Let's break down the problem step by step.

1. Understanding the Variables:

   • Let $C$ be the cost price.
   • Let $S$ be the selling price before the discount.
   • The discount is 25% of the selling price.
   • The profit after the discount is $\frac{50}{3} \%$ of the selling price.

2. Express the Discount:

   • The discount given is $25\%$ of the selling price $S$. So, the discount amount $D$ is: $D = 0.25S$

3. Selling Price After Discount:

   • The selling price after the discount is: $S_{\text{after}} = S - D = S - 0.25S = 0.75S$

4. Profit Equation:

   • The problem states that the profit is $\frac{50}{3} \%$ of the selling price $S$. So, the profit $P$ is: $P = \frac{50}{3} \times \frac{S}{100} = \frac{S}{6}$
   • Profit is also equal to the difference between the selling price after discount and the cost price $C$: $P = S_{\text{after}} - C = 0.75S - C$
   • Equating the two expressions for profit: $0.75S - C = \frac{S}{6}$

5. Solving for $C$:

   • Rearranging the equation: $0.75S - \frac{S}{6} = C$
   • To simplify, find a common denominator for the fractions: $\frac{9S}{12} - \frac{2S}{12} = C$ $\frac{7S}{12} = C$

6. Markup Percent Calculation:

   • The markup percentage is the percentage increase from the cost price to the initial selling price. This is calculated as: $\text{Markup \%} = \frac{S - C}{C} \times 100$
   • Substitute $C = \frac{7S}{12}$ into the equation: $\text{Markup \%} = \frac{S - \frac{7S}{12}}{\frac{7S}{12}} \times 100 = \frac{\frac{5S}{12}}{\frac{7S}{12}} \times 100 = \frac{5}{7} \times 100$ $\text{Markup \%} = \frac{500}{7} \approx 71.43\%$

Answer: The trader marked up the cost price by approximately 71.43% before offering the discount.

Would you like more details on any part of the solution, or have any other questions?

Here are some related questions you might find interesting:

1. How does the markup percentage affect the overall profit in a business?
2. What would be the selling price if the cost price and markup are known?
3. How can you calculate the break-even point in sales?
4. How does the discount percentage affect profit margins?
5. What is the relationship between markup percentage and gross margin?

Tip: Understanding the relationship between cost price, selling price, and profit is crucial in making strategic pricing decisions in business.