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1) A fashion designer makes and sells hats. The material for each hat costs $5.50. The hats sell for $12.50 each. The designer spends $1400 on advertising. How many hats must the designer sell to break even?


2) A puzzle expert wrote a new Sudoku puzzle book. His initial costs are $864. Binding and packaging each cost $0.80. The price of the book is $2. How many copies must be sold to break even?


Assign two variables and equations. Please solve.

Verified Answer :

1) If the hats cost $12.50 each and the materials cost $5.50, then the fashion designer makes  ($12.50 - $5.50 =) $7 per hat. So, to work out how many hats you must make, you need to divide how much you spend on advertising by how much you make per hat so, $1,400 ÷ $7 = 200 hats. 
$12.50 - $5.50 = $7
$1,400 ÷ $7 = 200 hats

and your variable and equation is:

y = number of hats needed to sell
advertising = earnings of 1 hat x number of hats
1,400 = 7y
1,400÷7 = y 
200 = y
∴ The fashion designer must make and sell 200 hats to break even and get back her $1,400 spent on advertising.

2) So, you do the same on this one as well. If he makes $2 per book, and binding and packaging cost $0.80, then you need to $2 - $0.80 to find how much each book will make you, the answer is $1.20 per book. Then you need to $864 ÷ $1.20 which equals 720 books. 
$2 - $0.80 = $1.20 
$864 ÷ $1.20 = 720 books.

and your variable and equation would be:

y = unknown number of books needed to sell
initial costs = earnings of 1 book x number of books
864 = 1.20y
864 ÷ 1.20 = y 
720 = y
∴ The puzzle expert must sell 720 books to break even and earn back his initial costs of $864.

I hope that's what you meant (putting it in an algebraic equation) and I hope is correct 

1) 12.50-5.50=$7
1400/7=200
The designer has to sell 200 hats to break even. 

2) 2-0.80=1.20
864/1.20= 720
The puzzle expert must sell 720 books to break even.

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