Question 1021884
{{{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,C)
 )}}}
<pre>
Everybody in the red circle M have masters.
Everybody in the blue circle C have certification.

a = the number of employees who have a masters but no certification
b = the number of employees who have both a masters and a certification
c = the number of employees who have a certification but no masters
d = the number of employees who have neither a masters nor a 
certification
</pre>A company has 22 employees.<pre> 
a+b+c+d = 22</pre>
Twelve of these employees have masters degrees<pre> 
a+b = 12</pre>
and fifteen have at least one Microsoft Certification<pre> 
b+c = 15</pre>
and three have neither a Microsoft certification nor a masters 
degree.<pre> 
d = 3

So we have this system of four equations and 4 unknowns:

{{{system(a+b+c+d = 22,a+b = 12,b+c = 15,d = 3)}}}

Solve by substitution or elimination.

Subtract the 2nd equation from the first, get c+d=10.
Substitute d=3, get c=7.  Substitute c=7 in b+c=15,
get b=8.  Substitute b=8 in a+b=12, get a=4.</pre>
How many have both a Masters degree and a certification?<pre>
That's b = 8.

{{{drawing(300,200,-4,4,-2,4.8,rectangle(-4,-1.6,4,4.4), locate(-2,1.8,a=4),locate(1.5,1.7,c=7),locate(-3.7,-1,d=3), locate(-3.6,2.5,M), locate(-.35,1.8,b=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)),
locate(3.4,2.5,C)
 )}}}

Edwin</pre>