To gain $8\%$ after allowing a discount of $10\%$, by what percent must the marked price be hiked from the cost price?

Let the cost price ($CP$) of the article be $100$.
To gain a profit of $8\%$, the selling price ($SP$) must be:

$$SP=CP\times \left(1+\frac{\text{Profit}\%}{100}\right)=100\times \left(1+\frac{8}{100}\right)=108$$

A discount of $10\%$ is allowed on the marked price ($MP$). Therefore, the selling price is $90\%$ of the marked price:

$$SP=MP\times \left(1-\frac{\text{Discount}\%}{100}\right)$$

$$108=MP\times \left(1-\frac{10}{100}\right)$$

$$108=MP\times \frac{90}{100}$$

$$MP=\frac{108\times 100}{90}=12\times 10=120$$

The percentage by which the marked price is hiked from the cost price is:

$$\text{Percentage Hike}=\left(\frac{MP-CP}{CP}\right)\times 100\%$$

$$\text{Percentage Hike}=\left(\frac{120-100}{100}\right)\times 100\%=20\%$$