In a survey of 100 students, 60 students like to watch movies, 40 students like to play video games, and 20 students like to do both. How many students like neither movies nor video games?
The correct answer and explanation is:
To solve this problem, we can use the principle of inclusion and exclusion. The total number of students surveyed is 100, and we have the following information:
- 60 students like to watch movies.
- 40 students like to play video games.
- 20 students like both movies and video games.
We need to determine how many students like neither movies nor video games.
Step 1: Calculate the number of students who like either movies or video games (or both).
According to the principle of inclusion and exclusion:
- The number of students who like movies or video games (or both) is given by: Total who like movies or video games=(students who like movies)+(students who like video games)−(students who like both)\text{Total who like movies or video games} = \text{(students who like movies)} + \text{(students who like video games)} – \text{(students who like both)}
Substitute the given values: Total who like movies or video games=60+40−20=80\text{Total who like movies or video games} = 60 + 40 – 20 = 80
Step 2: Calculate the number of students who like neither movies nor video games.
Since the total number of students is 100, the number of students who like neither movies nor video games is: Students who like neither=100−Total who like movies or video games\text{Students who like neither} = 100 – \text{Total who like movies or video games}
Substitute the value we found: Students who like neither=100−80=20\text{Students who like neither} = 100 – 80 = 20
Conclusion:
So, 20 students like neither movies nor video games.
Explanation:
In this problem, we used the inclusion-exclusion principle to avoid counting students who like both movies and video games more than once. By adding the number of students who like movies and the number who like video games, we accounted for the students who like both. Then, subtracting this overlap gave us the correct number of students who like either one or both. Finally, subtracting this result from the total survey population gave us the number of students who like neither activity.