A store in Vancouver has operating expenses of 25% of the selling price and the operating profit is 35% of the selling price. During a sale, their watches were marked down by 30%. What is the profit or loss at the sale price if they purchased the watches at $619 each?

To determine the profit or loss at the sale price, we need to follow these steps:

1. **Determine the original selling price of the watches**:
   Since the operating profit is 35% of the selling price, we can denote the original selling price as $ S $.

2. **Calculate the sale price**:
   The sale price is obtained by marking down the original selling price by 30%.

3. **Calculate the profit or loss at the sale price**:
   Subtract the purchase cost and operating expenses from the sale price to find the profit or loss.

Let's go through each step in detail.

### Step 1: Determine the Original Selling Price

We know that the operating profit is 35% of the selling price $ S $, and the operating expenses are 25% of $ S $.

Given:
- Purchase price $ P = 619 $
- Operating expenses $ = 0.25S $
- Operating profit $ = 0.35S $

The cost of the watch to the store can be expressed as:
\[ \text{Cost} = \text{Purchase price} + \text{Operating expenses} \]
\[ \text{Cost} = 619 + 0.25S \]

The store's revenue (original selling price $ S $) should equal the cost plus the operating profit:
\[ S = 619 + 0.25S + 0.35S \]
\[ S = 619 + 0.60S \]

To find $ S $, we solve the equation:
\[ S - 0.60S = 619 \]
\[ 0.40S = 619 \]
\[ S = \frac{619}{0.40} \]
\[ S = 1547.50 \]

### Step 2: Calculate the Sale Price

The sale price after a 30% markdown is:
\[ \text{Sale Price} = S \times (1 - 0.30) \]
\[ \text{Sale Price} = 1547.50 \times 0.70 \]
\[ \text{Sale Price} = 1083.25 \]

### Step 3: Calculate the Profit or Loss at the Sale Price

The profit or loss at the sale price is calculated by subtracting the purchase price and operating expenses from the sale price.

\[ \text{Profit/Loss} = \text{Sale Price} - \text{Purchase Price} - \text{Operating Expenses} \]
\[ \text{Profit/Loss} = 1083.25 - 619 - (0.25 \times 1547.50) \]
\[ \text{Profit/Loss} = 1083.25 - 619 - 386.875 \]
\[ \text{Profit/Loss} = 77.375 \]

### Conclusion

The profit at the sale price is $77.375 per watch.

Do you want any more details or have any questions?

Here are 5 numbered questions for you to ask:
1. How do I calculate the break-even point for my business?
2. Can you explain how to derive the quadratic formula?
3. What are the steps to solve a system of linear equations?
4. How do I find the derivative of a composite function?
5. What is the difference between permutations and combinations?

Tip: Always double-check your calculations to ensure accuracy, especially when dealing with percentage changes and financial figures.

Explore how to calculate profit or loss after a 30% markdown on watches sold by a store in Vancouver. Includes detailed steps and formulas for determining sale price, operating expenses, and purchase cost. Suitable for Grades 10-12.

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