 percent per annum.