Let b = the number of boys.
Let g = the number of girls.

Each boy has one less brother than the number of boys, since he is not his 
own brother, so

b-1 = the number of brothers each boy has.

However

b = the number of brothers each girl has.

Each girl has one less sister than the number of girls, since she is not 
her own sister, so

g-1 = the number of sisters each girl has.

However

g = the number of sisters each boy has.

>>...Each boy has as many brother as sisters...<<

            b-1 = g


>>...each girl has half as many sisters as brothers...<<

           g-1 = ½b

Clear that equation of fractions by multiplying through by 2

          2g-2 = b

Substitute b-1 for g

      2(b-1)-2 = b
        2b-2-2 = b  
          2b-4 = b
            2b = b+4
             b = 4

Substitute 4 for b in

            b-1 = g
            4-1 = g
              3 = g

So there are 4 boys and 3 girls in the family.  Each boy has 
3 brothers and 3 sisters.  Each girl has 2 sisters and 
4 brothers.

Edwin