(i) A has at least one girl and twice as many boys as girls.

(ii) B has at least one girl and three times as many boys as girls.

(iii) C has at least one girl and three more boys than girls.

(iv) When I tell you the number of children we have – a number less than 25 – you will know how many children I have, but not how many children each of the others has. Altogether we have……..”

Who is the speaker and how many children does the speaker have?

I agree with your post. 16 is not ruled out by (iv). Neither is 19 as the total number of children.

This does not contradict A as the speaker.