SOLUTION: Matt brought several cartons of large eggs for $112. If he had bought extra-large eggs costing $1 more per carton, he would have 2 less cartons of eggs for the same money. How much

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Question 1143524: Matt brought several cartons of large eggs for $112. If he had bought extra-large eggs costing $1 more per carton, he would have 2 less cartons of eggs for the same money. How much did he pay for a carton of large eggs?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


There are many ways to solve the problem using algebra. Here is an outline of one of them (not necessarily the best).

Let the number of cartons be x and the price of each carton be y. Then

xy+=+112
%28x-2%29%28y%2B1%29+=+112

Since both expressions are equal to 112, set them equal to each other and simplify:

xy+=+%28x-2%29%28y%2B1%29
xy+=+xy%2Bx-2y-2
x+=+2y%2B2

Substitute that expression for x in the first equation:

%282y%2B2%29y+=+112

Solve for y and then use that value to determine the value of x.

By far the easiest way to solve the problem is simply to play with numbers:

112 = 16*7
112 = 14*8

Those pairs of numbers satisfy the conditions of the problem:
16 cartons of eggs for $7 each;
14 cartons of eggs for $8 each

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

            I will show you another way to solve the problem using only one unknown from the very beginning. 


Let x be the price for a carton of large eggs (the unknown value under the question).


Then Matt bought  112%2Fx  cartoons of large eggs.


The price extra-large eggs is then  (x+1) dollars per cartoons, so he could buy  112%2F%28x%2B1%29  cartoons of extra-large eggs.


The condition says that


    112%2Fx - 112%2F%28x%2B1%29 = 2.


The setup is completed.

To solve the equation, multiply both sides by x*(x+1).


    112*(x+1) - 112x = 2x*(x+1)

    112x + 112 - 112x = 2x^2 + 2x

    2x^2 + 2x - 112 = 0

    x^2 + x - 56 = 0


Factor left side

    (x-7)*(x+8) = 0


It has two roots, but only positive root  x= 7  makes sense.


ANSWER.  The price of a cartoon of large eggs is $7.


CHECK.   At $7 per cartoon, Matt bough  112%2F7 = 16 cartoons of large eggs.

         At the price (7+1) = 8 dollars per cartoon, Matt could buy  112%2F8 = 14 cartoons of extra-large eggs.

         The difference is 16-14 = 2, in accordance with the condition  (!)


You can solve the problem by EITHER of the two ways.

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    - Had they sold . . .
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