SOLUTION: A women bought a number of items for $48. She realizes that if she bought 6 more items for the same money, she would have paid $4 less per item. How many items did she buy?

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Question 1125874: A women bought a number of items for $48. She realizes that if she bought 6 more items for the same money, she would have paid $4 less per item. How many items did she buy?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


This is a good example of a problem where a formal algebraic solution is far more work than an informal solution using logical analysis.

Algebraically, we have

x = number of items
y = price per item
(1)
(2) = 48

Solve (1) for y and substitute in (2):








(nonsense) or

The woman bought 6 items for $8 each, for a total of $48. She could have bought 6+6 = 12 items for $8-$4 = $4 each and spent the same total of $48.

Whew!! That was rather ugly....

How can you solve the problem far more easily with logical reasoning? Simply find two pairs of numbers whose product is 48 that fit the conditions of the problem.

48 = 1*48
48 = 2*24
48 = 3*16
48 = 4*12
48 = 6*8

Those last two satisfy the conditions of the problem....
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.

            The original formulation in the post is TERRIBLE (from the common sense of view).

            And it is the main reason why the problem seems to be upside-down, staying on the head.

            So the first step is to transform it to the right form.

            Much more reasonable (classic form) formulation (after editing) is this: 


    A women bought a number of equally priced items for $48. Had the price be $4 less for each item,  she could buy 6 more items 
    for the same money. How many items did she buy?


In this formulation, the standard algebra solution is and works as follows. 

    x*p = 48,            (1)
    (x+6)*(p-4) = 48.    (2)


Simplify equation (2) :

    xp + 6p - 4x - 24 = 48.


Replace the term xp in the left side of the last equation by 48, based on (1) :

    48 + 6p - 4x - 24 = 48,

    6p - 4x = 24.        (3)


From equation (3), express p =   and substitute it into equation (1). You will get


     = 48,


    x*(24+4x) = 288


    4x^2 +24x - 288 = 0


    x^2 + 6x - 72 = 0


    (x+12)*(x-6) = 72.


The positive root (which only makes sense) is 6.


Answer.  The women bought 6 items.

Solved.

------------------

Was it so ugly ?

But of course,  not - if you know on how to do it in a correct way.
Simply a standard  (and a smooth)  algebra solution,  and you should learn on how to solve it algebraically -
simply because you can meet many of such problems in your student years.

To see many other similar solved problems,  look into the lesson
    - Had they sold . . .
in this site.


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