The printed price of an article is $40\%$ higher than its cost price. Then, what is the rate of discount so that the gain is $12\%$ profit?

Let the Cost Price ($\text{CP}$) be $100$.
The printed (marked) price is $40\%$ higher than $\text{CP}$:

$$\text{Marked Price (MP)}=100+40=140$$

To achieve a profit of $12\%$:

$$\text{Selling Price (SP)}=100+12=112$$

The amount of discount offered on the marked price:

$$\text{Discount}=\text{MP}-\text{SP}=140-112=28$$

The rate of discount is:

$$\text{Discount Percentage}=\frac{\text{Discount}}{\text{MP}}\times 100=\frac{28}{140}\times 100=\frac{1}{5}\times 100=20\%$$