Which of the following statements are correct?
A. A man purchased an item at 4/5​ of its selling price ‘X’, and sold it at 5% more than ‘X’. His gain is 25%.
B. A dealer professes to sell his goods at cost price but uses a weight of 900g for a kg weight. His gain is 11 1/9%.
C. If loss is one third of the selling price, the loss percentage is 25%.
D. By selling 100 pens, a vendor gains the selling price of 20 pens. His gain is 20%. Choose the correct answer from the options given below:

The correct answer is B and C only.

Let us verify each of the given statements step-by-step to understand why only statements B and C are correct.

Statement A: A man purchased an item at 4/5 of its selling price 'X', and sold it at 5% more than 'X'. His gain is 25%.
Let the original selling price be $X=100$.
The cost price (CP) is 4/5 of X:
$\mathrm{CP}=\frac{4}{5}\times 100=80$
He sold the item at 5% more than X. So, the new selling price (SP) is:
$\mathrm{SP}=100+5=105$
The gain amount is:
$\mathrm{Gain}=\mathrm{SP}-\mathrm{CP}=105-80=25$
The gain percentage is:
$\mathrm{Gain\; \%}=\left(\frac{25}{80}\right)\times 100=31.25\%$
Since the calculated gain is 31.25% (and not 25%), Statement A is incorrect.

Statement B: A dealer professes to sell his goods at cost price but uses a weight of 900g for a kg weight. His gain is 11 1/9%.
Let the cost price of 1g of goods be 1 unit. Thus, the cost price of 1 kg (1000g) of goods is 1000 units.
The dealer professes to sell 1 kg of goods at cost price, meaning the selling price (SP) is 1000 units.
However, he only gives 900g of goods. The actual cost price (CP) of the goods sold is:
$\mathrm{CP}=900\text{units}$
The gain amount is:
$\mathrm{Gain}=\mathrm{SP}-\mathrm{CP}=1000-900=100\text{units}$
The gain percentage is:
$\mathrm{Gain\; \%}=\left(\frac{\mathrm{Gain}}{\mathrm{CP}}\right)\times 100=\left(\frac{100}{900}\right)\times 100=\frac{100}{9}\%=11\frac{1}{9}\%$
Thus, Statement B is correct.

Statement C: If loss is one third of the selling price, the loss percentage is 25%.
Let the selling price (SP) be 3 units.
The loss is one third of the selling price:
$\mathrm{Loss}=\frac{1}{3}\times 3=1\text{unit}$
The cost price (CP) is:
$\mathrm{CP}=\mathrm{SP}+\mathrm{Loss}=3+1=4\text{units}$
The loss percentage is:
$\mathrm{Loss\; \%}=\left(\frac{\mathrm{Loss}}{\mathrm{CP}}\right)\times 100=\left(\frac{1}{4}\right)\times 100=25\%$
Thus, Statement C is correct.

Statement D: By selling 100 pens, a vendor gains the selling price of 20 pens. His gain is 20%.
Let the selling price (SP) of 1 pen be 1 unit.
The total selling price of 100 pens is:
${\mathrm{SP}}_{100}=100\text{units}$
The gain is the selling price of 20 pens:
$\mathrm{Gain}=20\text{units}$
The cost price (CP) of 100 pens is:
$\mathrm{CP}=\mathrm{SP}-\mathrm{Gain}=100-20=80\text{units}$
The gain percentage is:
$\mathrm{Gain\; \%}=\left(\frac{\mathrm{Gain}}{\mathrm{CP}}\right)\times 100=\left(\frac{20}{80}\right)\times 100=25\%$
Since the calculated gain is 25% (and not 20%), Statement D is incorrect.

Comparing our findings, only statements B and C are correct.