1. How to Approach Word Problems on Sets
Word problems involving sets usually deal with groups, overlaps, or choices. The aim is to translate the information into sets, use operations like union or intersection, and find the required values.
Drawing a simple Venn diagram often helps visualise the relationships clearly.
2. Two-Set Word Problems
These problems involve two groups and usually ask about totals, common parts, or exclusive parts.
2.1. Example 1
In a group of 40 people, 25 like tea and 20 like coffee. If 10 like both, how many like at least one of the two?
Using the formula:
|A \cup B| = |A| + |B| - |A \cap B|
Answer:
|A \cup B| = 25 + 20 - 10 = 35
2.2. Example 2
In a class of 60 students, 35 play football and 30 play basketball. If 12 play both, how many play only one of the games?
Total playing at least one:
35 + 30 - 12 = 53
Those who play only one game:
53 - 12 = 41
3. Three-Set Word Problems
These problems involve three groups and use the principle of inclusion and exclusion.
3.1. Example 1
In a survey of 100 people:
- 60 like tea
- 50 like coffee
- 30 like juice
- 20 like both tea and coffee
- 15 like both coffee and juice
- 10 like both tea and juice
- 5 like all three
How many like at least one drink?
Using PIE:
|A \cup B \cup C| = 60 + 50 + 30 - 20 - 15 - 10 + 5
Answer:
|A \cup B \cup C| = 100
4. Problems on "Only" and "Neither"
Words like "only" and "neither" often appear and must be handled carefully.
4.1. Example: Only
Out of 50 students:
- 30 like maths
- 20 like science
- 10 like both
How many like only maths?
30 - 10 = 20
4.2. Example: Neither
How many like neither subject?
Total liking at least one:
30 + 20 - 10 = 40
So neither:
50 - 40 = 10
5. Venn Diagram Method
Drawing Venn diagrams helps sort information clearly. Fill the central overlap first, then work outward. This avoids confusion and double-counting.
6. Mixed Examples
Some additional problems for practice:
6.1. Example 1
Out of 80 people, 50 have a bike, 30 have a car, and 20 have both. How many have only a bike?
50 - 20 = 30
6.2. Example 2
Out of 100 students, 70 read novels, 40 read magazines, and 25 read both. How many read neither?
At least one = 70 + 40 - 25 = 85
Neither = 100 - 85 = 15
7. Important Points
Key ideas for solving word problems:
7.1. Key Ideas
- Translate the problem into sets first.
- Use union, intersection, or complement as needed.
- PIE helps avoid double-counting.
- Venn diagrams simplify multi-step problems.
- Always check if the question asks for "only," "at least," or "neither".