Tricky Math problem, can you help !?

Question

In a group of 40 students, 25 applied to Columbia and 30 applied to Cornell. If 3 students applied to neither Columbia nor Cornell, how many students applied to both schools?

Answer 1

Total students=40
Total applications=30+25=55
not any=3
total applicants=40-3=37
Therefore who applied for both=55-37=18

Answer 2

For this problem, you'll want to use a Venn Diagram. Think of two circles that overlap slightly inside of a rectangle.
To find the number of students that are in the "Cornell only" region we subtract all of the students that aren't in that set from the Universal set, like this:
40 - (3 + 25) = 40 - 28 = 12
This value goes into the "Cornell Only" region. We do the same for the "Columbia Only" region:
40 - (3 + 30) = 40 - 33 = 7
We can now add up all the students who have not applied to both schools and subtract that from the Universal set:
40 - (7 + 12 + 3) = 40 - 22 = 18
The are the student who have applied to both schools.

So, now we have
7 students who applied only to Columbia
12 students who applied only to Cornell
18 students who applied to both
and
3 students who applied to neither
for a total of 40 students

Answer 3

Hi,
The best way to see this is to use Venn diagrams, but that is really not an option with this word processor. So, we’ll do it symbolically only. Let A = number in Cornell = 30 and B = number in Columbia = 25. Then we can write this equation:
A+B –(A∩B) = 40 -3 (Where ∩ means intersection or those in both colleges.)
–(A∩B) = 37-(A+B)
–(A∩B) = 37-(30+25)
–(A∩B) = -18
(A∩B) = 18

Hope this helps.
FE

Answer 4

37 applied to either Columbia or to Cornell. 7 of these did not apply to Cornell, so applied to Columbia only, and 12 of the 37 did not apply to Columbia, so applied to Cornell only. Therefore, 37 - 7 - 12 = 18 applied to both.

Answer 5

18

Answer 6

Total Students = 40
Total Applicants=40-3=37.
fomula : n(a intersection b)=n(a)+n(b)-n(aUb)
n(a intersection b)=25+30-37
=18.

Answer 7

f(u)=f(ca)+f(cl)-f(ca U cl)+f'(ca U cl)
where f(u) is the universal set =40 students

f(ca) is the no of students applied to columbia=25
f(cl) is the no of students applied to cornell=30
f'(ca U cl) is the no of students who havent applied to any state=3

f(ca U cl) is the no of students who have applied to both the states let it be x
so
substitute the values in the equation

40=25+30-x+3
x=58-40
x=18
so 18 students have applied to both the states