Analysis:
We are given that Sirish borrowed a sum of rupees 163840 at an interest rate of 12.5% per annum compounded annually. On the same day, he lent out the same amount to Sahej at the same rate of interest but compounded half-yearly. We need to find Sirish's gain after 2 years.

Calculating the gain with annual compounding:
To calculate Sirish's gain with annual compounding, we can use the formula for compound interest:

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

Where:
A = the final amount after interest
P = the principal amount (initial sum borrowed)
r = the rate of interest per period
n = the number of compounding periods per year
t = the number of years

In this case, P = 163840, r = 12.5% = 0.125, n = 1, and t = 2.

Using the formula, we can calculate the final amount:

A = 163840(1 + 0.125/1)^(1*2)
A = 163840(1.125)^2
A ≈ 184320.625

The gain with annual compounding is approximately 184320.625 - 163840 = 20480.625 rupees.

Calculating the gain with half-yearly compounding:
To calculate Sirish's gain with half-yearly compounding, we need to consider that the interest is compounded twice a year. In this case, n = 2.

Using the formula, we can calculate the final amount:

A = 163840(1 + 0.125/2)^(2*2)
A = 163840(1.0625)^4
A ≈ 186413.654

The gain with half-yearly compounding is approximately 186413.654 - 163840 = 22573.654 rupees.

Comparison:
Sirish's gain with annual compounding is 20480.625 rupees, while his gain with half-yearly compounding is 22573.654 rupees. The gain is higher with half-yearly compounding because the interest is compounded more frequently. This means that the interest is added to the principal more often, resulting in a higher final amount after 2 years.

Conclusion:
Sirish's gain after 2 years is 22573.654 rupees when the interest is compounded half-yearly.