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Let's break down the problem to determine the operating profit or loss from the sale of the dinner plate set.

### Given:
- **Cost to the department store (C):** $55.45
- **Overhead expense:** 21% of the regular selling price
- **Profit margin:** 14% of the regular selling price
- **Markdown during clearance sale:** 43%

We will find:
1. The regular selling price (RSP).
2. The sale price after markdown.
3. The total cost (cost of the item plus overhead).
4. The operating profit or loss.

### Step-by-step solution:

1. **Let the regular selling price be $ x $**.

   - Overhead expense = $ 0.21x $
   - Profit = $ 0.14x $
   - Hence, the total regular selling price can be expressed as:
     \[
     x = \text{Cost price} + \text{Overhead expense} + \text{Profit}
     \]
     \[
     x = 55.45 + 0.21x + 0.14x
     \]
     Simplify the equation:
     \[
     x = 55.45 + 0.35x
     \]
     Solving for $ x $:
     \[
     x - 0.35x = 55.45
     \]
     \[
     0.65x = 55.45
     \]
     \[
     x = \frac{55.45}{0.65} = 85.31
     \]
     So, the **regular selling price** is **$85.31**.

2. **Sale price after markdown:**
   The markdown is 43%, so the sale price after markdown is:
   \[
   \text{Sale price} = 85.31 \times (1 - 0.43) = 85.31 \times 0.57 = 48.63
   \]
   So, the **sale price** is **$48.63**.

3. **Total cost (Cost price + Overhead expense):**
   The total cost of the set is the cost to the department store plus the overhead expense. The overhead expense is 21% of the regular selling price:
   \[
   \text{Overhead expense} = 0.21 \times 85.31 = 17.91
   \]
   So, the total cost is:
   \[
   \text{Total cost} = 55.45 + 17.91 = 73.36
   \]

4. **Operating profit or loss:**
   The operating profit or loss is the difference between the sale price and the total cost:
   \[
   \text{Operating profit/loss} = \text{Sale price} - \text{Total cost}
   \]
   \[
   \text{Operating profit/loss} = 48.63 - 73.36 = -24.73
   \]
   Therefore, the operating **loss** on the sale is **$24.73**.

---

Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:
1. How would the result change if the markdown percentage was 50% instead of 43%?
2. What would the regular selling price be if the profit margin was 20% instead of 14%?
3. If the store wants to avoid a loss, what should be the minimum selling price during the sale?
4. How would the overall outcome be different if the overhead expenses increased to 25% of the regular selling price?
5. If the item was sold without any markdown, what would be the operating profit?

**Tip:** Understanding how overhead expenses impact profit margins can help in pricing decisions, especially during sales or discounts.

A department store paid $55.45 for a dinner plate set. Overhead expense is 21% of the regular selling price and profit is 14% of the regular selling price. During a clearance sale, the set was sold at a markdown of 43%. What was the operating profit or loss on the sale?

Learn how to calculate the operating profit or loss on a dinner plate set sold at a markdown. This problem involves determining the regular selling price, applying a markdown, and computing total cost and loss. Suitable for students in Grades 9-12.

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