Solution
Understanding the set :-
- Base group: There are 15 girls and some boys (number to be found).
- Event duration: The get-together can be 1-day, 2-day, or 3-day.
[This is the main dividing point because every other piece of information (boys/girls, singers/dancers, etc.) can be categorized by the number of days they want to attend.]
- Performers:
a) Singers: 6 in total → 4 boys + 2 girls.
b) Dancers: 10 in total → 6 boys + 4 girls.
c) No overlap: no dancer is a singer.
- Attendance interest:
a) Some students are not interested in attending at all.
b) Interest is nested:
3-day → also 2-day → also 1-day.
This is a “multi-division” set problem:
There are three independent categories:
Gender, Performer type, and Event duration.
Not all categories are directly related, so putting all raw data into a single table is messy.
The smart approach: pick the one category that cuts across all others — here, number of days — and make it the first split.
| Not interested | 1-day event | 2-day event | 3-day event | |
|---|---|---|---|---|
| Boys (x) | ||||
| Singers (4) | ||||
| Dancers (6) | ||||
| Neither singers nor dancers (x-10) | ||||
| Girls (15) | ||||
| Singers (2) | ||||
| Dancers (4) | ||||
| Neither singers nor dancers (9) |
Here,
-
Number of girls = 15
-
Let the number of boys be x
-
Number of singers = 6
-
Number of boys who are singers = 4
-
Therefore, number of girls who are singers = 2
-
Number of dancers = 10
-
Number of boys dancer = 6
-
Number of girls dancers = 4
-
Therefore, number of boys who are neither singers nor dancers = x – 4 – 6 = x – 10
-
Number of girls who are neither singer nor dancers = 15 – 2 – 4 = 9
| Not interested | 1-day event | 2-day event | 3-day event | |
|---|---|---|---|---|
| Boys (x) | 0.8x | 0.6x | ||
| Singers (4) | ||||
| Dancers (6) | ||||
| Neither singers nor dancers (x-10) | ||||
| Girls (15) | 0 | 15 | ||
| Singers (2) | 0 | 2 | ||
| Dancers (4) | 0 | 4 | ||
| Neither singers nor dancers (9) | 0 | 9 |
Statement 1 says,
All the girls and 80% of the boys are interested in attending a 1-day event
Therefore, all 15 girls are interested in attending 1 day event.
Therefore, no girl is not interested in attending the event.
Likewise, all singers, dancers, and neither will all be interested in attending 1 day event.
And 80% of boys = 0.8x are interested in one day event.
Also, 60% of the boys are interested in attending a 2-day event.
Which implies, 60% of x = 0.6x boys are interested in attending a 2 day event.
| Not interested | 1-day event | 2-day event | 3-day event | |
|---|---|---|---|---|
| Boys (x) | 0.8x | 0.6x | ||
| Singers (4) | ||||
| Dancers (6) | ||||
| Neither singers nor dancers (x-10) | ||||
| Girls (15) | 0 | 15 | a | |
| Singers (2) | 0 | 2 | ||
| Dancers (4) | 0 | 4 | ||
| Neither singers nor dancers (9) | 0 | 9 |
Using clue 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.
Which implies, not all girls are interested in attending a 2 day event.
So, Let the number of girls interested in attending a 2-day event be “a”
| Not interested | 1-day event | 2-day event | 3-day event | |
|---|---|---|---|---|
| Boys (x) | 0.8x | 0.6x | ||
| Singers (4) | 4 | 4 | 4 | |
| Dancers (6) | 2 or more then 2 | 2 or more then 2 | 2 | |
| Neither singers nor dancers (x-10) | ||||
| Girls (15) | 0 | 15 | a | 0 |
| Singers (2) | 0 | 2 | 0 | |
| Dancers (4) | 0 | 4 | 0 | |
| Neither singers nor dancers (9) | 0 | 9 | 0 |
Using clue 4,
No girl is interested in attending a 3-day event.
Which means, number of girls attending a 3 day event will be 0
Also, All male singers and 2 of the dancers are interested in attending a 3-day event.
Also, From the question we know that, 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.
Since, all the boys singers are interested in attending 3 day event
So, all will be interested in attending 2 and 1 day event as well
Also, since only 2 boy dancers are interested in attending 3 day event therefore for 1 day and 2 day event 2 or more then 2 boy dancer will be interested.
| Not interested | 1-day event | 2-day event | 3-day event | |
|---|---|---|---|---|
| Boys (x) | 0.8x | 0.6x | ||
| Singers (4) | 0 | 4 | 4 | 4 |
| Dancers (6) | 2 or more then 2 | 2 or more then 2 | 2 | |
| Neither singers nor dancers (x-10) | 0.42 | |||
| Girls (15) | 0 | 15 | a | 0 |
| Singers (2) | 0 | 2 | 0 | |
| Dancers (4) | 0 | 4 | 0 | |
| Neither singers nor dancers (9) | 0 | 9 | 0 |
Using clue 3,
70% of the boys who are interested in attending a 2-day event are neither singers nor dancers
We know, number of boys interested in attending 2 day event = 0.6x
70% of o.6 = 0.42
Therefore, 0.42x boys are neither singers nor dancer who are interested in 2 day event.
| Not interested | 1-day event | 2-day event | 3-day event | |
|---|---|---|---|---|
| Boys (x) | 0.8x | 0.6x | ||
| Singers (4) | 0 | 4 | 4 | 4 |
| Dancers (6) | 2 or more then 2 | 2 or more then 2 | 2 | |
| Neither singers nor dancers (x-10) | 0.42 | |||
| Girls (15) | 0 | 15 | a | 0 |
| Singers (2) | 0 | 2 | 0 | |
| Dancers (4) | 0 | 4 | 0 | |
| Neither singers nor dancers (9) | 0 | 9 | 0.6a | 0 |
Continuing with clue 3,
60% of the girls who are interested in attending a 2-day event are neither singers nor dancers.
We know, the number of interested in attending 2 day event is – “a”
Which implies, 60 % of a = 0.6a – are the girls neither singer nor dancer interested in attending a 2 day event.
| Not interested | 1-day event | 2-day event | 3-day event | |
|---|---|---|---|---|
| Boys (x) | 0.8x | 0.6x | ||
| Singers (4) | 4 | 4 | 4 | |
| Dancers (6) | 2 or more then 2 | 0.18x -4 | 2 | |
| Neither singers nor dancers (x-10) | 0.42 | |||
| Girls (15) | 0 | 15 | a | 0 |
| Singers (2) | 0 | 2 | 0 | |
| Dancers (4) | 0 | 4 | 0 | |
| Neither singers nor dancers (9) | 0 | 9 | 0.6a | 0 |
Now, we know
Number of boys singer interested in attending 2 day event = 4
Number of boys dancer interested in attending 2 day event = 2 or more then 2
Neither of singer and dancer (boys) interested in attending 2 day event = 0.42x
Also, total number of boys = 0.6 x
Now, boys dancer will be = 0.6x – 0.42x – 4
= 0.18x – 4
| Not interested | 1-day event | 2-day event | 3-day event | |
|---|---|---|---|---|
| Boys (50) | 40 | 30 | ||
| Singers (4) | 0 | 4 | 4 | 4 |
| Dancers (6) | 2 or more then 2 | 5 | 2 | |
| Neither singers nor dancers (40) | 21 | |||
| Girls (15) | 0 | 15 | a | 0 |
| Singers (2) | 0 | 2 | 0 | |
| Dancers (4) | 0 | 4 | 0 | |
| Neither singers nor dancers (9) | 0 | 9 | 0.6a | 0 |
since, 2 ≤ 0.18x−4 ≤ 6
= 6 ≤ 0.18 𝑥 ≤ 10
= 6 ≤ 0.18 x ≤ 10
= 33.33 ≤ x ≤ 55.55
Now we want a value of x to be a number which gives integers to all of 0.42x, 0.18x, 0.2x etc..
Therefore the number will be a multiple of 10
Which in this range is only 40 and 50
But for 40 , 0.42 won’t give an integer value
Therefore, only possible value of x will be 50
Substitute value of x in the table.
| Not interested | 1-day event | 2-day event | 3-day event | |
|---|---|---|---|---|
| Boys (50) | 10 | 40 | 30 | |
| Singers (4) | 0 | 4 | 4 | 4 |
| Dancers (6) | 0/1 | 6/5 | 5 | 2 |
| Neither singers nor dancers (40) | 10/9 | 30/31 | 21 | |
| Girls (15) | 0 | 15 | 5 | 0 |
| Singers (2) | 0 | 2 | 2 | 0 |
| Dancers (4) | 0 | 4 | 0 | 0 |
| Neither singers nor dancers (9) | 0 | 9 | 3 | 0 |
Using statement 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.
Let the number of girls dancer interested in attending 3 day event be “b”
Therefore, girls singer interested in attending 3 day event will be – 0.4 a – b
Now, 4+ 0.4a - b = 5+b +1
= 0.4a = 2+2b
= a = 5(1+b)
a should be a multiple of 5 as b is a whole number.
So possible values of a can be 5, 10 or 15.
Now, as the maximum value of b can be 4 and the maximum value of 0.4a-b can be 2, so the only possible value of a satisfying the conditions above is 5. If a= 5 then b=1
| Not interested | 1-day event | 2-day event | 3-day event | |
|---|---|---|---|---|
| Boys (50) | 10 | 40 | 30 | |
| Singers (4) | 0 | 4 | 4 | 4 |
| Dancers (6) | 0/1 | 6/5 | 5 | 2 |
| Neither singers nor dancers (40) | 10/9 | 30/31 | 21 | |
| Girls (15) | 0 | 15 | 5 | 0 |
| Singers (2) | 0 | 2 | 2 | 0 |
| Dancers (4) | 0 | 4 | 0 | 0 |
| Neither singers nor dancers (9) | 0 | 9 | 3 | 0 |
No of boys = 50