Question 1105262
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It can be solved with only one variable....<br>
Represent the numbers of chocolate cookies, chocolate muffins, chocolate cakes, butter cookies, butter muffins, and butter cakes, respectively, with the matrix
{{{matrix(1,6,a,b,c,d,e,f)}}}<br>
The fact that he sold 140 more cookies than cakes is the last piece of information we will use.<br>
Let's see what the other information tells us to reduce the number of variables.<br>
(1) 1/5 of the cookies sold were chocolate.  That means the number of butter cookies sold is 4 times the number of chocolate cookies:
{{{matrix(1,6,a,b,c,4a,e,f)}}}<br>
(2) 2/3 of the cakes sold were chocolate.  That means the number of chocolate cakes is 2 times the number of butter cakes:
{{{matrix(1,6,a,b,2f,4a,e,f)}}}<br>
(3) The number of chocolate cookies and chocolate cakes is the same:
{{{matrix(1,6,2f,b,2f,8f,e,f)}}}<br>
(4) The number of butter cakes and butter muffins is the same:
{{{matrix(1,6,2f,b,2f,8f,f,f)}}}<br>
(5) The number of cakes is 3/4 the number of muffins:
{{{matrix(1,6,2f,3f,2f,8f,f,f)}}}<br>
Everything is now in terms of a single variable.  We can use the fact that the number of cookies was 140 more than the number of cakes to find the number of each item he sold.<br>
{{{2f+8f = 2f+f+140}}}
{{{10f = 3f+140}}}
{{{7f = 140}}}
{{{f = 20}}}<br>
He sold...
40 chocolate cookies;
60 chocolate muffins;
40 chocolate cakes;
160 butter cookies;
20 butter muffins; and
20 butter cakes<br>
The answer to the specific question that was asked is (I think) the difference between the total number of butter items and the total number of chocolate items; that difference is
{{{(160+20+20) - (40+60+40) = 200-140 = 60}}}