A department store paid ​$40.51 for a wine set. Overhead expense is 16 % of the regular selling price and profit is 22% of the regular selling price. During a clearance​ sale, the set was sold at a markdown of 21%. What was the operating profit or loss on the​ sale?

To determine the operating profit or loss on the sale, we'll go step by step.

### 1. Define Variables:
Let $ P $ represent the regular selling price.

- Overhead expenses are 16% of the regular selling price.
- Profit is 22% of the regular selling price.
- During the clearance sale, the wine set is sold at a markdown of 21%.

We are given the **cost price** of the wine set: $ \$40.51 $.

### 2. Regular Selling Price:
From the profit formula, we know the following relationship:

\[
\text{Cost Price} = P - \text{Overhead Expense} - \text{Profit}
\]
where overhead expense is 16% of $ P $, and profit is 22% of $ P $.

Thus, the cost price is:

\[
40.51 = P - 0.16P - 0.22P
\]

Simplifying:

\[
40.51 = P(1 - 0.16 - 0.22) = P(0.62)
\]

Solving for $ P $:

\[
P = \frac{40.51}{0.62} \approx 65.34
\]

So, the regular selling price $ P $ is approximately $ \$65.34 $.

### 3. Clearance Sale Price:
Since the markdown is 21%, the clearance sale price is:

\[
\text{Sale Price} = P \times (1 - 0.21) = 65.34 \times 0.79 \approx 51.62
\]

So, the sale price during the clearance is approximately $ \$51.62 $.

### 4. Operating Profit or Loss:
The operating profit or loss is the difference between the sale price and the sum of the cost price and the overhead expense.

The overhead expense is 16% of the regular selling price:

\[
\text{Overhead Expense} = 0.16 \times 65.34 \approx 10.45
\]

Now, calculate the total expenses (cost price + overhead):

\[
\text{Total Expense} = 40.51 + 10.45 = 50.96
\]

Finally, the operating profit or loss is the difference between the sale price and total expenses:

\[
\text{Operating Profit/Loss} = 51.62 - 50.96 = 0.66
\]

Thus, the **operating profit** on the sale is approximately **\$0.66**.

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Would you like more details or have any questions?

### Related Questions:
1. How would the profit change if the markdown were increased to 30%?
2. What would the profit be if the cost price increased by 10%?
3. How would a decrease in overhead expenses to 10% affect the operating profit?
4. What would be the loss if the item was sold at a 50% markdown?
5. How does the percentage of profit compare with the overall sale price?

### Tip:
Always double-check calculations when working with percentages to avoid mistakes in real-world business scenarios!

Calculate the operating profit or loss for a wine set sold at a 21% markdown during a clearance sale. Learn how overhead (16%) and profit (22%) impact the final sale, with step-by-step guidance using algebra and percentage calculations. Suitable for Grades 9-12.

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