A sum of Rs 12,500 is invested for 1 year at 12% per annum, with the interest being compounded semi-annually (two times in a year). What is the total compound interest earned, in rupees, at the end of the year?

Introduction / Context:
Questions on compound interest commonly ask you to compare simple interest and compound interest, or to find the amount or interest when the compounding period is more than once a year. In this problem, a fixed sum is invested at a given annual rate, but the interest is compounded semi-annually, that is, twice in a year. The task is to compute the total compound interest earned in 1 year. This type of question tests your understanding of the compound interest formula when the compounding frequency is not yearly, as well as your ability to work carefully with percentages and time periods.

Given Data / Assumptions:

• Principal (P) = Rs 12,500.
• Nominal annual rate of interest (R) = 12% per annum.
• Interest is compounded semi-annually, so there are 2 compounding periods in one year.
• Time period (T) = 1 year.
• We assume standard compound interest, with no additional deposits or withdrawals.

Concept / Approach:
The compound interest formula when interest is compounded m times per year is:
Amount A = P * (1 + (R / 100) / m)^(m * T)Here, m is the number of compounding periods in a year. The compound interest (CI) is then:
CI = A - PWe will substitute the values of P, R, m, and T into this formula to find A, and then subtract the original principal to obtain the interest earned.

Step-by-Step Solution:
Step 1: Identify the number of compounding periods per year. Since the interest is compounded semi-annually, m = 2.Step 2: Compute the rate per half-year. Annual rate R = 12%, so rate per half-year = 12 / 2 = 6%.Step 3: Compute the number of compounding periods for 1 year. T = 1 year, so total periods = m * T = 2 * 1 = 2.Step 4: Use the formula A = P * (1 + r)^(n), where r is the periodic rate and n is the number of periods.Here, r = 6 / 100 = 0.06 and n = 2. So A = 12500 * (1 + 0.06)^2.Step 5: Calculate (1 + 0.06)^2 = 1.06^2 = 1.1236.Step 6: Multiply: A = 12500 * 1.1236 = 14045.Step 7: Compute the compound interest earned: CI = A - P = 14045 - 12500 = 1545.

Verification / Alternative check:
You can verify quickly by computing the interest for each half-year. First half-year interest is 12,500 * 6 / 100 = 750, giving an intermediate amount of 13,250. Second half-year interest is 13,250 * 6 / 100 = 795. Total interest over the year is 750 + 795 = 1,545, which matches the detailed calculation. Hence, the computed answer is consistent and reliable.

Why Other Options Are Wrong:
Option Rs.1505 is lower than the correct interest and would correspond to a slightly smaller effective rate or a rounding error. Option Rs.1535 is also close but still does not match the precise result of compounding at 6% each half-year. Option Rs.1550 is slightly larger than the correct value and again does not align with the exact formula. Option Rs.1520 is another distractor that does not arise from a correct application of semi-annual compounding. Only Rs.1545 matches the accurate calculation.

Common Pitfalls:
Students often mistakenly treat semi-annual compounding as if it were simple interest at 12% for one year, which gives an interest of only Rs.1,500. Others forget to divide the annual rate by the number of compounding periods, or they incorrectly multiply the annual rate by 2. Another common error is to round intermediate values too early, causing small discrepancies in the final answer. It is safer to keep exact values until the final step and then round only if needed.

Final Answer:
The total compound interest earned on Rs 12,500 at 12% per annum, compounded semi-annually for 1 year, is Rs.1545.