a = the number who have been to Germany but not Spain or France
b = the number who have been to Germany and Spain, but not France
c = the number who have been to Spain, but not Germany or France
d = the number who have been to Germany and France, but not Spain
e = the number who have been to Germany, France and Spain
h = the number who have been to Spain and France, but not Germany
i = the number who have been to France, but not Germany or Spain
j = the number who have been to not been to Germany, Spain, or France.
Most Venn diagram question give the clues in the exact
reverse order from the order you need to consider them in.
This one is no exception. So we use them in reverse order.
You will usually find reversing the clues to be the best
way to do a Venn diagram problem.
5 have been to all three countries.
So e=5:
3 have been to only Spain and France.
So h=3
5 have only been to Germany.
So a=5
8 have only been to Spain.
So c=8
12 have been to Germany and France.
So d+e = 12. And since e=5, d=12-5=7
So d=7
21 have visited France.
So d+e+h+i=21. And since d=7,
e=5, and h=3, i=21-7-5-3=6,
So i=6
18 have been to Spain.
So b+c+e+h=18. And since c=8,
e=5, and h=3, b=18-8-5-3=2,
So b=2
19 members have visited Germany.
So a+b+d+e=19. But we didn't need that
information, for a=5, b=2, d=7 and e=5,
and 5+2+7+5=19. But it provides a good
check on what we have done so far.
There are 45 students in the University Travel Club.
So a+b+c+d+e+h+i+j=45. And since a=5, b=2, c=8,
d=7, e=5, h=3, and i=6,
j=45-5-2-8-7-5-3-6=9
So j=9
Since the Venn diagram is complete, we can
now answer any question about it:
How many students have not been to Germany, Spain, or France?
That's j=9
Edwin