CAT 2022 Slot 1 · DILR · Question 6

CAT 2022 Slot 1 previous year question on Venn Diagram DI, with the answer and a step-by-step solution.

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Set 2: Get-together of Graduating Students — this question belongs to a set. Read the full set with all 5 questions →

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.

How many boys are there in the class?

Answer & solution

Answer: 50

Solving the set

Step 1 – Class make-up. Singers: 4 boys, 2 girls. Dancers: 6 boys, 4 girls. No one is both.

Step 2 – Number of boys b. 0.8b and 0.6b are whole numbers. Among the 0.6b boys interested in a 2-day event, 30% (= 0.18b) are singers or dancers, so 0.18b is a whole number ⇒ b is a multiple of 50. Only 10 boys sing or dance, so 0.18b ≤ 10 ⇒ b = 50. Boys: 40 (1-day), 30 (2-day), and 9 singers/dancers among the 2-day boys.

Step 3 – The 3-day group includes all 4 male singers and 2 dancers (boys, as no girl is interested). Since 3-day ⊂ 2-day, the 9 boy singers/dancers interested in a 2-day event are 4 singers + 5 dancers.

Step 4 – Girls. Let g girls be interested in a 2-day event (0 < g < 15). 40% of them are singers/dancers, so g = 5 or 10, giving 2 or 4 singer/dancer girls.

Step 5 – Fact 5: singers (2-day) = dancers (2-day) + 1 ⇒ (4 + female singers) = (5 + female dancers) + 1 ⇒ female singers − female dancers = 2 ⇒ 2 female singers and 0 female dancers. So 2 singer/dancer girls ⇒ g = 5.

Group	Total	1-day	2-day	3-day
Boys	50	40	30	≥ 6 (4 male singers + 2 male dancers)
– male singers	4	4	4	4
– male dancers	6	5 or 6	5	2
– other boys	40	30 or 31	21	—
Girls	15	15	5	0
– female singers	2	2	2	0
– female dancers	4	4	0	0
– other girls	9	9	3	0
Class	65	55	35

This question

From Step 2, there are 50 boys.

Hence, the answer is 50.

CAT 2022 Slot 1 · DILR · Question 6

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