Venn diagrams word problems | TPT - Free Printable

Problem 1: Communications Class Survey

Question: A communications class surveyed 150 students and found that 115 had seen the movie "Titanic," while 65 had seen "Forest Gump." Of these, 45 had seen both movies. How many people had not seen either movie?

#### Solution:
1. Define the sets:
- Let \( T \) be the set of students who saw "Titanic."
- Let \( F \) be the set of students who saw "Forest Gump."

2. Given information:
- Total number of students surveyed: \( |U| = 150 \)
- Number of students who saw "Titanic": \( |T| = 115 \)
- Number of students who saw "Forest Gump": \( |F| = 65 \)
- Number of students who saw both movies: \( |T \cap F| = 45 \)

3. Use the principle of inclusion-exclusion to find the number of students who saw at least one of the movies:
\[
|T \cup F| = |T| + |F| - |T \cap F|
\]
Substituting the given values:
\[
|T \cup F| = 115 + 65 - 45 = 135
\]

4. Find the number of students who did not see either movie:
- The total number of students is 150.
- The number of students who saw at least one movie is 135.
- Therefore, the number of students who did not see either movie is:
\[
|U| - |T \cup F| = 150 - 135 = 15
\]

Answer for Problem 1:
\[
\boxed{15}
\]

---

Problem 2: Campers Club

Question: Of 45 members of the Campers Club, 27 had camped at Yellowstone, 23 had camped at Yosemite, and 18 had camped at both. How many had camped at neither?

#### Solution:
1. Define the sets:
- Let \( Y \) be the set of members who camped at Yellowstone.
- Let \( S \) be the set of members who camped at Yosemite.

2. Given information:
- Total number of members: \( |U| = 45 \)
- Number of members who camped at Yellowstone: \( |Y| = 27 \)
- Number of members who camped at Yosemite: \( |S| = 23 \)
- Number of members who camped at both: \( |Y \cap S| = 18 \)

3. Use the principle of inclusion-exclusion to find the number of members who camped at least once:
\[
|Y \cup S| = |Y| + |S| - |Y \cap S|
\]
Substituting the given values:
\[
|Y \cup S| = 27 + 23 - 18 = 32
\]

4. Find the number of members who did not camp at either location:
- The total number of members is 45.
- The number of members who camped at least once is 32.
- Therefore, the number of members who did not camp at either location is:
\[
|U| - |Y \cup S| = 45 - 32 = 13
\]

Answer for Problem 2:
\[
\boxed{13}
\]

---

Problem 3: Coffee Survey

Question: A survey showed that 43 people like cream in their coffee, 20 like both cream and sugar, and 16 like neither. How many people were in this survey?

#### Solution:
1. Define the sets:
- Let \( C \) be the set of people who like cream in their coffee.
- Let \( S \) be the set of people who like sugar in their coffee.

2. Given information:
- Number of people who like cream: \( |C| = 43 \)
- Number of people who like both cream and sugar: \( |C \cap S| = 20 \)
- Number of people who like neither: 16

3. Let \( x \) be the total number of people surveyed.

4. Use the principle of inclusion-exclusion to express the total number of people:
- The total number of people can be divided into four groups:
1. People who like only cream.
2. People who like only sugar.
3. People who like both cream and sugar.
4. People who like neither.

- The number of people who like only cream is:
\[
|C| - |C \cap S| = 43 - 20 = 23
\]

- Let \( y \) be the number of people who like only sugar. Then the total number of people surveyed is:
\[
x = (\text{only cream}) + (\text{only sugar}) + (\text{both}) + (\text{neither})
\]
Substituting the known values:
\[
x = 23 + y + 20 + 16
\]
Simplifying:
\[
x = 59 + y
\]

5. To find \( y \), we need the total number of people who like sugar. However, we do not have this information directly. Instead, we use the fact that the total number of people surveyed is \( x \).

6. Since we are asked to find the total number of people surveyed, we can use the equation directly:
\[
x = 59 + y
\]
But we do not need \( y \) explicitly because the problem only asks for the total number of people surveyed. The equation \( x = 59 + y \) shows that the total number of people is the sum of all disjoint groups.

7. The total number of people surveyed is:
\[
x = 59 + 16 = 75
\]

Answer for Problem 3:
\[
\boxed{75}
\]

---

Final Answers:

1. \(\boxed{15}\)
2. \(\boxed{13}\)
3. \(\boxed{75}\)