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 analyze each statement step-by-step to determine its correctness:

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 initial selling price be $X$.
The cost price (CP) is given as:
$\mathrm{CP}=\frac{4}{5}X=0.8X$
The new selling price (SP) is 5% more than $X$:
$\mathrm{SP}=X+0.05X=1.05X$
The gain is:
$\mathrm{Gain}=\mathrm{SP}-\mathrm{CP}=1.05X-0.8X=0.25X$
The gain percentage is calculated as:
$\mathrm{Gain\; \%}=\left(\frac{\mathrm{Gain}}{\mathrm{CP}}\right)\times 100$
$\mathrm{Gain\; \%}=\left(\frac{0.25X}{0.8X}\right)\times 100=\frac{25}{80}\times 100=31.25\%$
Since the statement claims the gain is 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%.
The dealer professes to sell 1 kg (1000g) of goods, but actually gives only 900g.
Let the cost price of 1g of goods be 1 unit.
The cost price (CP) of the goods actually sold (900g) is 900 units.
The selling price (SP) charged is for 1000g at cost price, which is 1000 units.
The gain made by the dealer is:
$\mathrm{Gain}=1000-900=\mathrm{100\; units}$
The gain percentage is:
$\mathrm{Gain\; \%}=\left(\frac{100}{900}\right)\times 100=\frac{100}{9}\%=11\frac{1}{9}\%$
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 be $\mathrm{SP}$.
The loss is given by:
$\mathrm{Loss}=\frac{1}{3}\mathrm{SP}$
We know that Cost Price (CP) is the sum of Selling Price and Loss:
$\mathrm{CP}=\mathrm{SP}+\mathrm{Loss}=\mathrm{SP}+\frac{1}{3}\mathrm{SP}=\frac{4}{3}\mathrm{SP}$
The loss percentage is calculated relative to the cost price:
$\mathrm{Loss\; \%}=\left(\frac{\mathrm{Loss}}{\mathrm{CP}}\right)\times 100$
$\mathrm{Loss\; \%}=\left(\frac{\frac{1}{3}\mathrm{SP}}{\frac{4}{3}\mathrm{SP}}\right)\times 100=\frac{1}{4}\times 100=25\%$
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 of 1 pen be $S$ and the cost price of 1 pen be $C$.
For selling 100 pens:
Total Selling Price = $100S$
Total Cost Price = $100C$
The gain is equal to the selling price of 20 pens, which is $20S$.
We express gain as:
$\mathrm{Gain}=\mathrm{Total\; SP}-\mathrm{Total\; CP}$
$20S=100S-100C$
Rearranging the terms:
$100C=80S$
This gives the cost price of 100 pens as equivalent to the selling price of 80 pens.
The gain percentage is:
$\mathrm{Gain\; \%}=\left(\frac{\mathrm{Gain}}{\mathrm{Total\; CP}}\right)\times 100=\left(\frac{20S}{80S}\right)\times 100=\frac{1}{4}\times 100=25\%$
Since the statement claims the gain is 20%, Statement D is incorrect.

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