Question

A survey of 100 people asked if they have a dog or a cat. The answers were: 56 people answered yes for dog and 51 people answered yes for cat. Some people may own neither and some people may own both animals. What is the maximum number of people who could answer yes for both, cat and dog?

Answer 1

Explanation:

both cats and dogs a

Answer 2

Answer: The maximum number of people who could answer yes for both cat and dog is 7.

Step-by-step explanation: Given that a survey of 100 people asked if they have a dog or a cat.

Out of those 100 people, 56 people answered yes for dog and 51 people answered yes for cat. Some people may own neither and some people may own both animals.

We are to find the maximum number of people that could answer yes for both cat and dog.

Let A and B represents the set of people that answered Yes for dog and Yes for cat respectively.

Then, according to the given information, we have

`n(A)=56,~~n(B)=51.`

From Set Theory, we have

`n(A\cup B)=n(A)+n(B)-n(A\cap B)\\\\\Rightarrow n(A\cup B)=56+51-n(A\cap B)\\\\\Rightarrow n(A\cup B)=107-n(A\cap B)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)`

Since the total number of people is 100, so we get

`n(A\cup B)\geq 100.`

Therefore, from equation (i), we get

`107-n(A\cap B)\geq 100\\\\\Rightarrow n(A\cap B)\leq 107-100\\\\\Rightarrow n(A\cap B)\leq 7.`

Thus, the maximum number of people who could answer yes for both cat and dog is 7.