venn issue

please someone could explain clearly?

of the 200 students at college T majoring in one or more of the sciences, 130 are majoring in chemistry
and 150 are majoring in biology. If at least 30 of the students are not majoring in either chemistry or biology,
the the number of students majoring in both chemistry and biology could be any number from?

A 20 to 50
B 40 to 70
C 50 to 130
D 110 to 130
E 110 to 150

so maximum numberof student majoring in chemistry and biology can be 200 -30 =170

so now

t(aUb)=t(a)+t(b)-t(a*b) i.e a union b= a+b-a intersection b
170=130+150-t(a*b)
t(a*b)=110 this is minimum number of student graduating in both chemistry and biology

now let us take a case where 130 student graduating in chemistry also does graduation in bilogy
so maximum number of student majoring in both chemistry and biology can be 130
so range is 110 to 130

Hey Francois,

There are 130 students majoring in chem and 150 in bio, but we don't know the overlap.

There's a formula that can help here:

(Total # in group 1) + (Total # in group 2) - (# in the overlap) + (# in neither) = Total # of students

However, we're told that AT LEAST 30 are in the neither group, so let's do some rearranging:

Total # of students - (Total # in group 1) - (Total # in group 2) + (# in the overlap) = (# in neither) >= 30

200 - 130 - 150 + (# in overlap) >= 30

(# in overlap) >= 30 - 200 + 130 + 150

(# in overlap) >= 110

At this point, you've got it narrowed down to D and E.

This is where logic comes into play. You have to realize that there's no way the overlap can be greater than 130, because there are only 130 chem students, so that 130 represents the maximum possible overlap.

Ans: D

Make sense?

thks