document.write( "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? \n" ); document.write( "

Algebra.Com's Answer #742196 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "This is a good example of a problem where a formal algebraic solution is far more work than an informal solution using logical analysis.

\n" ); document.write( "Algebraically, we have

\n" ); document.write( "x = number of items
\n" ); document.write( "y = price per item
\n" ); document.write( "(1) $\"xy+=+48\"$
\n" ); document.write( "(2) $\"%28x%2B6%29%28y-4%29\"$ = 48

\n" ); document.write( "Solve (1) for y and substitute in (2):

\n" ); document.write( " $\"y+=+48%2Fx\"$
\n" ); document.write( " $\"%28x%2B6%29%28%2848%2Fx%29-4%29+=+48\"$
\n" ); document.write( " $\"48-4x%2B288%2Fx-24+=+48\"$
\n" ); document.write( " $\"-4x-24%2B288%2Fx+=+0\"$
\n" ); document.write( " $\"x%2B6-72%2Fx+=+0\"$
\n" ); document.write( " $\"x%5E2%2B6x-72+=+0\"$
\n" ); document.write( " $\"%28x%2B12%29%28x-6%29+=+0\"$
\n" ); document.write( " $\"x+=+-12\"$ (nonsense) or $\"x+=+6\"$

\n" ); document.write( "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.

\n" ); document.write( "Whew!! That was rather ugly....

\n" ); document.write( "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.

\n" ); document.write( "48 = 1*48
\n" ); document.write( "48 = 2*24
\n" ); document.write( "48 = 3*16
\n" ); document.write( "48 = 4*12
\n" ); document.write( "48 = 6*8

\n" ); document.write( "Those last two satisfy the conditions of the problem....
\n" ); document.write( " \n" ); document.write( "

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