CAT 2022 Slot 1 DILR Set 1 Q1–Q5

A DILR caselet from the Data Interpretation & Logical Reasoning (DILR) section of CAT 2022 Slot 1, with 5 questions (Venn & Set Puzzles). Answers are CAT's official key.

The set

There are 15 girls and some boys among the graduating students in a class. They are planning a get-together, which can be either a 1-day event, or a 2-day event, or a 3-day event. There are 6 singers in the class, 4 of them are boys. There are 10 dancers in the class, 4 of them are girls. No dancer in the class is a singer.

Some students are not interested in attending the get-together. Those students who are interested in attending a 3-day event are also interested in attending a 2-day event; those who are interested in attending a 2-day event are also interested in attending a 1-day event.

The following facts are also known:

1. All the girls and 80% of the boys are interested in attending a 1-day event. 60% of the boys are interested in attending a 2-day event.
2. Some of the girls are interested in attending a 1-day event, but not a 2-day event; some of the other girls are interested in attending both.
3. 70% of the boys who are interested in attending a 2-day event are neither singers nor dancers. 60% of the girls who are interested in attending a 2-day event are neither singers nor dancers.
4. No girl is interested in attending a 3-day event. All male singers and 2 of the dancers are interested in attending a 3-day event.
5. The number of singers interested in attending a 2-day event is one more than the number of dancers interested in attending a 2-day event.

Question 1

How many boys are there in the class?

TITA: type in the answer. There are no options and no negative marking.

Answer: 50 Venn & Set Puzzles TITA

Solution

  1. Boys: 60% of boys go to 2-day, and 70% of those are neither singers nor dancers, so \(0.42B\) is an integer: \(B\) is a multiple of 50. The 30% who are singers/dancers number \(0.18B \le 10\) (only 4 male singers + 6 male dancers), so \(B = 50\): 40 boys 1-day, 30 boys 2-day, 9 of them singers/dancers.
  2. So there are 50 boys.

Shortcut. \(0.6 \times 0.7 = 0.42\) of the boys must be a whole number, so boys are a multiple of 50.

Question 2

Which of the following can be determined from the given information?

I. The number of boys who are interested in attending a 1-day event and are neither dancers nor singers.
II. The number of female dancers who are interested in attending a 1-day event.
  1. A
    Neither I nor II
  2. B
    Both I and II
  3. C
    Only I
  4. D
    Only II

Answer: (D) Only II Venn & Set Puzzles MCQ

Solution

  1. Boys: 60% of boys go to 2-day, and 70% of those are neither singers nor dancers, so \(0.42B\) is an integer: \(B\) is a multiple of 50. The 30% who are singers/dancers number \(0.18B \le 10\) (only 4 male singers + 6 male dancers), so \(B = 50\): 40 boys 1-day, 30 boys 2-day, 9 of them singers/dancers.
  2. No girl goes 3-day, so the 2 dancers going 3-day are boys; all 4 male singers go 3-day (hence 2-day). So the 9 boy singers/dancers at 2-day are 4 singers + 5 dancers.
  3. Fact 5: singers at 2-day \(= 4 + f_s\), dancers at 2-day \(= 5 + f_d\), so \(f_s = f_d + 2\). Only 2 female singers, so \(f_s = 2, f_d = 0\).
  4. Girls at 2-day: 40% are singers/dancers \(= 2\), so 5 girls go 2-day (and 10 girls go only 1-day).
  5. I: 1-day boys who are neither \(= 40 - 4 - (\text{male dancers at 1-day})\); the 6th male dancer may or may not be at 1-day, so this is 30 or 31: not determined.
  6. II: every girl attends 1-day, so all 4 female dancers do: determined. Only II.

Question 3

What fraction of the class are interested in attending a 2-day event?
  1. A
    7/10
  2. B
    9/13
  3. C
    7/13
  4. D
    2/3

Answer: (C) 7/13 Venn & Set Puzzles MCQ

Solution

  1. Boys: 60% of boys go to 2-day, and 70% of those are neither singers nor dancers, so \(0.42B\) is an integer: \(B\) is a multiple of 50. The 30% who are singers/dancers number \(0.18B \le 10\) (only 4 male singers + 6 male dancers), so \(B = 50\): 40 boys 1-day, 30 boys 2-day, 9 of them singers/dancers.
  2. No girl goes 3-day, so the 2 dancers going 3-day are boys; all 4 male singers go 3-day (hence 2-day). So the 9 boy singers/dancers at 2-day are 4 singers + 5 dancers.
  3. Fact 5: singers at 2-day \(= 4 + f_s\), dancers at 2-day \(= 5 + f_d\), so \(f_s = f_d + 2\). Only 2 female singers, so \(f_s = 2, f_d = 0\).
  4. Girls at 2-day: 40% are singers/dancers \(= 2\), so 5 girls go 2-day (and 10 girls go only 1-day).
  5. 2-day attendees \(= 30 + 5 = 35\) out of \(50 + 15 = 65\), i.e. \(\frac{35}{65} = \frac{7}{13}\).

Question 4

What BEST can be concluded about the number of male dancers who are interested in attending a 1-day event?
  1. A
    5 or 6
  2. B
    4 or 6
  3. C
    5
  4. D
    6

Answer: (A) 5 or 6 Venn & Set Puzzles MCQ

Solution

  1. Boys: 60% of boys go to 2-day, and 70% of those are neither singers nor dancers, so \(0.42B\) is an integer: \(B\) is a multiple of 50. The 30% who are singers/dancers number \(0.18B \le 10\) (only 4 male singers + 6 male dancers), so \(B = 50\): 40 boys 1-day, 30 boys 2-day, 9 of them singers/dancers.
  2. No girl goes 3-day, so the 2 dancers going 3-day are boys; all 4 male singers go 3-day (hence 2-day). So the 9 boy singers/dancers at 2-day are 4 singers + 5 dancers.
  3. Fact 5: singers at 2-day \(= 4 + f_s\), dancers at 2-day \(= 5 + f_d\), so \(f_s = f_d + 2\). Only 2 female singers, so \(f_s = 2, f_d = 0\).
  4. Girls at 2-day: 40% are singers/dancers \(= 2\), so 5 girls go 2-day (and 10 girls go only 1-day).
  5. 5 male dancers attend 2-day, hence 1-day; the 6th male dancer is unconstrained. So 5 or 6.

Question 5

How many female dancers are interested in attending a 2-day event?
  1. A
    0
  2. B
    2
  3. C
    Cannot be determined
  4. D
    1

Answer: (A) 0 Venn & Set Puzzles MCQ

Solution

  1. Boys: 60% of boys go to 2-day, and 70% of those are neither singers nor dancers, so \(0.42B\) is an integer: \(B\) is a multiple of 50. The 30% who are singers/dancers number \(0.18B \le 10\) (only 4 male singers + 6 male dancers), so \(B = 50\): 40 boys 1-day, 30 boys 2-day, 9 of them singers/dancers.
  2. No girl goes 3-day, so the 2 dancers going 3-day are boys; all 4 male singers go 3-day (hence 2-day). So the 9 boy singers/dancers at 2-day are 4 singers + 5 dancers.
  3. Fact 5: singers at 2-day \(= 4 + f_s\), dancers at 2-day \(= 5 + f_d\), so \(f_s = f_d + 2\). Only 2 female singers, so \(f_s = 2, f_d = 0\).
  4. Girls at 2-day: 40% are singers/dancers \(= 2\), so 5 girls go 2-day (and 10 girls go only 1-day).
  5. From fact 5, \(f_d = 0\): no female dancer is interested in the 2-day event.

Same concept in other years

See every question on Venn diagram: counts of items belonging to overlapping categories (3-4 sets).

More Venn & Set Puzzles questions from CAT

All 60 Venn & Set Puzzles questions asked since 2017.