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 ($\text{CP}$) of the article be $\text{₹}100$.
1. **Calculate the Printed Price (Marked Price, $\text{MP}$):**
The printed price is $40\%$ higher than $\text{CP}$:

$$\text{MP}=100+40\%\text{ of }100=\text{₹}140$$

2. **Calculate the Selling Price ($\text{SP}$):**
The required profit gain is $12\%$:

$$\text{SP}=100+12\%\text{ of }100=\text{₹}112$$

3. Calculate the Discount Rate:
The discount amount is given by $\text{MP}-\text{SP}$:

$$\text{Discount}=140-112=\text{₹}28$$

$$\text{Rate of Discount}=\left(\frac{\text{Discount}}{\text{MP}}\right)\times 100\%$$

$$\text{Rate of Discount}=\left(\frac{28}{140}\right)\times 100\%=\frac{1}{5}\times 100\%=20\%$$

Thus, the rate of discount is $20\%$.