Question
According to the survey made among 200 students, 140 students like cold drinks, 120 students like milkshake and 80 like both. Draw a Venn diagram and calculate how many students like at least one of the drinks?
Answers
GPT-4o mini
To solve this problem, we can use the principle of inclusion-exclusion to find out how many students like at least one of the drinks (cold drinks or milkshakes).
Let's denote:
- \( A \): the set of students who like cold drinks.
- \( B \): the set of students who like milkshakes.
From the problem:
- \( |A| = 140 \) (students who like cold drinks)
- \( |B| = 120 \) (students who like milkshakes)
- \( |A \cap B| = 80 \) (students who like both)
We want to find \( |A \cup B| \), the number of students who like at least one of the drinks, which is given by the inclusion-exclusion principle:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
Substituting in the values we have:
\[
|A \cup B| = 140 + 120 - 80
\]
\[
|A \cup B| = 260 - 80
\]
\[
|A \cup B| = 180
\]
Thus, **180 students like at least one of the drinks** (either cold drinks, milkshakes, or both).
### Venn Diagram
To draw the Venn diagram:
1. Draw two overlapping circles.
2. Label one circle \( A \) for cold drinks and the other circle \( B \) for milkshakes.
3. In the area where both circles overlap (representing students who like both drinks), write 80.
4. In the part of circle \( A \) that does not overlap, calculate the number of students who only like cold drinks:
\[
|A| - |A \cap B| = 140 - 80 = 60
\]
5. In the part of circle \( B \) that does not overlap, calculate the number of students who only like milkshakes:
\[
|B| - |A \cap B| = 120 - 80 = 40
\]
Your completed Venn diagram will have:
- 60 in the cold drinks circle not overlapping.
- 40 in the milkshakes circle not overlapping.
- 80 in the overlapping area.
This provides a visual representation of the distribution among the students regarding their preferences for cold drinks and milkshakes.
Let's denote:
- \( A \): the set of students who like cold drinks.
- \( B \): the set of students who like milkshakes.
From the problem:
- \( |A| = 140 \) (students who like cold drinks)
- \( |B| = 120 \) (students who like milkshakes)
- \( |A \cap B| = 80 \) (students who like both)
We want to find \( |A \cup B| \), the number of students who like at least one of the drinks, which is given by the inclusion-exclusion principle:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
Substituting in the values we have:
\[
|A \cup B| = 140 + 120 - 80
\]
\[
|A \cup B| = 260 - 80
\]
\[
|A \cup B| = 180
\]
Thus, **180 students like at least one of the drinks** (either cold drinks, milkshakes, or both).
### Venn Diagram
To draw the Venn diagram:
1. Draw two overlapping circles.
2. Label one circle \( A \) for cold drinks and the other circle \( B \) for milkshakes.
3. In the area where both circles overlap (representing students who like both drinks), write 80.
4. In the part of circle \( A \) that does not overlap, calculate the number of students who only like cold drinks:
\[
|A| - |A \cap B| = 140 - 80 = 60
\]
5. In the part of circle \( B \) that does not overlap, calculate the number of students who only like milkshakes:
\[
|B| - |A \cap B| = 120 - 80 = 40
\]
Your completed Venn diagram will have:
- 60 in the cold drinks circle not overlapping.
- 40 in the milkshakes circle not overlapping.
- 80 in the overlapping area.
This provides a visual representation of the distribution among the students regarding their preferences for cold drinks and milkshakes.