The main issue with this problem is that the question is formulated unclear. The question: After some time, what is the proportion of boys to girls in the country? is rather ambiguous. What is the meaning of “after some time”? A second, two days, two million years? Since it is not clear, the mind rushes to leave the time as undetermined, hence a variable, say T. Right then, the logical path is to compute the expected ratio of boy to girls at a given time T. Notice, how the solution will be a function of the time in this case. However, this is not what the problem is pursuing since the answer should be an real number, and you are left wondering what is the meaning of life.
The question will be much clear if the problem only refers to a single family (the case of multiple families is easily generalized assuming independence) and in addition asks “What is the expected proportion of boys to girls after the family had a kid with the expected gender”.
The problem then becomes clear. A random variable describing the first time the family have a kid with the expected gender is then a reasonable construction. All other calculations follow.