Let’s solve each problem one by one, step by step. We’ll use Venn diagrams logic — that means we think about groups that overlap (both) and groups that don’t (only one or neither).
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Problem 1: Big Red’s Pizza Emporium
Total customers = 100
Mushrooms (P) = 80
Pepperoni (M) = 72
Both mushrooms AND pepperoni = 60
We want to find:
a) Mushrooms but no pepperoni → That’s total mushrooms minus those who got both.
→ 80 - 60 =
20
b) Pepperoni but no mushrooms → Total pepperoni minus those who got both.
→ 72 - 60 =
12
c) Neither topping → First, add up everyone who got at least one topping:
Only mushrooms + only pepperoni + both = 20 + 12 + 60 = 92
Then subtract from total: 100 - 92 =
8
✔ Check: 20 + 12 + 60 + 8 = 100 → Correct!
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Problem 2: Dan’s Automotive Shop
Total cars = 50
New brakes (B) = 23
New exhaust (E) = 34
Neither = 6
First, find how many needed at least one repair:
50 - 6 = 44
Now, if we add brakes + exhaust = 23 + 34 = 57 → But this counts the “both” group twice!
So, both repairs = 57 - 44 =
13
a) Both repairs =
13
b) New brakes but not exhaust = Total brakes - both = 23 - 13 =
10
✔ Check: Only brakes (10) + only exhaust (34-13=21) + both (13) + neither (6) = 10+21+13+6=50 → Correct!
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Problem 3: Student Survey (Math & History)
Total students = 85
Math (H) = 35
History (M) = 37
Both math and history = 20
a) Math OR history = Only math + only history + both
Only math = 35 - 20 = 15
Only history = 37 - 20 = 17
So, math or history = 15 + 17 + 20 =
52
b) Neither = Total - (math or history) = 85 - 52 =
33
c) Math but not history = Only math = 35 - 20 =
15
✔ Check: 15 (only math) + 17 (only history) + 20 (both) + 33 (neither) = 85 → Correct!
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Problem 4: Extracurricular Sports
Total students = 80
Basketball (B) = 56
Volleyball (V) = 30
Both = 14
a) Didn’t play either sport → First, find how many played at least one:
Only basketball = 56 - 14 = 42
Only volleyball = 30 - 14 = 16
Both = 14
Total playing at least one = 42 + 16 + 14 = 72
Didn’t play either = 80 - 72 =
8
b) Played volleyball only = 30 - 14 =
16
c) Played basketball only = 56 - 14 =
42
✔ Check: 42 + 16 + 14 + 8 = 80 → Correct!
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Problem 5: Holiday Activities
Total people = 24
Swimming (F) = 15
Fishing (S) = 12
Neither = 6
First, find how many did at least one activity:
24 - 6 = 18
Now, swimming + fishing = 15 + 12 = 27 → This counts “both” twice.
So, both = 27 - 18 =
9
a) Go swimming and fishing =
9
b) Only went swimming = 15 - 9 =
6
c) Only went fishing = 12 - 9 =
3
✔ Check: 6 (only swim) + 3 (only fish) + 9 (both) + 6 (neither) = 24 → Correct!
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Final Answer:
1. a) 20
b) 12
c) 8
2. a) 13
b) 10
3. a) 52
b) 33
c) 15
4. a) 8
b) 16
c) 42
5. a) 9
b) 6
c) 3
Parent Tip: Review the logic above to help your child master the concept of venn diagram problems worksheet.
