Question 1206529
<pre>
Use this Venn diagram:

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2.3,2,"s=?"),
locate(-3.5,-2,"z=?"),
locate(0,-2.7,P),
locate(-.45,-1,"y=?"),
locate(.6,.4,"x=?"),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,D),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,C),
locate(-1.5,.5,"v=?"),
locate(-.4,2.3,"t=?"),
locate(1.8,2,"u=?"),
locate(-.4,1.1,"w=?") )}}}

We want to find y=?

a survey of 220 families showed that: 
s+t+u+v+w+x+y+z=220
63 had a dog; 
s+t+v+w=63
64 had a cat;
t+u+w+x=64 
28 had a dog and a cat; 
t+w=28
84 neither had a cat nor a dog and in addition did not have a parakeet; 
z=84
6 had a cat, a dog and a parakeet. 
w=6

how many families had a parakeet only?  That's y.

     s t u v w x y z
     | | | | | | | |
(1)  s+t+u+v+w+x+y+z=220
(2)  s+t  +v+w      = 63
(3)    t+u  +w+x    = 64
(4)    t    +w      = 28
(5)                z= 84
(6)          w      =  6

Substitute w=6 in eq. (4) to get t=22
Substitute t=22, w=6, z=84 in the first three 
equations and simplify:

     s t u v w x y z
     | | | | | | | |
(1)  s  +u+v  +x+y   =108
(2)  s    +v         = 35
(3)      u    +x     = 36

Subtract eqs. (2) and (3) from eq. (1) 

and get y = 37

Edwin</pre>