word problem

Question

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
$21,390
, and the variable costs will be
$20.25
per book. With the other method, the one-time fixed costs will total
$47,466
, and the variable costs will be
$10
per book. For how many books produced will the costs from the two methods be the same?

Answer 1

This is solved by creating two algebraic equations and solving for a common x.

 

You know that each method requires the one-time costs plus the variable costs. Therefore, the two methods can be expressed as:

 

y=21390+20.25x

and

y=47466+10x, with y representing the total cost for a given x (x represents the number of books).

 

You also know that the costs must be equal to each other, so:

 

21390+20.25x=y=47466+10x

21390+20.25x=47466+10x

 

Now, all you have to do is solve for x.

 

Combining like terms and getting x by itself,

 

20.25x-10x=47466-21390

10.25x=26076

x=26076/10.25

x=2544

 

So, the costs of these two methods will be equal if they both make 2544 books.

 

To check, plug 2544 back into both equations for x and you should get the same number.

 

y=21390+20.25(2544)               y=47466+10(2544)

y=21390+51516                        y=47466+25440

y=72906                                   y=72906

Answer 2

method 1 can be modeled by c = 21390 + 20.25b     c is total cost and b is number of books.

method 2 can be modeled by c = 47466 + 10b

 

So the two are equal when they have the same values for c and the same valus of b. so just substitute to get

47466 + 10b = 21390 + 20.25b

26076           = 10.25b

2544             =  b

 

So let's check....if b=2544, the cost of method 1 is c = 21390 + 20.25(2544) = 72906

method 2 = 47466 + 10(2544) = 72906...............the same cost.

So the number of books to print that will come out to the same price is 2544.

 

Hope this helped

Answer 3

b = number of books

 

21390 + 20.25b = 47466 + 10b

21390 + 10.25b = 47466

10.25b = 26076

b = 2544

 

Answer 4

Since there are two production methods, there will be two equations to calculate the total cost. Let T be the total cost and B be the number of books. 

 

The first production method would be T = Fixed Cost + Variable Cost * B T = 21390 + 20.25*B

The second production method would be T = 47466 + 10*B

 

To find the number of books where the total cost is equal, set the two formulas equal to each other and solve for B.

21390 + 20.25*B = 47466 + 10*B

-21390                    -21390

 

20.25*B = 10*B + 26076

-10*B       -10*B

 

10.25*B = 26076

/10.25       /10.25

 

B = 2544

 

Always go back and check your answer. Using the found B, calculate the total cost for the two methods and make sure they are equal.