A pet store surveyed 683 people to see how many own a cat, how many own a bird, and how many own a fish. The Venn diagram below shows the results. (Each number gives the number of... A pet store surveyed 683 people to see how many own a cat, how many own a bird, and how many own a fish. The Venn diagram below shows the results. (Each number gives the number of people who fall into that Venn diagram. (a) How many of the people own a fish but don't own a cat? (b) How many of the people own neither a bird nor a fish? (c) How many of the people own exactly one of the three types of animals?
Understand the Problem
The question presents a Venn diagram displaying the results of a survey about pet ownership (cats, birds, and fish). The task is to analyze the diagram to answer three questions: (a) How many people own a fish but not a cat? (b) How many people own neither a bird nor a fish? (c) How many people own exactly one of the three types of animals?
Answer
(a) 201 (b) 52 (c) 250
Answer for screen readers
(a) 201 people (b) 52 people (c) 250 people
Steps to Solve
- Find the number of people who own a fish but not a cat
To find the number of people who own a fish but not a cat, we look at the section of the "fish" circle that does not overlap with the "cat" circle. This includes those who own only fish and those who own fish and birds but not cats. From the Venn diagram, these numbers are 77 (fish only) and 124 (fish and bird, but not cat), so we add them together:
$77 + 124 = 201$
- Find the number of people who own neither a bird nor a fish
To find the number of people who own neither a bird nor a fish, we look for the number of people outside of both the "bird" and "fish" circles. This includes those who own only cats, and those who own cats but not birds or fish. From the Venn diagram, these are the sections of the cat circle that do not overlap with either the bird or fish circles. We add these numbers together:
$52 + 77 = 129$
- Find the number of people who own exactly one type of animal
To find the number of people who own exactly one of the three types of animals, we need to find the numbers in each circle that do not overlap with any other circle. This means they only own that particular animal. From the Venn diagram, these numbers are 52 (cat only), 121 (bird only), and 77 (fish only). We add these numbers together:
$52 + 121 + 77 = 250$
(a) 201 people (b) 52 people (c) 250 people
More Information
Venn diagrams are a useful way to visualize the relationships between different groups of data. By carefully examining the overlaps and non-overlaps, we can answer various questions about the data.
Tips
A common mistake is to misinterpret the regions of the Venn diagram. For example, including the intersection of "fish" and "cat" when the question asks for "fish but not cat". Similarly, including the intersection of two or more circles when the question asks for "exactly one" is a common mistake.
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