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Question 1105453: A woman bought a number of items for $36. She realized that if she had bought 6 more items for the same money, she would have paid $3 less per item. How many items did she buy?
Found 2 solutions by Theo, Boreal:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x is the number of items bought.
y is the price per item.
x * y = 36
if she bought 6 more items, she would have paid 3 dollars less per item and the total cost would still have been 36 dollars.
(x+6) * (y-3) = 36
from the first equation, solve for y to get y = 36/x.
in the second equation, replace y with 36/x to get (x + 6) * (36/x - 3) = 36
multiply both sides of the equation by x to get x * (x + 6) * (36/x - 3) = 36 * x
simplify to get (x^2 + 6x) * (36/x - 3) = 36x.
distribute the multiplication to get 36x - 3x^2 + 216 - 18x = 36x.
the 36x on both sides of the equation cancels out and you are left with -3x^2 + 216 - 18x = 0
reorder the terms in descending order of degree to get -3x^2 - 18x + 216 = 0
factor this equation to get x = 6 or x = -12.
x can't be negative, so x = 6.
i the equation of xy = 36, replace x with 6 to get 6y = 36.
solve for y to get y = 6.
x is the number of items bought.
y is the price per item.
she bought 6 items at 6 dollars each for a total of 36 dollars.
if she bought 6 more items, then the price per item would have been 3 dollars less.
(x+6) * (y-3) = 36
x = 6 and y = 6, therefore (6+6) * (6-3) = 36 which becomes 12 * 3 = 36 which is true.
everything checks out so your solution is that she bought 6 items.
Answer by Boreal(15235)  (Show Source):
You can put this solution on YOUR website! bought x items at a price of $36/x per item.
if she bought x=6 items for $36, the price would have been 36/(x+6)
36/(x+6)+3=(36/x)
multiply everything by x(x+6) to clear fractions
36x+3x^2+18x=36x+216
3x^2+18x-216=0
x^2+6x-72=0
(x+12)(x-6)=0
x=6 items, only positive root. ANSWER
They were $6 each, but had she bought 6 more items for 36, they would have been $36/12=$3 each.
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