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 \$22,857, and the variable costs will be $\$ 24.75$ per book. With the other method, the one-time fixed costs will total $\$ 62,709$, and the variable costs will be $\$ 11.25$ per book. For how many books produced will the costs from the two methods be the same?
[1] books
$\times 5$

Step 1 ：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 \$22,857, and the variable costs will be \$24.75 per book. With the other method, the one-time fixed costs will total \$62,709, and the variable costs will be \$11.25 per book. We need to find the number of books for which the total cost of both methods will be the same.

Step 2 ：This can be done by setting the total cost equations for both methods equal to each other and solving for the number of books.

Step 3 ：The total cost for the first method is given by \$22,857 + \$24.75 \times \text{number of books} and for the second method it is \$62,709 + \$11.25 \times \text{number of books}.

Step 4 ：Setting these two equations equal to each other gives us: \$22,857 + \$24.75 \times \text{number of books} = \$62,709 + \$11.25 \times \text{number of books}

Step 5 ：Solving this equation for the number of books gives us 2952 books.

Step 6 ：Final Answer: The costs from the two methods will be the same when \(\boxed{2952}\) books are produced.