A bank offers $5\%$ compound interest per annum, calculated on half-yearly basis. A customer deposits $₹3200$ each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is:

$5\%$ per annum calculated half-yearly means $\dfrac{5}{2} = 2.5\%$ for every half year.

So after each half year, the amount becomes $1.025$ times.

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The deposit of 1st January earns interest for $2$ half years:

Amount $= 3200 \times (1.025)^2$

$= 3200 \times 1.050625$

$= 3362$

Interest $= 3362 - 3200 = 162$

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The deposit of 1st July earns interest for only $1$ half year:

Amount $= 3200 \times 1.025$

$= 3280$

Interest $= 3280 - 3200 = 80$

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Total interest $= 162 + 80 = 242$

Therefore, the interest gained is Rs. $242$.