Question 1206529: a survey of 220 families showed that: 63 had a dog; 64 had a cat; 28 had a dog and a cat; 84 neither had a cat nor a dog and in addition did not have a parakeet; 6 had a cat, a dog and a parakeet. how many families had a parakeet only?
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website!
Use this Venn diagram:
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
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(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
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(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
Answer by ikleyn(52775)  (Show Source):
You can put this solution on YOUR website! .
a survey of 220 families showed that:
63 had a dog; 64 had a cat; 28 had a dog and a cat;
84 neither had a cat nor a dog and in addition did not have a parakeet;
6 had a cat, a dog and a parakeet.
how many families had a parakeet only?
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We have a universal set of 220 families and three its basic subsets
d of 63 families having a dog;
c of 64 families having a cat;
p of unknown number of families having a parakeet.
The number of families in the union (d U c) is 63 + 64 - 28 = 99.
The set of families that neither have a cat nor a dog is the complement to the set (d U c),
and the number of such families is 220 - 99 = 121.
We are told, that the number of families that "in addition" do not have parakeet is 84.
From it, we conclude that 121-84 = 37 families do have a parakeet, but have neither a cat nor a dog.
So, the answer to the problem's question is 37 families.
Solved.
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Notice that the last condition in the post
"6 had a cat, a dog and a parakeet. "
is excessive and unnecessary. I did not use it in my solution.
This problem is DEFINITELY not to write and/or to use Venn diagram
and accompanied system of linear equations in 8 unknowns.
This problem's true destination and true purpose is to solve it MENTALLY
and to teach young students to think logically about the sets and their subsets.
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