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The problem asks for the number of books produced where the total costs from two different production methods will be the same.

### Let’s define the variables:
- For Method 1:
  - Fixed costs = $57,644
  - Variable costs = $10.25 per book

- For Method 2:
  - Fixed costs = $19,826
  - Variable costs = $18.50 per book

### Step-by-Step Solution:
We want to find the number of books $ x $ where the total costs for both methods are the same.

1. **Total cost for Method 1**:
   \[
   \text{Total cost (Method 1)} = 57,644 + 10.25x
   \]

2. **Total cost for Method 2**:
   \[
   \text{Total cost (Method 2)} = 19,826 + 18.50x
   \]

3. **Set the two total costs equal**:
   \[
   57,644 + 10.25x = 19,826 + 18.50x
   \]

4. **Solve for $ x $**:
   - First, subtract 10.25x from both sides:
     \[
     57,644 = 19,826 + 8.25x
     \]
   - Next, subtract 19,826 from both sides:
     \[
     37,818 = 8.25x
     \]
   - Finally, divide both sides by 8.25:
     \[
     x = \frac{37,818}{8.25} = 4,584
     \]

### Conclusion:
The two production methods will have the same cost when 4,584 books are produced.

Would you like further details or have any questions?

Here are 5 related questions:
1. How does the number of books affect the total cost for each method?
2. What happens if the fixed costs of Method 2 increase?
3. How would the outcome change if the variable cost of Method 1 decreases to $9 per book?
4. What is the break-even point for each method individually?
5. How can you represent these cost functions graphically?

**Tip:** When comparing different options involving fixed and variable costs, finding the break-even point helps in choosing the more cost-effective option for different production quantities.

A small publishing company is planning to publish a new book. The production costs will include one-time fixed costs (such as editing) and variable costs (such as printing). There are two production methods it could use. With one method, the one-time fixed costs will total $57,644, and the variable costs will be $10.25 per book. With the other method, the one-time fixed costs will total $19,826, and the variable costs will be $18.50 per book. For how many books produced will the costs from the two methods be the same?

Learn how to calculate the number of books produced where two different production methods yield the same cost. This problem involves linear equations with fixed and variable costs, ideal for students in Grades 9-11.

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