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:
Clue #1. 110 students,
Clue #2. 60 take math,
Clue #3. 50 take English,
Clue #4. 40 take history,
Clue #5. 25 take English and math,
Clue #6. 20 take English and history,
Clue #7. 20 take math and history,
Clue #8. 15 take all three subjects.
We begin with clue #8 and go backward:
Clue #8. 15 take all three subjects.
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.
Clue #7. 20 take math and history,
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.
Clue #6. 20 take English and history,
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.
Clue #5. 25 take English and math,
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.
Clue #4. 40 take history,
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.
Clue #3. 50 take English,
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.
Clue #2. 60 take math,
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.
Clue #1. 110 students,
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:
a. The number of students taking math
but not the other two subjects.
These are the students is the region with a=30.
Answer: 30
b. The number of students taking English
or math but not history.
These are the students in the region with a=30,
b=10 and c=20.
Answer: 30+10+20 = 60
c. The number of students taking none of
these subjects.
These students are the ones in the region outside
all three circles, that is, h=10.
Answer: 10.
Edwn
