Problem Solving

In a college, where every student follows at least one of the three activities- drama, sports, or arts- 65% follow drama, 86% follow sports, and 57% follow arts. What can be the maximum and minimum percentage of students who follow
· all three activities
· exactly two activities

Let's make several assumptions.

Total number of students: 100.
Students with all three activities: x:
Students with drama and sports but not all three activities: a
Students with arts and drama but not all three activities: b
Students with arts and sports but not all three activities: c

Which means, only drama= 65-a-b-x
Only sports= 86-a-c-x
Only arts= 57-b-c-x

All students must add to 100 as stated by the first statement.

So, 65+ 86-a-c-x+c + 57-b-c-x=100
This gives, (a+b+c)+2x=108

You get max of all three activities when a+b+c=0, which gives x=54%
You get min of all three activities when a+b+c=92, which gives x=8% (this needs a bit of trial and error, but this is the only possible combination for the above equation to hold good for min value of x).