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). The one-time fixed costs will total $41,870 The variable costs will be $12.50 per book The publisher will sell the finished product to bookstores at a price of $25.75 per book. How many books must the publisher produce and sell so that the production costs will equal the money from sales?
Let's represent the number of books the publisher must produce and sell as "x".
The one-time fixed costs total $41,870.
The variable costs are $12.50 per book, so the total variable costs will be 12.50x.
The publisher will sell each book to bookstores for $25.75, so the total revenue from sales will be 25.75x.
To find the number of books the publisher must produce and sell to break even, we need to find the value of "x" that satisfies the equation:
41,870 + 12.50x = 25.75x
To solve for "x", let's isolate the variable terms on one side:
41,870 = 25.75x - 12.50x
Combine like terms:
41,870 = 13.25x
Divide both sides by 13.25:
x = 41,870 / 13.25
Simplify the expression:
x ≈ 3,159.25
Since the publisher cannot produce and sell a fraction of a book, the publisher must produce and sell at least 3,160 books in order for the production costs to equal the money from sales.
To find out how many books the publisher must produce and sell so that the production costs will equal the money from sales, we need to determine the break-even point.
The one-time fixed costs are $41,870.
The variable costs per book are $12.50.
The selling price per book is $25.75.
Let's denote the number of books as 'x'.
The fixed costs are one-time costs, so they will not change with the number of books produced.
The total fixed costs will be $41,870.
The variable costs will change with the number of books produced and sold.
The total variable costs will be given by the formula:
Total Variable Costs = Variable Cost per Book * Number of Books
Total Variable Costs = $12.50 * x
The total production costs will be the sum of the fixed costs and the variable costs:
Total Production Costs = Total Fixed Costs + Total Variable Costs
Total Production Costs = $41,870 + ($12.50 * x)
The revenue from sales will be the selling price per book multiplied by the number of books sold:
Total Sales Revenue = Selling Price per Book * Number of Books
Total Sales Revenue = $25.75 * x
To find the break-even point, the production costs must equal the sales revenue:
Total Production Costs = Total Sales Revenue
$41,870 + ($12.50 * x) = $25.75 * x
Now, we can solve this equation to find the value of 'x' (the number of books):
$41,870 + ($12.50 * x) = $25.75 * x
Simplifying the equation:
$41,870 + $12.50 * x = $25.75 * x
$41,870 = $25.75 * x - $12.50 * x
$41,870 = $13.25 * x
Dividing both sides of the equation by $13.25:
$x = $41,870 / $13.25
x ≈ 3162.6415
Therefore, the publisher must produce and sell approximately 3163 books in order for the production costs to equal the money from sales.
To find the number of books the publisher must produce and sell in order to cover the production costs, we need to set up and solve an equation.
Let's denote the number of books as "x".
The fixed costs are given as $41,870.
The variable costs are $12.50 per book, so for "x" books, the total variable costs would be 12.50 * x.
The selling price of each book is $25.75, so the revenue from selling "x" books would be 25.75 * x.
To break even, the total production costs should equal the revenue from sales. Hence, we can set up the equation:
Fixed costs + Variable costs = Revenue from sales
41,870 + 12.50 * x = 25.75 * x
Now we can solve for "x".
41,870 = 13.25 * x
Dividing both sides by 13.25:
x = 41,870 / 13.25
Calculating the equation:
x ≈ 3158.49
Therefore, the publisher must produce and sell approximately 3159 books to cover the production costs.