Calculation of Compound Interest:
To calculate the compound interest, we need to use the formula:

A = P(1 + r/n)^(nt)

Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (Rs.10000 in this case)
r = annual interest rate (8% or 0.08)
n = number of times that interest is compounded per year
t = number of years the money is invested for (1 year in this case)

If the interest is compounded annually, n = 1:
A = 10000(1 + 0.08/1)^(1*1)
A = 10000(1 + 0.08)^1
A = 10000(1.08)
A = Rs.10800

If the interest is compounded half-yearly, n = 2:
A = 10000(1 + 0.08/2)^(2*1)
A = 10000(1 + 0.04)^2
A = 10000(1.04)^2
A = Rs.10816

Comparison of Gains:
By compounding the interest half-yearly instead of annually, the man would have gained an additional Rs.16.

Rate Compounded Annually or Half Yearly?
Definition:
The interest rate is the cost of borrowing money or the return on investment. It is expressed as a percentage of the principal amount.

Compounding Annually:
When the interest is compounded annually, the interest is added to the principal once a year. The interest earned in the previous year does not contribute to the interest calculation for the current year. This results in a lower overall return compared to compounding more frequently.

Compounding Half-Yearly:
When the interest is compounded half-yearly, the interest is added to the principal twice a year. The interest earned in the first half-year contributes to the interest calculation for the second half-year. This leads to a higher overall return compared to compounding annually.

Explanation:
By compounding the interest half-yearly, the man would have gained more money compared to compounding annually because the interest is added more frequently. This allows the interest to earn additional interest, resulting in higher returns.

In this case, the difference in gains is Rs.16. By compounding half-yearly, the man earned an additional Rs.16 compared to compounding annually.

Conclusion:
In general, if given the option, it is more beneficial to compound interest more frequently (e.g., quarterly, monthly, daily) as it leads to higher returns. However, the frequency of compounding should be balanced with other factors such as convenience and liquidity needs.

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