Question Find the number of books the publisher must produce and sell to break even, given fixed costs of $\$77,142$ , variable costs of $\$8.25$ per book, and a selling price of $\$25.50$ per book.

Studdy Solution

STEP 1

Assumptions
1. The one-time fixed costs are $\$77,142$ .
2. The variable cost per book is $\$8.25$ .
3. The selling price per book is $\$25.50$ .
4. The publisher wants to break even, meaning the production costs equal the money from sales.

STEP 2

Let's denote the number of books the publisher needs to produce and sell to break even as $x$ .

STEP 3

The total production costs can be calculated by adding the one-time fixed costs to the product of the variable cost per book and the number of books.
$Total\, production\, costs = Fixed\, costs + (Variable\, cost\, per\, book \times Number\, of\, books)$

STEP 4

The total money from sales can be calculated by multiplying the selling price per book by the number of books.
$Total\, money\, from\, sales = Selling\, price\, per\, book \times Number\, of\, books$

STEP 5

To break even, the total production costs must equal the total money from sales.
$Fixed\, costs + (Variable\, cost\, per\, book \times x) = Selling\, price\, per\, book \times x$

STEP 6

Now, we can plug in the given values for the fixed costs, variable cost per book, and selling price per book into the equation.
$\$77,142 + (\$8.25 \times x) = \$25.50 \times x$

STEP 7

To solve for $x$ , we need to isolate $x$ on one side of the equation. Let's distribute $x$ on both sides of the equation.
$\$77,142 + \$8.25x = \$25.50x$

STEP 8

Subtract $\$8.25x$ from both sides of the equation to get all the terms involving $x$ on one side.
$\$77,142 = \$25.50x - \$8.25x$

STEP 9

Combine like terms on the right side of the equation.
$\$77,142 = (\$25.50 - \$8.25)x$

STEP 10

Calculate the difference in the parentheses.
$\$77,142 = (\$17.25)x$

STEP 11

Now, divide both sides of the equation by $\$17.25$ to solve for $x$ .
$x = \frac{\$77,142}{\$17.25}$

STEP 12

Calculate the value of $x$ .
$x = \frac{\$77,142}{\$17.25} \approx 4472.4058$

STEP 13

Since the publisher cannot produce and sell a fraction of a book, we round up to the next whole number.
$x = 4473$
The publisher must produce and sell 4,473 books to break even.

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Question Find the number of books the publisher must produce and sell to break even, given fixed costs of $\$77,142$ , variable costs of $\$8.25$ per book, and a selling price of $\$25.50$ per book.

Studdy Solution

STEP 1

Assumptions
1. The one-time fixed costs are $\$77,142$ .
2. The variable cost per book is $\$8.25$ .
3. The selling price per book is $\$25.50$ .
4. The publisher wants to break even, meaning the production costs equal the money from sales.

STEP 2

Let's denote the number of books the publisher needs to produce and sell to break even as $x$ .

STEP 3

The total production costs can be calculated by adding the one-time fixed costs to the product of the variable cost per book and the number of books.
$Total\, production\, costs = Fixed\, costs + (Variable\, cost\, per\, book \times Number\, of\, books)$

STEP 4

The total money from sales can be calculated by multiplying the selling price per book by the number of books.
$Total\, money\, from\, sales = Selling\, price\, per\, book \times Number\, of\, books$

STEP 5

To break even, the total production costs must equal the total money from sales.
$Fixed\, costs + (Variable\, cost\, per\, book \times x) = Selling\, price\, per\, book \times x$

STEP 6

Now, we can plug in the given values for the fixed costs, variable cost per book, and selling price per book into the equation.
$\$77,142 + (\$8.25 \times x) = \$25.50 \times x$

STEP 7

To solve for $x$ , we need to isolate $x$ on one side of the equation. Let's distribute $x$ on both sides of the equation.
$\$77,142 + \$8.25x = \$25.50x$

STEP 8

Subtract $\$8.25x$ from both sides of the equation to get all the terms involving $x$ on one side.
$\$77,142 = \$25.50x - \$8.25x$

STEP 9

Combine like terms on the right side of the equation.
$\$77,142 = (\$25.50 - \$8.25)x$

STEP 10

Calculate the difference in the parentheses.
$\$77,142 = (\$17.25)x$

STEP 11

Now, divide both sides of the equation by $\$17.25$ to solve for $x$ .
$x = \frac{\$77,142}{\$17.25}$

STEP 12

Calculate the value of $x$ .
$x = \frac{\$77,142}{\$17.25} \approx 4472.4058$

STEP 13

Since the publisher cannot produce and sell a fraction of a book, we round up to the next whole number.
$x = 4473$
The publisher must produce and sell 4,473 books to break even.

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Question Find the number of books the publisher must produce and sell to break even, given fixed costs of $77,142\$77,142$77,142, variable costs of $8.25\$8.25$8.25 per book, and a selling price of $25.50\$25.50$25.50 per book.

Studdy Solution

STEP 1

Assumptions1. The one-time fixed costs are $77,142\$77,142$77,142.2. The variable cost per book is $8.25\$8.25$8.25.3. The selling price per book is $25.50\$25.50$25.50.4. The publisher wants to break even, meaning the production costs equal the money from sales.

STEP 2

Let's denote the number of books the publisher needs to produce and sell to break even as xxx.

STEP 3

STEP 4

STEP 5

To break even, the total production costs must equal the total money from sales.Fixed costs+(Variable cost per book×x)=Selling price per book×xFixed\, costs + (Variable\, cost\, per\, book \times x) = Selling\, price\, per\, book \times xFixedcosts+(Variablecostperbook×x)=Sellingpriceperbook×x

STEP 6

Now, we can plug in the given values for the fixed costs, variable cost per book, and selling price per book into the equation.$77,142+($8.25×x)=$25.50×x\$77,142 + (\$8.25 \times x) = \$25.50 \times x$77,142+($8.25×x)=$25.50×x

STEP 7

To solve for xxx, we need to isolate xxx on one side of the equation. Let's distribute xxx on both sides of the equation.$77,142+$8.25x=$25.50x\$77,142 + \$8.25x = \$25.50x$77,142+$8.25x=$25.50x

STEP 8

Subtract $8.25x\$8.25x$8.25x from both sides of the equation to get all the terms involving xxx on one side.$77,142=$25.50x−$8.25x\$77,142 = \$25.50x - \$8.25x$77,142=$25.50x−$8.25x

STEP 9

Combine like terms on the right side of the equation.$77,142=($25.50−$8.25)x\$77,142 = (\$25.50 - \$8.25)x$77,142=($25.50−$8.25)x

STEP 10

Calculate the difference in the parentheses.$77,142=($17.25)x\$77,142 = (\$17.25)x$77,142=($17.25)x

STEP 11

Now, divide both sides of the equation by $17.25\$17.25$17.25 to solve for xxx.x=$77,142$17.25x = \frac{\$77,142}{\$17.25}x=$17.25$77,142​

STEP 12

Calculate the value of xxx.x=$77,142$17.25≈4472.4058x = \frac{\$77,142}{\$17.25} \approx 4472.4058x=$17.25$77,142​≈4472.4058

STEP 13

Since the publisher cannot produce and sell a fraction of a book, we round up to the next whole number.x=4473x = 4473x=4473The publisher must produce and sell 4,473 books to break even.

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Question Find the number of books the publisher must produce and sell to break even, given fixed costs of $\$77,142$ , variable costs of $\$8.25$ per book, and a selling price of $\$25.50$ per book.

Assumptions
1. The one-time fixed costs are $\$77,142$ .
2. The variable cost per book is $\$8.25$ .
3. The selling price per book is $\$25.50$ .
4. The publisher wants to break even, meaning the production costs equal the money from sales.

Let's denote the number of books the publisher needs to produce and sell to break even as $x$ .

To break even, the total production costs must equal the total money from sales.
$Fixed\, costs + (Variable\, cost\, per\, book \times x) = Selling\, price\, per\, book \times x$

Now, we can plug in the given values for the fixed costs, variable cost per book, and selling price per book into the equation.
$\$77,142 + (\$8.25 \times x) = \$25.50 \times x$

To solve for $x$ , we need to isolate $x$ on one side of the equation. Let's distribute $x$ on both sides of the equation.
$\$77,142 + \$8.25x = \$25.50x$

Subtract $\$8.25x$ from both sides of the equation to get all the terms involving $x$ on one side.
$\$77,142 = \$25.50x - \$8.25x$

Combine like terms on the right side of the equation.
$\$77,142 = (\$25.50 - \$8.25)x$

Calculate the difference in the parentheses.
$\$77,142 = (\$17.25)x$

Now, divide both sides of the equation by $\$17.25$ to solve for $x$ .
$x = \frac{\$77,142}{\$17.25}$

Calculate the value of $x$ .
$x = \frac{\$77,142}{\$17.25} \approx 4472.4058$

Since the publisher cannot produce and sell a fraction of a book, we round up to the next whole number.
$x = 4473$
The publisher must produce and sell 4,473 books to break even.