Let b = the number of boys in the family
Let g = the number of girls in the family

The boy has b-1 brothers.  We must subtract 1 from b,
the number of boys, since the boy who spoke is not his own brother.

The boy has g sisters.  So

b-1 = g

The girl has g-1 sisters.  We must subtract 1 from g, the number of 
girls, since the girl who spoke is not her own sister.

The girl has b brothers.  So

b = 2(g-1)

So we have this system of equations:

b-1 = g
  b = 2(g-1)

Use substitution:

2(g-1)-1 = g

  2g-2-1 = g

    2g-3 = g
      
       g = 3

b = 2(3-1)
b = 2(2)
b = 4

There are 3 girls and 4 boys.

Each boy has 3 brothers and 3 sisters
Each girl has 4 brothers and 2 sisters.  

Edwin