The usual approximation for arithmatic of quantities with uncorrelated uncertainties that for small(ish) uncertainties $\Delta x_i$ or measurements $x_i$ let's us write for multiplicative operation
$$
\begin{array}~
y &=& \left(\frac{x_1}{x_2}\right) \, \text{ or }\, \left(x_1 x_2\right) \\
\frac{\Delta y}{y} &=& \sqrt{ \left(\frac{\Delta x_1}{x_1}\right)^2 + \left(\frac{\Delta x_2}{x_2}\right)^2 }
\end{array}
$$
(i.e. add relative uncertainties in quadrature).
You can get this kind of result from a bastardization of the chain the rule.
Given $y = f(x_1, x_2 , \dots)$ or each input measurement $x_i$, compute $\left(\frac{\partial f(x_1, x_2 , \dots)}{\partial x_i}\right) \Delta x_i$ and add all the resulting terms in quadrature.
You have also been a little free with your nomenclature here. Call $s$ the underling signal and $n$ the random noise (this can be counting statistics or any other random process such as shot noise in the detector, but not a constant bias which must be subtracted off--our noise is assumed to have a mean of zero) with a distribution whose width is characterized by $\sigma$. A single measurement is then $m_i = s_i + n_i$, and the population has a signal to noise ration of $\frac{s}{\sigma}$.
The win from addition is that the sum of $N$ such measurements is $$ M = \sum_{i=1}^N m_i = Ns + \sqrt{N}\sigma $$ meaning that the signal to noise ratio of the sum is $\frac{Ns}{\sqrt{N}\sigma} = \sqrt{N}\left(\frac{s}{\sigma}\right)$, an improvement.
