We draw this Venn diagram.  



The capital letters  M, E, and H represent the 
three subjects Math, English and History.  The 
small letters a through j represent the number 
of elements (students) in each of the 8 regions.

In these problems the clues are usually given 
in the reverse order that you need them to
place the proper numbers in the 8 regions of
the Venn diagram.

Here are the clues:

We begin with clue #8 and go backward:

This is the most inclusive of all the clues, so
the 15 are placed in the region that is common
to all three circles, the region with "e=?".
So we put "e=15" in the middle region.

This is the pair of regions that are common
to circle M and circle H, which are regions
with e=15 and d=?. So e+d=20, and since e=15,
d=20-15 or 5.  So we put d=5 in that region.

This is the pair of regions that are common
to circle E and circle H, which are regions
with e=15 and f=?. So e+f=20, and since e=15,
f=20-15 or 5.  So we put f=5 in that region.

This is the pair of regions that are common
to circle E and circle M, which are regions
with e=15 and b=?. So e+b=25, and since e=15,
b=25-15 or 10.  So we put b=10 in that region.

This is the 4 regions of circle H, which are regions
with e=15,d=5,f=5,g=?. So e+d+f+g=40, and since e=15,
d=5,f=5, the g=40-15-5-5=15 So we put g=15 in that region.

This is the 4 regions of circle E, which are regions
with e=15,b=10,f=5,c=?. So e+b+f+c=50, and since e=15,
b=10,f=5, then c=50-15-10-5=20 So we put c=20 in that region.

This is the 4 regions of circle M, which are regions
with e=15,b=10,d=5,a=?. So e+b+d+a=50, and since e=15,
b=10,d=5, then a=60-15-10-5=30 So we put a=30 in that region.

This is the total of all the students in all the 8 regions.
So this is a+b+c+d+e+f+g+h=110.  We have the numbers in all
the regions except the region with h=?,  which is outside 
all the three circles yet inside the big rectangle.
So h=110-30-10-20-5-15-5-15 = 10.  So we put h=10 in that 
last region.



Now our Venn diagram is complete.
We can answer any question asked:

These are the students is the region with a=30.
Answer: 30

These are the students in the region with a=30,
b=10 and c=20. 
Answer: 30+10+20 = 60

These students are the ones in the region outside 
all three circles, that is, h=10.
Answer: 10.

Edwn