Puzzle ZYVH 

Workings

Hint
The answer isn't very large at all.

Answer
There are four boys and six girls in the family.

If there were B boys, and G girls we know that Billy has twice as many sisters as brothers, so 2 x (b - 1) = g.

And his sister, Frankie, has one more sister than she has brothers, so g - 1 = b + 1.
    2(b - 1) = g       [1]
       g - 1 = b + 1   [2]

Using [1] in [2] gives:

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

Expanding the bracket gives:

  2b - 2 - 1 = b + 1

Simplifying the left hand side gives:

     2b - 3  = b + 1

Add 3 to each side gives:

          2b = b + 4

Subtract b from both sides gives:

           b = 4

Using b = 4 in [1] gives:

    2(4 - 1) = g

           6 = g