The Cook's Paradise paid $102 for a kitchen knife set. Expenses are 23% of selling price and the required profit is 24% of selling price. Round ALL answers to the nearest cent if applicable.

1) What is the regular selling price?

$


2) What is the break-even selling price?

$


3) During an inventory sale, the knife set was marked down 19% on the regular selling price. What is the sale price?

$


4) What is the operating profit or loss during the inventory sale (use a negative sign (-) for a loss)?

$

To solve these questions, we will use similar steps as before, adjusting the calculations based on the new percentages provided.

### 1) Regular Selling Price

Given:
- Cost price of the kitchen knife set: $102
- Expenses: 23% of selling price
- Required profit: 24% of selling price

Let $ SP $ be the regular selling price.

We know:
\[ \text{Expenses} = 0.23 \times SP \]
\[ \text{Profit} = 0.24 \times SP \]
\[ \text{Cost Price} + \text{Expenses} + \text{Profit} = SP \]

Substituting the expenses and profit:
\[ 102 + 0.23 \times SP + 0.24 \times SP = SP \]

Simplifying:
\[ 102 + 0.47 \times SP = SP \]
\[ 102 = SP - 0.47 \times SP \]
\[ 102 = 0.53 \times SP \]
\[ SP = \frac{102}{0.53} \]
\[ SP = 192.45 \]

So, the regular selling price is $ \boxed{192.45} $.

### 2) Break-even Selling Price

Break-even selling price is where the cost price and expenses are covered, but no profit is made.

Let $ BE $ be the break-even selling price.
\[ \text{Cost Price} + \text{Expenses} = BE \]

We know expenses are 23% of the break-even selling price:
\[ 102 + 0.23 \times BE = BE \]
\[ 102 = BE - 0.23 \times BE \]
\[ 102 = 0.77 \times BE \]
\[ BE = \frac{102}{0.77} \]
\[ BE = 132.47 \]

So, the break-even selling price is $ \boxed{132.47} $.

### 3) Sale Price During Inventory Sale

During an inventory sale, the knife set is marked down by 19% on the regular selling price.

Sale price $ SP_{sale} $ is calculated as:
\[ SP_{sale} = SP - 0.19 \times SP \]
\[ SP_{sale} = 192.45 - 0.19 \times 192.45 \]
\[ SP_{sale} = 192.45 - 36.57 \]
\[ SP_{sale} = 155.88 \]

So, the sale price is $ \boxed{155.88} $.

### 4) Operating Profit or Loss During the Inventory Sale

Operating profit or loss during the sale is calculated by subtracting the cost price and expenses from the sale price.

We know the sale price is $ 155.88 $, cost price is $ 102 $, and expenses are $ 0.23 \times 155.88 $.

\[ \text{Expenses} = 0.23 \times 155.88 = 35.85 \]

Operating profit/loss is:
\[ \text{Profit/Loss} = SP_{sale} - \text{Cost Price} - \text{Expenses} \]
\[ \text{Profit/Loss} = 155.88 - 102 - 35.85 \]
\[ \text{Profit/Loss} = 18.03 \]

So, the operating profit during the inventory sale is $ \boxed{18.03} $.

### Summary:
1. Regular Selling Price: $192.45
2. Break-even Selling Price: $132.47
3. Sale Price: $155.88
4. Operating Profit during Sale: $18.03

Do you want details or have any questions?

### Relative Questions:
1. What would be the new regular selling price if expenses were increased to 25% of the selling price?
2. How would the break-even selling price change if the cost price increased to $110?
3. What is the percentage markdown if the sale price was set to $160?
4. If the required profit was increased to 30% of the selling price, what would be the new selling price?
5. How much profit or loss would occur if the knife set was sold at a 25% discount?
6. What is the net profit percentage during the inventory sale?
7. How would the selling price change if there were no expenses?
8. What would be the profit or loss if the knife set was sold for $170 during the sale?

### Tip:
When solving percentage-based problems, always verify the accuracy of the percentage calculations to avoid errors. Double-check each step for precision.

Learn how to calculate the regular selling price, break-even selling price, sale price during an inventory sale, and operating profit or loss with detailed step-by-step solutions. This problem covers percentage calculations, profit and loss concepts, and markdown calculations. Suitable for Grades 10-12 students.

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