Logical Reasoning · Micro-test
Venn Diagrams and Set Counting
Venn diagram questions count people in overlapping groups. The one idea underneath is that anyone in two groups gets counted twice when you add the groups, so you take them off once.
10 questions · 5 minutes · instant scoring
What this topic actually tests
For two groups A and B, the number in at least one is A + B − (both). Those in neither are the total minus that figure, and those in only A are A − (both). For three groups, add the three, subtract the three pairwise overlaps and add back those in all three: A + B + C − (A∩B) − (B∩C) − (A∩C) + (A∩B∩C). Only A is A − (A∩B) − (A∩C) + (all three), because the people in all three were taken off twice. Exactly two of the three is the sum of the pairwise overlaps minus three times those in all three. For the smallest possible overlap of two groups inside a total, take A + B − total (or zero, if that is negative); the largest possible overlap is the smaller group. Worked example: of 100 people, 50 read A, 40 read B and 30 read C; 15 read A and B, 10 read B and C, 12 read A and C, and 5 read all three. At least one: 50 + 40 + 30 − 15 − 10 − 12 + 5 = 88. None: 12.
The common trap on this topic
The trap is treating a pairwise figure as 'only those two'. When a question says 15 read A and B, that 15 includes the 5 who read all three. Taking it as 15 who read exactly A and B gives the wrong 'only' and 'exactly two' counts. Draw the three circles and fill the centre first, then the pairs, then the singles.
Take the micro-test
In a class of 60 students, 35 like tea, 30 like coffee and 10 like both. How many like neither?
40 students play cricket and 25 play football. 55 play at least one of the two. How many play both?
Of 100 people, 50 read newspaper A, 40 read B and 30 read C. 15 read A and B, 10 read B and C, 12 read A and C, and 5 read all three. How many read at least one?
Of 100 people, 50 read newspaper A, 40 read B and 30 read C. 15 read A and B, 10 read B and C, 12 read A and C, and 5 read all three. How many read none of the three?
Of 100 people, 50 read newspaper A, 40 read B and 30 read C. 15 read A and B, 10 read B and C, 12 read A and C, and 5 read all three. How many read only A?
In an examination, 70 students passed English, 60 passed Mathematics, 40 passed both and 10 failed both. How many students sat the examination?
In a group, 45 people speak Hindi, 30 speak English and 15 speak both. How many speak only English?
Of 200 people, 120 own a car, 90 own a motorcycle and 30 own neither. How many own both?
Of 100 people, 50 read newspaper A, 40 read B and 30 read C. 15 read A and B, 10 read B and C, 12 read A and C, and 5 read all three. How many read exactly two of the three?
Of 100 people, 70 like tea and 50 like coffee. What is the least possible number who like both?
Venn Diagrams and Set Counting: answers and explanations
In a class of 60 students, 35 like tea, 30 like coffee and 10 like both. How many like neither?
Answer: D. 5
At least one: 35 + 30 − 10 = 55. Neither: 60 − 55 = 5.
40 students play cricket and 25 play football. 55 play at least one of the two. How many play both?
Answer: D. 10
Both = 40 + 25 − 55 = 10.
Of 100 people, 50 read newspaper A, 40 read B and 30 read C. 15 read A and B, 10 read B and C, 12 read A and C, and 5 read all three. How many read at least one?
Answer: B. 88
50 + 40 + 30 − 15 − 10 − 12 + 5 = 88.
Of 100 people, 50 read newspaper A, 40 read B and 30 read C. 15 read A and B, 10 read B and C, 12 read A and C, and 5 read all three. How many read none of the three?
Answer: D. 12
88 read at least one (50 + 40 + 30 − 15 − 10 − 12 + 5), so 100 − 88 = 12 read none.
Of 100 people, 50 read newspaper A, 40 read B and 30 read C. 15 read A and B, 10 read B and C, 12 read A and C, and 5 read all three. How many read only A?
Answer: C. 28
Only A = 50 − 15 − 12 + 5 = 28. The 5 who read all three were taken off twice, so they are added back once.
In an examination, 70 students passed English, 60 passed Mathematics, 40 passed both and 10 failed both. How many students sat the examination?
Answer: A. 100
Passed at least one: 70 + 60 − 40 = 90. Add the 10 who failed both: 100.
In a group, 45 people speak Hindi, 30 speak English and 15 speak both. How many speak only English?
Answer: D. 15
Only English = 30 − 15 = 15.
Of 200 people, 120 own a car, 90 own a motorcycle and 30 own neither. How many own both?
Answer: D. 40
170 own at least one (200 − 30). Both = 120 + 90 − 170 = 40.
Of 100 people, 50 read newspaper A, 40 read B and 30 read C. 15 read A and B, 10 read B and C, 12 read A and C, and 5 read all three. How many read exactly two of the three?
Answer: B. 22
Exactly two = (15 + 10 + 12) − 3 × 5 = 37 − 15 = 22.
Of 100 people, 70 like tea and 50 like coffee. What is the least possible number who like both?
Answer: A. 20
At most 100 people can like at least one, so at least 70 + 50 − 100 = 20 must like both.
FAQ
What is the formula for two overlapping groups?
At least one = A + B − both. Neither = total − (A + B − both).
What is the formula for three overlapping groups?
At least one = A + B + C − (A∩B) − (B∩C) − (A∩C) + (A∩B∩C).
How do I find 'exactly two' of three groups?
Add the three pairwise overlaps and subtract three times the number in all three groups.
Keep practising
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