A department store paid ​$37.52 for a wine set. Overhead expense is 14 % of the regular selling price and profit is 10 % of the regular selling price. During a clearance​ sale, the set was sold at a markdown of 18 %. What was the operating profit or loss on the​ sale?

Let's break down the information to find the operating profit or loss from the sale.

### Step 1: Define the Variables
- **Cost price** (what the store paid for the wine set) = $37.52
- **Overhead expenses** = 14% of the regular selling price
- **Profit margin** = 10% of the regular selling price
- **Markdown during clearance sale** = 18% of the regular selling price

### Step 2: Determine the Regular Selling Price
The regular selling price can be denoted as $ R $. According to the problem:
- The total cost for the store is the sum of the cost price, overhead, and profit.
  
The equation for the store's total cost (which includes overhead and profit) is:

\[
\text{Cost Price} = R - (\text{Overhead} + \text{Profit}) 
\]

Overhead is 14% of $ R $ and profit is 10% of $ R $, so:

\[
\text{Cost Price} = R - (0.14R + 0.10R)
\]

Simplifying:

\[
37.52 = R - 0.24R
\]

\[
37.52 = 0.76R
\]

Solving for $ R $:

\[
R = \frac{37.52}{0.76} = 49.37
\]

So, the regular selling price $ R $ is **$49.37**.

### Step 3: Find the Clearance Sale Price
During the clearance sale, the markdown was 18%. So the sale price $ S $ is:

\[
S = R - 0.18R
\]

\[
S = 49.37 - 0.18 \times 49.37 = 49.37 - 8.89 = 40.48
\]

So, the clearance sale price is **$40.48**.

### Step 4: Calculate the Operating Profit or Loss
Now, we need to find the operating profit or loss. The operating profit or loss is the difference between the clearance sale price and the total cost (cost price + overhead expense).

Overhead expense is 14% of the regular selling price:

\[
\text{Overhead Expense} = 0.14 \times 49.37 = 6.91
\]

The total cost to the store (Cost Price + Overhead Expense) is:

\[
37.52 + 6.91 = 44.43
\]

Finally, calculate the operating profit or loss:

\[
\text{Operating Profit or Loss} = \text{Clearance Sale Price} - \text{Total Cost}
\]

\[
\text{Operating Profit or Loss} = 40.48 - 44.43 = -3.95
\]

The store incurred an operating **loss of $3.95** on the sale.

---

Would you like to dive into more details or have any questions?

Here are some related questions:
1. What if the markdown was 25% instead of 18%?
2. How would a change in the overhead percentage affect the result?
3. What if the profit margin was increased to 15%?
4. How could the store avoid an operating loss during the sale?
5. How do markdowns typically affect operating profits in retail?

**Tip:** Always calculate overhead and profit based on the regular selling price, not the cost price.

Learn how to calculate the operating profit or loss for a wine set sold at a markdown. The department store paid $37.52, with a 14% overhead and 10% profit on the regular price. The sale price was reduced by 18%. Detailed step-by-step solution included.

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