Question 1157583
A class has 30 students. 21 students like math and 9 like english . If 16 students like math only and 5 students like neither math nor English, how many students like both math and English? Draw a Venn diagram to represent the information.
<pre>
{{{drawing(300,200,-4,4,-2,4.8,
rectangle(-4,-1.6,4,4.4), locate(-2,1.8,a),locate(1.5,1.7,c),
locate(-3.7,-1,d),
 locate(-3.6,2.5,M), locate(-.1,1.8,b),
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)),
locate(3.4,2.5,E)
 )}}}

Everybody in the red circle likes math.
Everybody in the blue circle likes English.
a = the number who like math only
b = the number who like both math and English.
c = the number who like English only.
d = the number who do not like either math or English.

A class has 30 students.  So a+b+c+d = 10

21 students like math. So a+b = 21 

9 like English.  So b+c = 9

16 students like math only.  So a = 16 

5 students like neither math nor English.  So d = 5 

</pre>how many students like both math and English?<pre> 

Since 21 like math of which 16 like math only, the other 5 must like both math
and English.  So we have the answer immediately, 5 like both.

Draw a Venn diagram to represent the information.

a=16, b=5 (which is the answer).

{{{drawing(300,200,-4,4,-2,4.8,
rectangle(-4,-1.6,4,4.4), locate(-2,1.8,16),locate(1.5,1.7,c),
locate(-3.7,-1,5),
 locate(-3.6,2.5,M), locate(-.1,1.8,5),
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)),
locate(3.4,2.5,E)
 )}}}

We have everything in the Venn diagram except c. But since 9 like English,
and since 5 like both, the other 4 must like English only. So c = 4.

{{{drawing(300,200,-4,4,-2,4.8,
rectangle(-4,-1.6,4,4.4), locate(-2,1.8,16),locate(1.5,1.7,4),
locate(-3.7,-1,5),
 locate(-3.6,2.5,M), locate(-.1,1.8,5),
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)),
locate(3.4,2.5,E)
 )}}}

We didn't need to be told that there are 30 students.  We could have figured
that out just by adding 16+5+4+5 = 30. 

Edwin</pre>