CAT 2022 Slot 1 QA Q17 Set Theory

Question 17 of the Quantitative Ability (QA) section, CAT 2022 Slot 1, with CAT's official answer and a worked solution.

In a class of 100 students, 73 like coffee, 80 like tea and 52 like lemonade. It may be possible that some students do not like any of these three drinks. Then the difference between the maximum and minimum possible number of students who like all the three drinks is
  1. A
    48
  2. B
    53
  3. C
    47
  4. D
    52

Answer: (C) 47 Set Theory MCQ

Solution

  1. Maximum liking all three is at most the smallest group, 52, and that is achievable (the 52 lemonade lovers like everything; coffee and tea fit within 100).
  2. For the minimum, total likes \(= 73 + 80 + 52 = 205 = a_1 + 2a_2 + 3a_3\), where \(a_i\) = students liking exactly \(i\) drinks and \(a_1 + a_2 + a_3 \le 100\).
  3. \(205 \le 2(a_1 + a_2 + a_3) + a_3 \le 200 + a_3\), so \(a_3 \ge 5\). It is achievable: \(a_3 = 5\), and exactly-two counts coffee-tea 48, coffee-lemonade 20, tea-lemonade 27.
  4. Difference \(= 52 - 5 = 47\).

Same concept in other years

See every question on Three-set Venn diagrams: maximise or minimise a region.

More Set Theory questions from CAT

All 9 Set Theory questions asked since 2017.