In a survey of 436 people, a pet food manufacturer found that 140 owned a dog but not a cat, 76 owned a cat but not a dog, and 122 owned neither a dog or a cat. How many owned both a cat and a dog?

We can solve this using the principle of inclusion-exclusion, a common method in set theory.

Let:
- \( D \) represent the set of people who own a dog.
- \( C \) represent the set of people who own a cat.
- The total number of surveyed people is 436.

We are given the following:
- 140 people own only a dog (\( D - C \)).
- 76 people own only a cat (\( C - D \)).
- 122 people own neither a dog nor a cat.

We need to find how many people own both a dog and a cat (\( D \cap C \)).

### Step-by-step solution:

1. The total number of people who own a dog or a cat or both is:
   \[
   436 - 122 = 314
   \]
   So, 314 people own either a dog, a cat, or both.

2. The total number of people who own only a dog or only a cat is:
   \[
   140 + 76 = 216
   \]
   
3. Let \( x \) be the number of people who own both a dog and a cat. We know that the number of people who own either a dog or a cat (or both) is the sum of those who own only a dog, only a cat, and both a dog and a cat:
   \[
   140 + 76 + x = 314
   \]
   
4. Solving for \( x \):
   \[
   x = 314 - (140 + 76) = 314 - 216 = 98
   \]

### Conclusion:
98 people own both a dog and a cat.

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. What is the total number of people who own a dog (including those who own both)?
2. How many people own a cat (including those who own both)?
3. What percentage of people own only a dog?
4. What percentage of the surveyed population own neither a dog nor a cat?
5. If 50 more people were surveyed and all owned a dog, how would this affect the total?

**Tip:** Always use diagrams like Venn diagrams for such set-related problems to visualize relationships easily.

In a survey of 436 people, a pet food manufacturer found that 140 owned a dog but not a cat, 76 owned a cat but not a dog, and 122 owned neither a dog or a cat. How many owned both a cat and a dog?

Learn how to use the principle of inclusion-exclusion to determine how many people own both a dog and a cat from a survey of 436 people. Explore key set theory concepts with a detailed solution. Suitable for students in Grades 9-12.

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