In these Venn diagram problems, the clues are
always given in the reverse order in which you
use them:

1. 4000 students in the college
2. 2000 take French (F)
3. 3000 take Spanish (S)
4. 500 take Latin (L)
5. 1500 take both French and Spanish
6. 300 take both French and Latin
7. 200 take Spanish and Latin
8. 50 take all three languages 



8. 50 take all three languages 
So e=50, for it is the region 
common to all three circles.



7. 200 take Spanish and Latin
This is e+f, for they are the
regions common the S and L. We
already have e=50, so f=200-50
or f=150.



6. 300 take both French and Latin 
This is d+e, for they are the
regions common the F and L. We
already have e=50, so d=300-50
or d=250.



5. 1500 take both French and Spanish
This is b+e, for they are the
regions common the F and S. We
already have e=50, so b=1500-50
or b=1450. [There isn't enough
room to write that in the region,
so it'll have to stick outside of 
the region]



4. 500 take Latin (L)
This means that there are 500 in the
whole circle L = d+e+f+g. we already 
have that d=250, e=50, f=150, so
g=500-250-50-150=50.  So g=50.



3. 3000 take Spanish (S)  
This means that there are 3000 in the
whole circle S = b+c+e+f. we already 
have that b=1450, e=50, f=150, so
c=3000-1450-50-150=1350.  So c=1350.



2. 2000 take French (F)
This means that there are 2000 in the
whole circle F = a+b+d+e. we already 
have that b=1450, d=250, e=50, so
c=2000-1450-250-50=250.  So a=250.



1. 4000 students in the college
This means that there are 4000
in the whole "universe", which
is a+b+c+d+e+f+g+h.  We now have
all but h. So 
h=4000-250-1450-1350-250-50-150-50=450



1. L intersect (F union S)
   L ᑎ (F ᑌ S)

defg ᑎ (abde ᑌ bcef)

defg ᑎ (abcdef)

  def

d+e+f = 250+50+150 = 450


2. L' (complement)

That's everything but L:

a+b+c+h = 250+1450+1350+450 = 3500

3. F intersect S' (complement) intersect L' (complement)

(F ᑌ S') ᑎ L'

(abde ᑌ bcef') ᑎ defg'

(abde ᑌ adgh) ᑎ abch

abdegh ᑎ abch

abh

a+b+h = 250+1450+450 = 2150

Edwin