An amount of Rs 5,000 is invested at a fixed nominal rate of 8% per annum, with interest compounded every six months (semi annually). What will be the value of this investment after 5 years under compound interest with half yearly compounding?

Introduction / Context:
This question tests understanding of compound interest with semi annual compounding. In many bank deposits and financial products, interest is not added just once per year but more frequently, such as twice a year. When the interest is compounded, each period's interest is added to the principal, and future interest is calculated on this increased amount. This produces exponential growth rather than the linear growth seen with simple interest.

Given Data / Assumptions:

Principal P = Rs 5,000
Nominal annual interest rate = 8 percent per annum
Compounding frequency = semi annually (twice per year)
Number of years = 5 years
Interest is compounded at each half year interval

Concept / Approach:
For compound interest with m compounding periods per year, the amount A after T years is:
A = P * (1 + r / m)^(m * T) Here r is the nominal annual rate as a decimal, m is the number of compounding periods per year, and T is the number of years. In this problem, r = 0.08 and m = 2 because the interest is compounded every six months. Over 5 years, there will be 2 * 5 = 10 compounding periods. After calculating A, we choose the option that matches the obtained value.

Step-by-Step Solution:
Step 1: Convert annual rate to decimal. r = 8 percent = 8 / 100 = 0.08. Step 2: Identify the number of compounding periods per year. m = 2 (semi annual compounding). Step 3: Compute the periodic rate per half year. Periodic rate = r / m = 0.08 / 2 = 0.04. Step 4: Determine the total number of compounding periods. Total periods = m * T = 2 * 5 = 10. Step 5: Apply the compound interest formula. A = 5,000 * (1 + 0.04)^10. Step 6: Evaluate the factor (1.04)^10, which is approximately 1.480244. Step 7: Multiply: A ≈ 5,000 * 1.480244 = 7,401.22 rupees (approximately).

Verification / Alternative check:
To verify, we can approximate by noting that a simple interest approximation at 8 percent for 5 years would give 5,000 * 0.08 * 5 = 2,000 rupees of interest, for a total of 7,000 rupees. Compound interest should give a slightly higher figure than simple interest because each half year's interest is added to the principal. The calculated amount of about 7,401.22 rupees is somewhat above 7,000 and therefore fits this expectation. It is also very close to the given option Rs 7,401.22.

Why Other Options Are Wrong:
Option Rs 3,456.00 is far below the original principal and clearly impossible after earning positive interest. Option Rs 4,567.00 is less than the starting amount and therefore cannot be correct. Option Rs 7,890.00 is higher than the computed compound amount for 8 percent with semi annual compounding over 5 years and corresponds to a higher effective rate or longer duration. Only Rs 7,401.22 matches the correct compound interest calculation.

Common Pitfalls:
Learners sometimes confuse simple and compound interest and wrongly use I = P * r * T, which would underestimate the final amount. Another common mistake is to treat the rate per compounding period as 8 percent instead of 4 percent, which leads to a large overestimate. Some also miscount the number of compounding periods, forgetting that 5 years with semi annual compounding gives 10 periods. Careful attention to periodic rate and total number of periods is essential in compound interest questions.

Final Answer:
The value of the investment after 5 years with semi annual compounding is approximately Rs 7,401.22.