Learning Objectives
You should be able to:
- translate a two-set word problem into a Venn diagram;
- fill the intersection before the “only” regions;
- calculate the number in at least one set and the number in neither;
- check that all regions add to the universal total.
Introduction
This lesson develops Two-Set Problems from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🌍 Why This Matters
Two-set problems model real surveys—for example, learners who study Geography, History, both subjects or neither. A systematic method prevents double counting.
Two-Set Word Problems
Venn-diagram questions often provide totals for two groups and their overlap. The safest approach is to fill the overlap first.
Fill a two-set problem from the overlap outward
A Reliable Method
- Write the overlap first.
- Subtract the overlap from each set total.
- Add all regions inside the circles.
- Subtract from n(U) to find neither.
- Check the total.
Worked Example
In a class of 40 learners, 24 play football, 18 play netball and 9 play both.
Both = 9.
Football only = 24−9=15.
Netball only = 18−9=9.
At least one = 15+9+9=33.
Neither = 40−33=7.
💡 Exam Tip
Start with the intersection. Subtract it from each total to find the “only” regions, then use the universal total to find neither.
🧠 Let's Think Together: Place Before You Count
Let A = {6, 12, 18}. Select a number and let the notation update.
Practice Questions
Foundation- Of 35 learners, 20 like tea, 17 like coffee and 8 like both. Find tea only, coffee only and neither.
- In a survey of 60 people, 38 use WhatsApp, 27 use Telegram and 15 use both. Find the number using at least one.
- In a group of 80 students, 46 study Biology, 39 study Chemistry and 12 study neither. How many study both?
- In a survey, 28 people chose A only, 19 chose B only, 11 chose both and 7 chose neither. Find n(U), n(A) and n(B).
- n(U)=100, n(A)=x+20, n(B)=2x and n(A∩B)=x−5. If 15 belong to neither set, find x.
✅ Check Your Work
Show practice answers
- Tea only 12; coffee only 9; at least one 29; neither 6.
- 38+27−15=50.
-
At least one = 80−12=68.
Both = 46+39−68=17. - n(U)=65; n(A)=39; n(B)=30.
-
At least one = 85.
(x+20)+2x−(x−5)=85
2x+25=85, so x=30.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Fill the overlap first.
- Subtract overlap to find “only” regions.
- Add inside-circle regions to find at least one.
- Subtract from U to find neither.