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Reasoning practice

Venn Diagrams Questions with Answers

Venn diagrams questions come in two kinds. In the first, you choose the diagram that shows how given items relate, such as tiger, mammal and animal. In the second, you use numbers in overlapping sets to find how many people fall in a region. SSC CGL, CHSL and MTS and RRB NTPC and Group D ask both kinds. The main tip is to fill the centre of the diagram first and work outward.

Rules to remember

  • One class fully inside another: one circle inside the other (doctors inside professionals).
  • Two classes with some common members: two overlapping circles (teachers and women).
  • Two different classes of one bigger class: two separate circles inside a large circle (cats and dogs inside animals).
  • Unrelated items: separate circles with no overlap (pen, cow, river).
  • Two sets: n(A or B) = n(A) + n(B) - n(A and B).
  • Three sets: n(A or B or C) = n(A) + n(B) + n(C) - n(A and B) - n(B and C) - n(A and C) + n(A and B and C).
  • Only A = n(A) - n(A and B) - n(A and C) + n(A and B and C).
  • Neither = Total - n(union). When a 'both A and B' count includes the triple overlap, subtract the triple overlap once to get 'only A and B'.

Worked examples

Q1. Which diagram best shows the relation among Tiger, Mammal and Animal?

  1. Every tiger is a mammal, so the tiger circle lies inside the mammal circle.
  2. Every mammal is an animal, so the mammal circle lies inside the animal circle.
  3. The diagram is three circles, one inside another.

Answer: Three circles, one inside the next (Tiger inside Mammal inside Animal)

Q2. In a class of 50 students, 30 like tea, 25 like coffee and 10 like neither. How many like both?

  1. Students who like at least one drink = 50 - 10 = 40.
  2. Use n(T or C) = n(T) + n(C) - n(T and C).
  3. 40 = 30 + 25 - n(both), so n(both) = 55 - 40 = 15.
  4. Check: only tea 15, only coffee 10, both 15, neither 10. The total is 50.

Answer: 15 students

Q3. In a group of 120 people, 60 read paper A, 50 read paper B and 40 read paper C. Of these, 25 read both A and B, 20 read both B and C, and 15 read both A and C (each count includes those who read all three). 10 people read all three. How many people read exactly one paper?

  1. Only A = 60 - 25 - 15 + 10 = 30.
  2. Only B = 50 - 25 - 20 + 10 = 15.
  3. Only C = 40 - 20 - 15 + 10 = 15.
  4. Exactly one paper = 30 + 15 + 15 = 60.
  5. Check: exactly two = (25 - 10) + (20 - 10) + (15 - 10) = 30 and all three = 10, so at least one paper = 60 + 30 + 10 = 100. The other 120 - 100 = 20 read none. All counts are consistent.

Answer: 60 people

Shortcuts that save time

  • For numerical questions, fill the triple overlap (centre) first, then the two-set overlaps, then the single regions. A diagram filled in this order rarely gives a negative number, which is a quick sign of an error.
  • For relation questions, ask whether one item is fully inside another, partly overlapping, or separate. Test every option against these three ideas before you choose, to avoid a wrong mark.
  • Check that your region numbers add up to the total before you mark. This one-line check catches most calculation slips.

Where this appears

SSC CGL Tier-1General Intelligence & Reasoning section
SSC CHSL Tier-1General Intelligence section
SSC MTSReasoning section
RRB NTPC CBT-1General Intelligence & Reasoning section
RRB Group D CBTGeneral Intelligence & Reasoning section

Venn Diagrams

Practice Set
Attempted: 0Correct: 0Incorrect: 0
1 of 1 (adapting live)Medium
In a survey of 144 people, 40 read newspaper A, 105 read newspaper B, and 6 read both newspaper A and newspaper B. How many people read only newspaper B?

Reasoning · stress-free practice · answers reveal instantly · difficulty adapts as you go

Venn Diagrams questions with answers

10 sample questions from this topic with the correct option and the solution.

  1. Q1. In a group, 15 people are only Doctors, 29 are only Engineers, and 11 are both Doctors and Engineers. How many people are there in total?

    1. 55
    2. 77
    3. 44
    4. 66

    Answer: (A) 55

    Solution
    Total = only Doctors + only Engineers + both = 15+29+11 = 55. कुल = केवल डॉक्टर + केवल इंजीनियर + दोनों = 15+29+11 = 55।

  2. Q2. In a survey of 69 people, 42 read newspaper A, 37 read newspaper B, and 20 read both newspaper A and newspaper B. How many people read only newspaper A?

    1. 22
    2. 18
    3. 17
    4. 24

    Answer: (A) 22

    Solution
    Only A = Total reading A − Both = 42 − 20 = 22.

  3. Q3. In a group of 76 people, 45 are Players and 51 are Singers. How many people are both Players and Singers?

    1. 20
    2. 18
    3. 40
    4. 25

    Answer: (A) 20

    Solution
    Both = (Players) + (Singers) - total = 45+51-76 = 20. दोनों = खिलाड़ी + गायक - कुल = 45+51-76 = 20।

  4. Q4. In a group, 31 people are only Students, 24 are only Athletes, and 10 are both Students and Athletes. How many people are there in total?

    1. 48.75
    2. 55
    3. 75
    4. 65

    Answer: (D) 65

    Solution
    Total = only Students + only Athletes + both = 31+24+10 = 65. कुल = केवल छात्र + केवल एथलीट + दोनों = 31+24+10 = 65।

  5. Q5. In a survey of 114 people, 81 read newspaper A, 65 read newspaper B, and 44 read both newspaper A and newspaper B. How many people read neither newspaper A nor newspaper B?

    1. 9
    2. 12
    3. 16
    4. 15

    Answer: (B) 12

    Solution
    At least one = A + B − Both = 81 + 65 − 44 = 102. Neither = Total − At least one = 114 − 102 = 12.

  6. Q6. In a group of 61 people, 35 are Students and 45 are Athletes. How many people are both Students and Athletes?

    1. 19
    2. 24
    3. 17
    4. 23.75

    Answer: (A) 19

    Solution
    Both = (Students) + (Athletes) - total = 35+45-61 = 19. दोनों = छात्र + एथलीट - कुल = 35+45-61 = 19।

  7. Q7. In a group, 23 people are only Players, 28 are only Singers, and 10 are both Players and Singers. How many people are there in total?

    1. 51
    2. 71
    3. 61
    4. 85.4

    Answer: (C) 61

    Solution
    Total = only Players + only Singers + both = 23+28+10 = 61. कुल = केवल खिलाड़ी + केवल गायक + दोनों = 23+28+10 = 61।

  8. Q8. In a survey of 134 people, 36 read newspaper A, 60 read newspaper B, and 29 read both newspaper A and newspaper B. How many people read at least one of the two newspapers?

    1. 65
    2. 69
    3. 68
    4. 67

    Answer: (D) 67

    Solution
    At least one = A + B − Both = 36 + 60 − 29 = 67.

  9. Q9. In a group of 62 people, 44 are Students and 31 are Athletes. How many people are both Students and Athletes?

    1. 14.95
    2. 18
    3. 11
    4. 13

    Answer: (D) 13

    Solution
    Both = (Students) + (Athletes) - total = 44+31-62 = 13. दोनों = छात्र + एथलीट - कुल = 44+31-62 = 13।

  10. Q10. In a group, 25 people are only Students, 27 are only Athletes, and 9 are both Students and Athletes. How many people are there in total?

    1. 70
    2. 91.5
    3. 61
    4. 52

    Answer: (C) 61

    Solution
    Total = only Students + only Athletes + both = 25+27+9 = 61. कुल = केवल छात्र + केवल एथलीट + दोनों = 25+27+9 = 61।