It depends on the number of families.
Let $X$ be the number of children in a family, it is geometric random variable with $p=0.5$, i.e., $$ P(X = x) = 0.5^x, x=1,2,3... $$ which implies (E(X) = 2)
For $N$ families, the ratio is $$ \frac{\sum (X_i - 1)}{ \sum X_i} = \frac{\sum X_i - N}{\sum X_i} = 1 - \frac{N}{ \sum X_i} $$
Since $\sum X_i /N \longrightarrow E(X) = 2$ (law of large number), the ratio coverages to 1/2 if $N \longrightarrow \infty$.
If there are only finite families, let $T$ be the total number of children of the country: $T = \sum X_i$, then
$T$ has a negative binomial distribution with pmf
$$ P(T=t) = C^{t-1}_{N-1} 0.5^t, t = N, N+1... $$
therefore $$ E\left[ \frac{N}{\sum X_i} \right] = E\left[ \frac{N}{T} \right] = \sum_{t=N}^{\infty} \frac{N}{t} C^{t-1}_{N-1} 0.5^t = {_2F_1} (N, 1, N+1, -1) $$ where $_2F_1$ is the hypergeometric function.
Therefore the ratio is $ 1- {_2F_1} (N, 1, N+1, -1) $.