Fill the overlap first, then work outwards — never the other way round.
The problem
In a class of 40, 25 like Math, 20 like Science, and 8 like both. How many like neither?
Those liking at least one number 25 + 20 − 8 = 37, so 40 − 37 = 3 like neither. B skips the subtraction and counts the 8 twice, pushing the union to 25 + 20 = 45 — more students than the class holds, which is the giveaway — and then reads 45 − 40 = 5 as the answer.
This one sentence is the whole topic.
25 like Math only
25 like Math, some of them Science too
Adding gives 45, more than the whole class. The eight who like both were counted twice — once in each group.
both = 8
the middle, written first
Math only = 25 − 8 = 17
take the middle out
Science only = 20 − 8 = 12
same again
17 + 8 + 12 = 37
inside the circles
40 − 37 = 3
outside both
The three regions inside plus the one outside must account for all 40. That is the check, built into the method.
Same reasoning, no diagram.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
n(A ∪ B) = 25 + 20 − 8
add, then remove the double count
= 37
like at least one
neither = 40 − 37 = 3
the complement
in one, the other, or both
in both at once
Seven regions, and they only fill correctly from the middle outwards.
1. all three
the very centre
2. each pair − centre
the three lens shapes
3. each set − everything already in it
the three outer petals
4. total − all seven
outside
In a three-set problem it usually means Math and Science, some of whom also like the third. Read it as including the centre unless the question says “only”.
40 − 7 = 33
play at least one
22 + 19 − both = 33
the union formula
both = 41 − 33 = 8
the overlap
Then the regions: basketball only 22 − 8 = 14 and volleyball only 19 − 8 = 11; 14 + 8 + 11 + 7 = 40 ✓.
centre: 3
all three
lenses: 10 − 3 = 7, 8 − 3 = 5, 6 − 3 = 3
each pair minus the centre
Art only: 30 − 7 − 3 − 3 = 17
take out every region already inside Art
Music only: 25 − 7 − 5 − 3 = 10
Dance only: 20 − 5 − 3 − 3 = 9
inside: 17 + 10 + 9 + 7 + 5 + 3 + 3 = 54
none = 60 − 54 = 6
Now any region question is a matter of adding the right pieces:
| Asked | Regions | Here |
|---|---|---|
| exactly one | the three outer petals | 17 + 10 + 9 = 36 |
| exactly two | the three lenses, centre left out | 7 + 5 + 3 = 15 |
| all three | the centre | 3 |
| Art or Dance, not Music | everything in Art or Dance, minus anything in Music | 17 + 3 + 9 = 29 |
| none | outside all three circles | 6 |
The centre students take all three subjects. Adding the 3 to the lenses answers “at least two”, which is a different question.
if no one liked both: 35 + 30 = 65 people
more than the group holds
at most 50 can like at least one
only 50 people exist
both ≥ 65 − 50 = 15
the excess must overlap
The greatest possible overlap is the smaller set, 30 — when every ice-cream fan also likes rice cakes.
An overlap of 0 needs room for 65 different people. Whenever the two counts add to more than the total, the excess is forced into both.
Venn items come as surveys: two sets with “neither” given, three sets with the pairs and the centre given, region questions such as “exactly two” or “A but not B”, or the least possible overlap. Fill from the centre out and subtract every region already inside a circle. Ten-second check: all the regions, outside included, must add up to the total surveyed.
The problem
Of 60 students: 30 play basketball, 27 play volleyball, 22 play badminton, 12 play basketball and volleyball, 10 play volleyball and badminton, 9 play basketball and badminton, and 5 play all three. How many play none?
Inclusion–exclusion: add the singles (79), subtract the pairs (31), then add the triple back (5), giving 53. So 60 − 53 = 7 play none. The five all-rounders get removed three times by the pair subtractions, so one has to be restored.
Hint: write the 5 in the centre first, then take it out of each pair figure before you fill the lens regions. Every region must be a number you have actually subtracted down to.
