How do I solve problems with venn diagrams? Example of a problem at description.

Question

Here's the problem:

 

A veterinarian surveys 26 of his patrons. He discovers that 14 have dogs, 10 have cats, and 5 have fish. Four have dogs and cats, 3 have dogs and fish, and one has a cat and fish. If no one has all three kinds of pets, how many patrons have none of these pets?

 

I need the answer, as well as an explanation, if you can manage.

Answer 1

Mario,

 

My figure would have 3 overlapping regions, each representing dogs, cats, or fish.  The overlapping parts represent those areas where owners have more than one pet.

 

Now, we know the number owners that have dogs is 14 (place this number outside one of the 3 large regions and call it dogs.  Do the same for cats (10) and fish (5).  At this point it's important to remember that only some of the 14 dog owners have only dogs.  Same for cats and fish.

 

In fact, we know that 4 owners have both dogs and cats.  So, on the figure where the dog region and cat region overlap, write 4.  Do the same where the dog and fish regions overlap and write 3.

 

What this means is that of the 14 dog owners, 7 of them own more than a dog.  So, 7 dog owners own only dogs:

 

14 - 7 = 7

 

Use the same approach for each of the other pets:

 

Cats: 10 owners have cats; 4 have dogs and cats and 1 has a cat and a fish.  So, 5 owners have only cats (10 - 5).

 

Fish: 5 owners have fish; 3 have dogs and fish and 1 has a cat and a fish.  So 1 owner has only fish (5 - 4).

 

Then, in summary:

 

Owners that have:

 

Only Dogs - 7

 

Only Cats - 5

 

Only Fish - 1

 

Dogs AND Cats - 4

 

Dogs AND Fish - 3

 

Cats AND Fish - 1

 

The total number of owners that have at least one pet then is:

 

7 + 5 + 1 + 4 + 3 + 1 = 21

 

So, since there are 26 total owners, 5 must not own a dog, cat, or fish.

 

 

 

 

 

 

Answer 2

I can't actually draw a venn diagram here, but let's see what we can do.

 

Dogs                                            Dogs but no cats                Dogs but no cats or fish

-----                                             ------------------                 -------------------------

  14                                                        10                                           7

                Dogs and Cats

                ----------------

Cats                   4                         Cats but no dogs               Cats but no dogs or fish

-----                                              ------------------                --------------------------

  10                                                        6                                          5

                Cats and Fish

                ---------------

Fish                   1                         Fish but no cats                  Fish but not dogs or cats

-----                                             ------------------                 ----------------------------

  5                                                          4                                           1

                Dogs and Fish

                ---------------

                        3

 

To summarize we have

7 with only dogs

5 with only cats

1 with only a fish

4 with cats and dogs

1 with cats and fish

3 with dogs and fish

0 with all three

---

21 patrons with dogs, cats and fish.

 

Since 26 patrons were surveyed, this leaves 5 with none of these pets, implying these 5 have some other kind of pet.