SOLUTION: Please show me how to solve this problem ( it was a homework problem (1 complete problem) and I didn't know where to start). thanks! 200 Students 70 Students take English cl

Draw a big square to put all 200 students in:

Now inside the square, draw a circle to put the
70 English students in and label it E

Now inside the rectangle, draw a second circle which
overlaps the first circle to put the 54 Math stdents
in. Label it M:


Notice that the "football" shaped region where those two
circles overlap is where we will put the 28 students who
take English and Math.

Now inside the square, draw a third circle which
overlaps the other two circles to put the 45 Social
Studies students in. Label it S:



You will notice there are 8 regions formed within the
square. I will number them I, II, III, IV, V,
VI, VII, and VIII



Let's go through the seven given statements and identify which
regions they are talking about:

#1. 70 Students take English class.  These 70 students are in the
E circle, which is made up of regions I,II,IV,and V.

#2. 54 Students take Math class. These 54 students are in the
M circle, which is made up of regions II,III,V,and VI. 

#3. 45 Students take Social Studies. These 45 students are in the
S circle, which is made up of regions IV,V,VI,and VII.

#4. 28 Students take English and Math.  These 28 students are in the
overlapping parts of circles E and M, which is made up of regions
II and V.

#5. 19 Students take English and Social Studies. These 19 students are 
in the overlapping parts of circles E and S, which is made up of regions
IV and V.

#6. 15 Students take Math and Social Studies. These 15 students are in the
overlapping parts of circles M and S, which is made up of regions
V and VI.

#7. 10 Students take English, Math and Social Studies. These 10 students are
in the region where all three circles overlap.  This consists of only one
region, namely region V.  It's shaped like a triangle with curved sides.

Now since we know how many students region V contains, namely 10, we will
write 10 in region V.

 
 
Next we go back to #6.  It tells us there are 15 students 
altogether in regions V and VI.  So since there are 10
students in region V, there have to be 15-10 or 5 students
in region VI.  So we write 5 in region VI.


 
Next we go back to #5.  It tells us there are 19 students 
altogether in regions IV and V.  So since there are 10
students in region V, there have to be 19-10 or 9 students
in region IV.  So we write 9 in region IV.



Next we go back to #4.  It tells us there are 28 students 
altogether in regions II and V.  So since there are 10
students in region V, there have to be 28-10 or 18 students
in region II.  So we write 18 in region II.



Next we go back to #3.  It tells us there are 45 students 
altogether in regions IV, V, VI and VII.  So since there are 9
students in region IV, 10 in V, and 5 in VI, which amounts to
9+10+5 or 24, so the leaves 45-24 or 21 for region VII.  So
we write 21 in region VII
 


Next we go back to #2.  It tells us there are 54 students 
altogether in regions II, III, V and VI.  So since there are 18
students in region II, 10 in V, and 5 in VI, which amounts to
18+10+5 or 33, so the leaves 54-33 or 21 for region III.  So
we write 21 in region III:
 


Next we go back to #1.  It tells us there are 70 students 
altogether in regions I, II, IV and V.  So since there are 18
students in region II, 10 in V, and 9 in IV, which amounts to
18+10+9 or 37, so the leaves 70-37 or 33 for region I.  So
we write 33 in region I:
 


There is still one region we haven't yet found, that is
the number of students in the region which is inside
the big square, but outside all the circles.  This is
region VIII.

Now tthere are respectively, 33, 18, 21, 9, 10, 5, and
21 students in I, II, III, IV, V, VI, and VII.  They
account for 33+18+21+9+10+5+21 or 117 students. So that
leaves 200-117 or 83 students in outer region VIII.  So 
we write 83 in region VIII:




Now that we know how many students are in each of the eight
regions, we are ready to answer any questions.

The first question is:

1.) How many students take English only?

That's the 33 students in region I.

2.) How many Students take math only.

That's the 21 in region III 

3.) How many Students take English, social studies but not math

That's the 9 in region IV

4.) How many students take math and social studies but not English class

That's the 5 in region VI.

5.) How many do not take any class.

That's the 83 in the outer region VIII.

Edwin