Question 1125874
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This is a good example of a problem where a formal algebraic solution is far more work than an informal solution using logical analysis.<br>
Algebraically, we have<br>
x = number of items
y = price per item
(1) {{{xy = 48}}}
(2) {{{(x+6)(y-4)}}} = 48<br>
Solve (1) for y and substitute in (2):<br>
{{{y = 48/x}}}
{{{(x+6)((48/x)-4) = 48}}}
{{{48-4x+288/x-24 = 48}}}
{{{-4x-24+288/x = 0}}}
{{{x+6-72/x = 0}}}
{{{x^2+6x-72 = 0}}}
{{{(x+12)(x-6) = 0}}}
{{{x = -12}}}  (nonsense)  or  {{{x = 6}}}<br>
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.<br>
Whew!!  That was rather ugly....<br>
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.<br>
48 = 1*48
48 = 2*24
48 = 3*16
48 = 4*12
48 = 6*8<br>
Those last two satisfy the conditions of the problem....