How can I compute
$$\text{Var}\left[\frac{a+X}{b+Y}\right]$$
for $a, b > 0$, $X\sim Bin(n,p)$, and $Y \sim Bin(m,p)$ and $X,Y$ independent? I know that adding some constant to a binomially distributed variable is no longer binomial, which is why the above becomes troublesome to evaluate analytically.
Some transformation that facilitates the approximation given in Katz et al. (1978) is also fine for me. Invoking the paper referenced would allow stating that for $X$ and $Y$ as given above (i.e., without the non-negative constants $a$ and $b$), then $$\log{\frac{X/n}{Y/m}} \overset{\text{approx}}{\sim} \mathcal{N}\left(0,\frac{m+n}{nm}\right).$$
I feel like the transformation is painfully obvious, but I'm just not seeing it.
By independence, $$\begin{align}\text{Var}\left[\frac{a+X}{b+Y}\right]&=E\left[\frac{a+X}{b+Y}\right]^2-\left(E\left[\frac{a+X}{b+Y}\right]\right)^2\\ &=E\left[a+X\right]^2E\left[b+Y\right]^{-2}-\left(E\left[a+X\right]\right)^2\left(E\left[b+Y\right]^{-1}\right)^{2}\end{align}.$$
Can you take it from here?