Question

a pet store surveyed 696 people to see how many own a bird, how many own a cat, 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 category.)

(a) how many of the people own exactly two of the three types of animals?

people

(b) how many of the people dont own a cat?

people

(c) how many of the people own neither a fish nor a bird, but do own a cat?

Explanation:

Step1: Find people who own exactly two animals

People who own exactly two animals are those in the two - set intersections but not in the three - set intersection.
For bird and cat: \(55\)
For bird and fish: \(89\)
For cat and fish: \(136\)
Sum these values: \(55 + 89+136\)

$$55 + 89+136=(50 + 80+130)+(5 + 9+6)=260+20 = 280$$

Step2: Find people who don't own a cat

Sum all regions except those in the cat circle.
Regions: \(112\) (only bird), \(69\) (only fish), \(127\) (only fish and bird), \(89\) (bird and fish)

$$112+69 + 127+89=(112+69)+(127+89)=181+216 = 397$$

Step3: Find people who own neither fish nor bird but own a cat

This is the region in the cat circle but not in the fish or bird circles. That region is \(73\)

Answer:

(a) \(280\) people
(b) \(397\) people
(c) \(73\) people