Worksheet featuring Venn diagrams to solve problems related to console ownership, pizza preferences, and travel destinations.

Let's solve each part of the Venn diagram problems step by step, using the information provided.

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Problem 1: Xbox, PS3, Wii Venn Diagram

We are told that 40 people were surveyed. The Venn diagram has three overlapping circles for Xbox, PS3, and Wii.

The numbers in the regions:

- Only Xbox: 7
- Xbox and PS3 only: 1
- Xbox and Wii only: 9
- PS3 and Wii only: 2
- All three (Xbox, PS3, Wii): 5
- Only PS3: 3
- Only Wii: 2

Let’s sum all these values to check total:

$$
7 + 1 + 9 + 2 + 5 + 3 + 2 = 30
$$

But we are told 40 people were surveyed → so 10 people own no console.

Now answer the questions:

#### 1) An Xbox
This includes all parts of the Xbox circle:
Only Xbox: 7
Xbox & PS3 only: 1
Xbox & Wii only: 9
All three: 5
→ $7 + 1 + 9 + 5 = \boxed{22}$

#### 2) An Xbox and a PS3
This is the intersection of Xbox and PS3 (includes those who also have Wii).
So: Xbox & PS3 only: 1
All three: 5
→ $1 + 5 = \boxed{6}$

#### 3) At least two consoles
This means people who own two or more consoles.

- Xbox & PS3 only: 1
- Xbox & Wii only: 9
- PS3 & Wii only: 2
- All three: 5
→ $1 + 9 + 2 + 5 = \boxed{17}$

#### 4) An Xbox or a Wii
"Or" means union of Xbox and Wii.
Add all regions in either Xbox or Wii (including overlap):

- Only Xbox: 7
- Xbox & PS3 only: 1
- Xbox & Wii only: 9
- All three: 5
- Only Wii: 2
- PS3 & Wii only: 2
- But wait — PS3 & Wii only is already included in Wii
- Only PS3: 3 → not in Xbox or Wii → exclude

So:
$7 + 1 + 9 + 5 + 2 + 2 = \boxed{26}$

Alternatively:
Total in Xbox: 22
Total in Wii: Only Wii (2) + Wii & Xbox only (9) + Wii & PS3 only (2) + all three (5) = $2+9+2+5=18$
But overlap (Xbox ∩ Wii) = 9 + 5 = 14
So:
$|Xbox ∪ Wii| = |Xbox| + |Wii| - |Xbox ∩ Wii| = 22 + 18 - 14 = \boxed{26}$ ✔

#### 5) An Xbox and PS3, but not a Wii
This is just the region where Xbox and PS3 intersect, but not in Wii → only the "Xbox & PS3 only" region:
→ $\boxed{1}$

#### 6) No console
Total surveyed: 40
People with at least one console: $30$ (from earlier sum)
→ $40 - 30 = \boxed{10}$

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Problem 2: Mushrooms and Cheese Pizza (Venn Diagram)

Given: 33 people at the party.

Diagram:
- Only Mushrooms: 5
- Both Mushrooms and Cheese: 8
- Only Cheese: 12
- Outside both (no pizza): 8

Check total:
$5 + 8 + 12 + 8 = 33$ ✔

Now answer:

#### 7) Mushrooms
Includes only mushrooms and both:
$5 + 8 = \boxed{13}$

#### 8) Cheese
Only cheese and both:
$12 + 8 = \boxed{20}$

#### 9) Just Mushrooms
Only mushrooms: $\boxed{5}$

#### 10) Both Mushrooms and Cheese
Intersection: $\boxed{8}$

#### 11) No pizzas
Outside both circles: $\boxed{8}$

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Problem 3: Paris and France Travel Survey

Given: 19 people surveyed.

Venn diagram:
- Only Paris: 2
- Both Paris and France: 6
- Only France: ? (let’s call it x)
- Outside both: 8

We know total = 19

So:

$$
\text{Only Paris} + \text{Both} + \text{Only France} + \text{Neither} = 19 \\
2 + 6 + x + 8 = 19 \\
16 + x = 19 \\
x = 3
$$

So Only France = 3

Now answer:

#### 12) Just France
→ Only France: $\boxed{3}$

#### 13) France
Includes only France and both:
$3 + 6 = \boxed{9}$

#### 14) Both Paris and France
→ Intersection: $\boxed{6}$

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✔ Final Answers:

#### Xbox/PS3/Wii (40 people):
1) An Xbox: 22
2) An Xbox and a PS3: 6
3) At least two consoles: 17
4) An Xbox or a Wii: 26
5) Xbox and PS3, but not Wii: 1
6) No console: 10

#### Mushrooms/Cheese (33 people):
7) Mushrooms: 13
8) Cheese: 20
9) Just Mushrooms: 5
10) Both: 8
11) No pizzas: 8

#### Paris/France (19 people):
12) Just France: 3
13) France: 9
14) Both Paris and France: 6

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✔ All answers match your handwritten red answers except possibly #4 and #6 which you didn't fill in — but now they're solved correctly.

Let me know if you'd like this explained visually!