A pet store conducted a survey to gather information about the types of pets people own. Here are the results: 686 people were surveyed in all 283 of the people own a fish 39 of th... A pet store conducted a survey to gather information about the types of pets people own. Here are the results: 686 people were surveyed in all 283 of the people own a fish 39 of the people own both a fish and a dog, but don't own a bird 128 of the people own both a fish and a bird, but don't own a dog 81 of the people own a dog, but own neither a fish nor a bird 52 of the people own a fish, but own neither a bird nor a dog 121 of the people own a bird, but own neither a fish nor a dog 609 of the people own at least one of these three types of pets (fish, bird, dog) Construct a Venn diagram illustrating these results. Then answer the question: How many of the people own a bird? Read more

Steps to Solve

1. Fill in known values in the Venn diagram We are given the following information to directly place into the Venn diagram:

   • Fish only: 52
   • Bird only: 121
   • Dog only: 81
   • Fish and Dog, but not Bird: 39
   • Fish and Bird, but not Dog: 128

2. Determine the number of people who own at least one pet type From the problem statement, we know that 609 people own at least one of these types of pets

3. Calculate the number of people who own all three pets (fish, bird, and dog) Let $x$ be the number of people who own all three pets. The total number of people who own at least one pet (609) is the sum of all the regions in the Venn diagram. So, we have:

   $52 + 121 + 81 + 39 + 128 + x + $(number of people who own bird and dog but not fish) = 609

   Now let $y$ be the number of people who own bird and dog but not fish.

   The number of people who own at least one pet (609 people) can also be expressed as:

   (Fish only) + (Bird only) + (Dog only) + (Fish and Dog only) + (Fish and Bird only) + (Dog and Bird only) + (All three pets). Substituting the values, we have:

   $52 + 121 + 81 + 39 + 128 + y + x = 609$ $421 + y + x = 609$. From this we can deduce $y + x = 609 - 421 = 188$.

4. Calculate the number of bird and dog owners who don't have fish We know that 283 people own a fish. We can write this as: (Fish Only) + (Fish and Dog only) + (Fish and Bird only) + (All three pets) = 283

Substituting values: $52 + 39 + 128 + x = 283\219 + x = 283\x = 283 - 219 = 64$

5. Calculate the number of people who own both a bird and a dog, but not a fish Now we know $x = 64$. Since $y + x = 188$, we deduce y = $188 - 64 = 124$ where $y$ is the number of people who own a bird and a dog but not fish.

6. Calculate the total number of people who own a bird To find the total number of bird owners, we sum all the regions within the "bird" circle: (Bird Only) + (Fish and Bird only) + (Dog and Bird only) + (All three pets) $= 121 + 128 + 124 + 64 = 437$