Question 264251
22 students in a classroom. 12 like grape juice, 15 like orange juice, and 10 like both juices. How many students like neither juice? Thanks for any help, Sophie.
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Two ways to do it.  

1. By Venn diagram
2. By formula

By Venn diagram:

Imagine you have all 22 students on a football field.

First draw a big rectangle to represent, say, the big football field, for
all 22 students to stand on.
 
{{{drawing(400,400,-3.2,3.2,-3.2,3.2, rectangle(-3,-2,3,2) )}}}
 
Now on that football field draw a big circle for the 12 students
to stand in who like grape juice and label it G:
 
{{{drawing(400,400,-4,4,-4,4, rectangle(-3.9,-2.9,3.9,2.9),
circle(-1,0,2), locate(-2.3,2,G)   )}}}
 
Next draw another big circle overlapping it for the 10 students
to stand in who like orange juice and label it O:
 
{{{drawing(400,400,-4,4,-4,4, rectangle(-3.9,-2.9,3.9,2.9),
circle(-1,0,2), circle(1,0,2), locate(-2.3,2,G), locate(2.3,2,O)  )}}}
 
Now we tell the 10 people who like both juices to stand in the overlapping
part of those two circles.  So we'll write {{{10}}} in there to represent the
10 students in that region.
 
{{{drawing(100,173,-1,1,-1.1,1.2, rectangle(-3.9,-2.9,3.9,2.9),
circle(-1,0,2), circle(1,0,2), locate(-2.3,2,G), locate(2.3,2,O),
locate(0,0,10) )}}} 

and in the main drawing:
 
{{{drawing(400,400,-4,4,-4,4, rectangle(-3.9,-2.9,3.9,2.9),
circle(-1,0,2), circle(1,0,2), locate(-2.3,2,G), locate(2.3,2,O),
 locate(0,0,10)
 
  )}}}

 
Since there are 12 students who like grape juice, the rest of those 12 
besides the 10 that like both, that is, 12-10, or 2, will have to stand in 
this region, in which we write 2:
 
{{{drawing(200,400,-4,-.1,-4,4, locate(-2,0,2),
circle(-1,0,2), circle(1,0,2)  )}}}  
 
and in the main drawing:
 
{{{drawing(400,400,-4,4,-4,4, rectangle(-3.9,-2.9,3.9,2.9),
circle(-1,0,2), circle(1,0,2), locate(-2.3,2,G), locate(2.3,2,O),
locate(0,0,10), locate(-2,0,2))}}}
 
---
 
Since there are 15 students who like orange juice, the rest of those 15 
besides the 10 that like both, that is, 15-10, or 5, will have to 
stand in this region, in which we write 5:
 
`                      `{{{drawing(200,400,0,4,-4,4, locate(2,0,5),
circle(-1,0,2), circle(1,0,2)  )}}}  
 
and in the main drawing:
 
{{{drawing(400,400,-4,4,-4,4, rectangle(-3.9,-2.9,3.9,2.9),
circle(-1,0,2), circle(1,0,2), locate(-2.3,2,G), locate(2.3,2,O),
locate(0,0,10), locate(-2,0,2), locate(2,0,5)
 
  )}}}
 
Now we have accounted for 2 + 10 + 5 or 17 of the 22 students.  That
leaves 22-17, or 5 students who like neither juice, and so they have 
to stand outside the two circles, like this:

{{{drawing(400,400,-4,4,-4,4, rectangle(-3.9,-2.9,3.9,2.9),
circle(-1,0,2), circle(1,0,2), locate(-2.3,2,G), locate(2.3,2,O),
locate(0,0,10), locate(-2,0,2), locate(2,0,5), locate(-3,-2,5)
 
  )}}}

So the answer is 5.

----------------------

We could also have done it with a couple formulas:

N(neither G nor O) = 22 - N(G or O). 
 
N(G or O) = N(G) + N(O) - N(G and O)
    
N(G or O) = 12 + 15 - 10 
 
N(G or O) = 17
 
N(neither G nor O) = 22 - N(G or O)

N(neither G nor O) = 22 - 17

N(neither G nor O) = 5
 
Edwin</pre>