if the discount given is equal to 25 percent of the selling price and the sale gives the trader a rfit of 50/3 percent when calculated on his selling price by what percent did he mark up the cost price before offering the discount

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.

Learn how to calculate markup percentage and profit using a detailed step-by-step solution. This problem involves understanding percentage calculations, cost price, selling price, and profit margins. Suitable for high school students.

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