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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.
What fraction of the class are interested in attending a 2-day event?
No. of girls = 15
Let the no. of boys be x
No. of singers = 6
No of boys who are singers = 4
Therefore, no of girls who are singers = 2
No of dancers = 10
No of boys who are dancers = 6
Therefore, no. of girls who are dancers = 4
No. of boys who are neither singers nor dancers = x-10
No. of girls who are neither singers nor dancers = 9
Now we fill the above table,
using statements 1 and 2, we get the following table
Let the number of girls who are interested in attending a 2-day event be a and the number of girls who are dancers and are interested in 2-day event be b.
Now using statements 3 and 4, we get
$$2\le0.18x-4\le6$$
$$6\le0.18x\le10$$
0.18x should be integer for which x should be a multiple of 50, and 0.18x lies between 6 and 10; therefore, the only possible value of x is 50.
From statement 5, we can say that,
4+ 0.4a - b = 5+b +1
or, 0.4a = 2+2b
or, 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.
Total students interested in 2-day event = 30+5 = 35
Total students = 15 + 50 = 65
Fraction = $$\frac{35}{65}=\frac{7}{13}$$
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