Question 1121053
In a group of students, 70 have a personal computers, 120 have a personal
stereo and 41 have both. How many own at least one of these devices? Draw an
appropriate Venn diagram
<pre>
{{{drawing(300,200,-4,4,-2,4.8,



red(circle(-sqrt(2),sqrt(2),2)),

red(circle(-sqrt(2),sqrt(2),1.95)), 

red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975))) )}}}


There are two overlapping circles.  We pretend that the red circle contains
all 70 students who have a personal computer. We pretend that the blue
circle contains all 120 students who have a personal stereo.  The
overlapping part is in both circles and we pretend that the overlapping
part contains only the 41 students who have both a computer and a stereo. So
we write 41 in the overlapping part:

{{{drawing(300,200,-4,4,-2,4.8,

 locate(-.2,1.8,41),

red(circle(-sqrt(2),sqrt(2),2)),

red(circle(-sqrt(2),sqrt(2),1.95)), 

red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975))) )}}}
 
The red circle contains all 70 students who have a personal computer.  and
the 41 who have both are part of the 70, so that leaves 70-41 or 29 in the
left part of the red circle, who only have a personal computer.  So we write
29 in the left part of the red circle:

{{{drawing(300,200,-4,4,-2,4.8,

 locate(-2,1.8,29),

 locate(-.2,1.8,41),

red(circle(-sqrt(2),sqrt(2),2)),

red(circle(-sqrt(2),sqrt(2),1.95)), 

red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975))) )}}}


The blue circle contains all 120 students who have a personal stereo, and
the 41 who have both are part of the 120, so that leaves 120-41 or 79 in the
right part of the blue circle, who only have a personal stereo.  So we write
79 in the right part of the blue circle:

{{{drawing(300,200,-4,4,-2,4.8,

 locate(-2,1.8,29),locate(1.5,1.7,79),

 locate(-.2,1.8,41),

red(circle(-sqrt(2),sqrt(2),2)),

red(circle(-sqrt(2),sqrt(2),1.95)), 

red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975))) )}}}

Notice that there are 29 in the red circle that are not in the blue circle.
So 29 students have a computer but no stereo. Notice that there are 79 in
the blue circle that are not in the red circle.  So 79 students have a
stereo but no computer.  And as we said earlier, the 41 in the middle that
are in both circles are the 41 that have both a computer and a stereo.

So to determine how many students have one or the other (or both), that is,
at least one, we add the three numbers together:

 29 students have a computer but no stereo
 41 students have both a computer and a stereo
+79 students have a stereo but no computer
-------------------------------------------------  
149 students have at least one of the two devices

Edwin</pre>