The kinds of Venn question
- Read a diagram. You are shown overlapping shapes with numbers and asked for a total, a difference or a share.
- Find the region. Several shapes overlap, regions are lettered, and you pick the letter for a described group.
- Choose the diagram. You are given statements and asked which of four diagrams fits them.
- Work without a diagram. The information is in words, and you draw your own.
Reading the regions
The diagram below shows how many students at a school take part in three activities. Each number counts the students in that region and no other.
| What is asked | Regions to add | Answer |
|---|---|---|
| Choir only | 14 | 14 |
| Everyone in Choir | 14 + 5 + 3 + 2 | 24 |
| Choir and Drama, but not Debating | 5 | 5 |
| Everyone in both Choir and Drama | 5 + 2 | 7 |
| Exactly two activities | 5 + 3 + 4 | 12 |
| At least two activities | 5 + 3 + 4 + 2 | 14 |
| Drama but not Debating | 9 + 5 | 14 |
| Everyone shown | 14 + 9 + 6 + 5 + 3 + 4 + 2 | 43 |
Only, both, exactly, at least
These four words decide which regions you add. Only means one region. Both includes the centre where all three meet. Exactly two leaves the centre out. At least two puts it back in.
Worked example: reading
Original example
Using the diagram above, what percentage of the students shown take part in exactly one activity, to the nearest whole number?
Answer: 67%
Exactly one activity: 14 + 9 + 6 = 29. Everyone shown: 43. 29 ÷ 43 = 0.674, which is 67%.
The tempting wrong answer divides the three set totals (24 + 20 + 15 = 59) by 43 and gets more than 100%, because students in the overlaps have been counted more than once.
Building a diagram from words
When the question gives totals and overlaps in words, fill the diagram from the inside out.
- Start with the centre: the number in all the sets.
- Then each overlap of two: the stated overlap minus the centre.
- Then each "only" region: the set's total minus every overlap region already filled inside it.
- Then anyone outside: the overall total minus everything inside.
Original example: two sets
Of 40 members of a sports club, 22 play football and 18 play basketball. 7 play both. How many play neither?
Answer: 7
Both: 7. Football only: 22 − 7 = 15. Basketball only: 18 − 7 = 11. Inside the diagram: 15 + 7 + 11 = 33. Neither: 40 − 33 = 7.
The shortcut is 22 + 18 − 7 = 33 who play at least one. Adding 22 and 18 without subtracting gives 40 and the wrong answer of none.
Original example: three sets
In a survey of 60 households, 30 have a dog, 25 have a cat and 12 have a bird. 8 have a dog and a cat, 5 have a dog and a bird, 4 have a cat and a bird, and 2 have all three. How many have a dog only?
Answer: 19
Centre: 2. Dog and cat but no bird: 8 − 2 = 6. Dog and bird but no cat: 5 − 2 = 3. Dog only: 30 − 6 − 3 − 2 = 19.
Subtracting the stated overlaps without removing the centre first (30 − 8 − 5 = 17) takes the two households with all three pets off twice.
When the shapes are not circles
Some questions use a circle, a rectangle and a triangle, or four or five different shapes, with a key saying what each represents. The logic is the same. To find a region described in words, take one shape at a time: is the region inside it or outside it? A region that is inside the circle and the triangle and no other shape is the only place that is inside exactly those two.
Where marks are lost
- Giving a set's "only" number when the question asks for its total, or the reverse.
- Counting the centre region twice, or leaving it out of "both".
- Forgetting people outside every set when a total is given.
- Not reading the key when shapes differ.
Practise them
Draw every one on the notebook, even when you think you can do it in your head. The drawing takes seconds and removes the commonest error. Try free Decision Making questions, or read the full Decision Making guide.
Official sources
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