SOLUTION: a survey of a school 0f 850 students found that 150 used a bus to get to school, 75 used a train and 650 user neither. using a venn diagram, calculate the number of students who us

Imagine you have all 850 students in the bleachers of football field.

First draw a big rectangle to represent, say, a big football field, for
all 850 students to come down and stand in.



Now on that football field draw a big circle for all 150 students
to stand in who used a bus and label it B:



Next draw another big circle overlapping it for all 75 students
to stand in who used a train and label it T:



Now the 650 students who use neither bus nor train will have to stand
on the field outside both circles, so we'll write 650 out there to
stand for the 650 students who will stand out there:



We need to know how many people will be standing in both circles, 
that is, how many will be standing in this overapping part of the 
two circles:

`                 

So we'll let  represent the number of students in that region,
so we'll write "x" in there:

`                   

and in the main drawing:



Since there are 150 students who used a bus, the rest of those 150 
besides the x that used both, that is, 150-x, will have to stand in 
this region, in which we write 150-x:

  

and in the main drawing:



---

Since there are 75 students who used a train, the rest of those 75 
besides the x that used both, that is, 75-x, will have to stand in 
this region, in which we write 75-x:

`                      `  

and in the main drawing:



Now we add up all the numbers in the four regions and
that sum must equal the entire 850 students standing somewhere
on the football field:



Solve that and get 

The answer is 25 students used both bus and train.

----

If you didn't have to use a Venn diagram, you could have
done it easier with this formula:

N(B or T) = N(B) + N(T) - N(B and T)
    
850-650 = 150 + 75 - N(B and T)

200 = 225 - N(B and T)

N(B and T) = 225 - 200

N(N and T) = 25.

Edwin