A pet store surveyed 696 people to see how many own a dog, how many own a bird, and how many own a cat. The Venn diagram below shows the results. (Each number gives the number of p... A pet store surveyed 696 people to see how many own a dog, how many own a bird, and how many own a cat. The Venn diagram below shows the results. (Each number gives the number of people who fall into that Venn diagram category) (a) How many of the people own a dog? (b) How many of the people own neither a dog nor a cat? (c) How many of the people own both a bird and a cat, but don't own a dog?
Understand the Problem
The question provides a Venn diagram representing the results of a survey about pet ownership (dogs, birds, and cats). The goal is to use the information in the Venn diagram to answer three specific questions:
- How many people own a dog?
- How many people own neither a dog nor a cat?
- How many people own both a bird and a cat, but don't own a dog?
Answer
(a) 291 (b) 73 (c) 136
Answer for screen readers
(a) 291 people (b) 73 people (c) 136 people
Steps to Solve
- Find the number of people who own a dog
To find the number of people who own a dog, sum all the numbers within the 'dog' circle in the Venn diagram. This includes those who own only a dog, those who own a dog and a bird, those who own a dog and a cat, and those who own a dog, a bird, and a cat.
$112 + 55 + 89 + 35 = 291$
- Find the number of people who own neither a dog nor a cat
To find the number of people who own neither a dog nor a cat, we need to look for the number of people outside of both the 'dog' and 'cat' circles. Note that we are looking for people who may own a bird, but not a dog or a cat. From the Venn diagram, this number is 73.
- Find the number of people who own both a bird and a cat, but don't own a dog
To find the number of people who own both a bird and a cat, but not a dog, we need to look at the intersection of the 'bird' and 'cat' circles, excluding the part that also intersects with the 'dog' circle. From the Venn diagram, this number is 136.
(a) 291 people (b) 73 people (c) 136 people
More Information
Venn diagrams are useful for visually representing set relationships and solving problems involving overlapping categories.
Tips
A common mistake when using a Venn diagram is to double-count elements that belong to the intersection of two or more sets. For example, when counting the number of dog owners, you MUST include those who own other pets in addition to the dog. A further common mistake is to misread the Venn diagram in terms of what each number represents.
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