SOLUTION: A survey of 220 families showed that 83 had a dog, 64 had a cat, 28 had a dog and cat, 84 had neither a dog nor a cat nor a parakeet, 6 had a dog, cat and parakeet. How many only

So we fill in l=84, and i=6

We have already accounted for 6 of these 28
who had all three pets. So the number who
had only a dog and cat but no parakeet is
found by subtracting the 6 from the 28, so
f=28-6=22, so we fill 22 for f:

We have accounted for 22+6 or 28 dog owners,
and since this is all we are told about dog
owners, all we know is that between e and h
there are 83-28 or 55 dog owners that either
do or don't own a parakeet. So let's leave h
unknown and express e in terms of h as 55-h,
so we leave h unknown and put 55-h for e.



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We have accounted for 22+6 or 28 cat owners,
and since this is all we are told about cat
owners, all we know is that between g and j
there are 64-28 or 36 cat owners that either
do or don't own a parakeet. So let's leave j
unknown and express g in terms of j as
36-j, so we leave j unknown and put 36-j for
g.





The only other thing we are given is:

So we add up the numbers or expressions in
all 8 regions and get:

(55-h)+22+(36-j)+h+6+j+k+84 = 220

Simplifying:

55-h+22+36-j+h+6+j+k+84 = 220
                  203+k = 220
                      k = 17

Answer: 17

Edwin