Among 300 students, 56% study sociology, 44% study mathematics and 40% study biology. If 30% of students study both mathematics and sociology, what is the largest possible number of students who study biology but do not study either mathematics or sociology? Answers:
A) 30
B) 90
C) 120
D) 172
E) 188
I would just like to add a bit of explanation after the step where you calculate that the number of students studying both M and S = 90   Using your analysis:   We see that the total number of students who study either Maths or Sociology = 132 + 168 - 90= 
<gadget id="calculator">132 + 168 - 90</gadget>
<output>210</output>
210   So, in the image we know that the number of students in the zone with the black boundary= 210   Let's assume the number of students who studyonlybiology to beb(this is the number that we have to maximize)   And, let's assume the number of students who study none of the three subjects, that is the number of students in the white space =w   Since the total number of students = 300, we can write:   210 +b+w= 300   Or,b+w= 300 - 210= 
<gadget id="calculator">300 - 210</gadget>
<output>90</output>
90   That is,b= 90 -w   So, the maximum value ofbwill happen forw= 0   This is how we get, the maximum value ofb= 90   90
<result>B</result>
Among 300 students, 56% study sociology, 44% study mathematics and 40% study biology. If 30% of students study both mathematics and sociology, what is the largest possible number of students who study biology but do not study either mathematics or sociology?