| RIGHT- |  LEFT- |        |
       | HANDED | HANDED | TOTALS |
-------------------------|--------|
MEN    |   A    |   B    |  A+B   |   
-------|-----------------|--------|
WOMEN  |   C    |   D    |  C+D   |
----------------------------------|
TOTALS |  A+C   |  B+D   | 100%=1 | 
-----------------------------------

Let A,B,C,D be the percentages of people in each 
of the four categories, the total of all being 
100% or 1.

So          A+B+C+D = 1

20% of a population are left-handed, 

       .20(A+B+C+D) = B+D
.20A+.20B+.20C+.20D = B+D
.20A-.80B+.20C-.80D = 0
        2A-8B+2C-8D = 0
          A-4B+C-4D = 0 

60% of the left-handed people are men,

           .60(B+D) = B
          .60B+.60D = B
         -.40B+.60D = 0
             -4B+6D = 0  
             -2B+3D = 0

 while 55% of the right-handed people are women. 

           .55(A+C) = C
          .55A+.55C = C
          .55A-.45C = 0
            55A-45C = 0
             11A-9C = 0

So we have this system of equations:

          A+ B +C +D = 1
          A-4B +C-4D = 0
           -2B   +3D = 0
        11A   -9C    = 0

Solve that system by any method, and get

A = 0.36, or 36% of the people are right-handed men. 

B = 0.12, or 12% of the people are left-handed men.  

C = 0.44, or 44% of the people are right-handed women.

D = 0.08, or  8% of the people are left-handed women. 

 8% of the people are left-handed women.
36% of the people are right-handed men.

Edwin