Venn diagrams- Problem Solving | Nikita Brooks- Burrows - Free Printable

Let's solve each problem step by step using Venn diagrams and set theory principles.

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Problem 1: Pizza Toppings

Given:
- Total customers = 100
- Customers who ordered mushrooms (M) = 80
- Customers who ordered pepperoni (P) = 72
- Customers who ordered both mushrooms and pepperoni = 60

#### Venn Diagram Setup:
- Let \( M \cap P \) represent the intersection (both toppings).
- \( M \setminus P \) represents mushrooms but no pepperoni.
- \( P \setminus M \) represents pepperoni but no mushrooms.
- \( M^c \cap P^c \) represents neither topping.

#### Step-by-Step Solution:
1. Both toppings (\( M \cap P \)):
\[
|M \cap P| = 60
\]

2. Mushrooms but no pepperoni (\( M \setminus P \)):
\[
|M \setminus P| = |M| - |M \cap P| = 80 - 60 = 20
\]

3. Pepperoni but no mushrooms (\( P \setminus M \)):
\[
|P \setminus M| = |P| - |M \cap P| = 72 - 60 = 12
\]

4. Neither topping (\( M^c \cap P^c \)):
\[
|M^c \cap P^c| = \text{Total} - (|M| + |P| - |M \cap P|)
\]
\[
|M^c \cap P^c| = 100 - (80 + 72 - 60) = 100 - 92 = 8
\]

#### Answers:
a) Mushrooms but no pepperoni: \( \boxed{20} \)
b) Pepperoni but no mushrooms: \( \boxed{12} \)
c) Neither topping: \( \boxed{8} \)

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Problem 2: Car Repairs

Given:
- Total cars inspected = 50
- Cars needing new brakes (B) = 23
- Cars needing a new exhaust system (E) = 34
- Cars needing neither = 6

#### Venn Diagram Setup:
- Let \( B \cap E \) represent the intersection (both repairs).
- \( B \setminus E \) represents new brakes but no exhaust.
- \( E \setminus B \) represents new exhaust but no brakes.
- \( B^c \cap E^c \) represents neither repair.

#### Step-by-Step Solution:
1. Cars needing either or both repairs:
\[
\text{Cars needing either or both} = \text{Total} - \text{Neither} = 50 - 6 = 44
\]

2. Both repairs (\( B \cap E \)):
Using the principle of inclusion-exclusion:
\[
|B \cup E| = |B| + |E| - |B \cap E|
\]
\[
44 = 23 + 34 - |B \cap E|
\]
\[
|B \cap E| = 23 + 34 - 44 = 13
\]

3. New brakes but not an exhaust system (\( B \setminus E \)):
\[
|B \setminus E| = |B| - |B \cap E| = 23 - 13 = 10
\]

#### Answers:
a) Both repairs: \( \boxed{13} \)
b) New brakes but not an exhaust system: \( \boxed{10} \)

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Problem 3: Subject Preferences

Given:
- Total students = 85
- Students who like math (M) = 35
- Students who like history (H) = 37
- Students who like both math and history = 20

#### Venn Diagram Setup:
- Let \( M \cap H \) represent the intersection (both subjects).
- \( M \setminus H \) represents math but not history.
- \( H \setminus M \) represents history but not math.
- \( M^c \cap H^c \) represents neither subject.

#### Step-by-Step Solution:
1. Math but not history (\( M \setminus H \)):
\[
|M \setminus H| = |M| - |M \cap H| = 35 - 20 = 15
\]

2. History but not math (\( H \setminus M \)):
\[
|H \setminus M| = |H| - |M \cap H| = 37 - 20 = 17
\]

3. Students who like math or history (\( M \cup H \)):
Using the principle of inclusion-exclusion:
\[
|M \cup H| = |M| + |H| - |M \cap H| = 35 + 37 - 20 = 52
\]

4. Students who like neither (\( M^c \cap H^c \)):
\[
|M^c \cap H^c| = \text{Total} - |M \cup H| = 85 - 52 = 33
\]

#### Answers:
a) Math or history: \( \boxed{52} \)
b) Neither of the two subjects: \( \boxed{33} \)
c) Math but not history: \( \boxed{15} \)

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Problem 4: Sports Participation

Given:
- Total students = 80
- Students who played basketball (B) = 56
- Students who played volleyball (V) = 30
- Students who played both sports = 14

#### Venn Diagram Setup:
- Let \( B \cap V \) represent the intersection (both sports).
- \( B \setminus V \) represents basketball but not volleyball.
- \( V \setminus B \) represents volleyball but not basketball.
- \( B^c \cap V^c \) represents neither sport.

#### Step-by-Step Solution:
1. Basketball but not volleyball (\( B \setminus V \)):
\[
|B \setminus V| = |B| - |B \cap V| = 56 - 14 = 42
\]

2. Volleyball but not basketball (\( V \setminus B \)):
\[
|V \setminus B| = |V| - |B \cap V| = 30 - 14 = 16
\]

3. Students who didn't play either sport (\( B^c \cap V^c \)):
Using the principle of inclusion-exclusion:
\[
|B \cup V| = |B| + |V| - |B \cap V| = 56 + 30 - 14 = 72
\]
\[
|B^c \cap V^c| = \text{Total} - |B \cup V| = 80 - 72 = 8
\]

#### Answers:
a) Didn't play either sport: \( \boxed{8} \)
b) Played volleyball only: \( \boxed{16} \)
c) Played basketball only: \( \boxed{42} \)

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Problem 5: Holiday Activities

Given:
- Total people = 24
- People who went swimming (S) = 15
- People who went fishing (F) = 12
- People who did neither = 6

#### Venn Diagram Setup:
- Let \( S \cap F \) represent the intersection (both activities).
- \( S \setminus F \) represents swimming but not fishing.
- \( F \setminus S \) represents fishing but not swimming.
- \( S^c \cap F^c \) represents neither activity.

#### Step-by-Step Solution:
1. People who went either or both activities:
\[
\text{People who went either or both} = \text{Total} - \text{Neither} = 24 - 6 = 18
\]

2. Both activities (\( S \cap F \)):
Using the principle of inclusion-exclusion:
\[
|S \cup F| = |S| + |F| - |S \cap F|
\]
\[
18 = 15 + 12 - |S \cap F|
\]
\[
|S \cap F| = 15 + 12 - 18 = 9
\]

3. Swimming but not fishing (\( S \setminus F \)):
\[
|S \setminus F| = |S| - |S \cap F| = 15 - 9 = 6
\]

4. Fishing but not swimming (\( F \setminus S \)):
\[
|F \setminus S| = |F| - |S \cap F| = 12 - 9 = 3
\]

#### Answers:
a) Went swimming and fishing: \( \boxed{9} \)
b) Only went swimming: \( \boxed{6} \)
c) Only went fishing: \( \boxed{3} \)

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Final Answers:

1. a) \( \boxed{20} \), b) \( \boxed{12} \), c) \( \boxed{8} \)
2. a) \( \boxed{13} \), b) \( \boxed{10} \)
3. a) \( \boxed{52} \), b) \( \boxed{33} \), c) \( \boxed{15} \)
4. a) \( \boxed{8} \), b) \( \boxed{16} \), c) \( \boxed{42} \)
5. a) \( \boxed{9} \), b) \( \boxed{6} \), c) \( \boxed{3} \)