At what percentage above the cost price must an article be marked so as to gain $38\%$ after allowing a customer a discount of $8\%$ on the marked price?

Let the Cost Price (CP) be $100$.Since the gain is $38\%$, the Selling Price (SP) must be:
$$\text{SP}=138$$
A discount of $8\%$ is allowed on the Marked Price (MP). Therefore, SP is $92\%$ of MP:
$$92\%\text{ of MP}=138$$
$$\text{MP}=\frac{138\times 100}{92}=\frac{3\times 100}{2}=150$$
Marked price is $150$ when CP is $100$.Thus, it is marked $50\%$ above the cost price.