I am ashamed to say that I cannot figure this one out: I am given two ratios $\dfrac{p_i}{q_i}$ where $i=1$, $2$. (We just know the ratios and not the numbers $p_i, q_i$. What I mean by this is simply that in the expression $\psi$ I cannot separate the $p_i$ and $q_i$)
I want an elementary expression $\psi:\mathbb R \to \mathbb R$ such that
$$\dfrac{p_1+p_2}{q_1+q_2}=\psi \left(\dfrac{p_1}{q_1},\dfrac{p_1}{q_2}\right)$$
NOTE
- $p_i, q_i \in \mathbb Z$
- $q_2 \nmid q_1$, since otherwise we cannot find such a function.
- $p_2 \nmid p_1$ since this would make things trivial.
- Let us assume $(p_i,q_i)=1$.
As Martin Sleziak said in his comment, the function you defined is called the mediant.
As others have suggested, there is no reason to expect there to be a simpler formula than the definition you give. There is however a natural and obvious way to represent $(p_1 + p_2)/(q_1 +q _2)$ as a weighted average of $p_1/q_1$ and $p_2/q_2$, where the weights depend on $q_1,q_2$:
$$\frac{p_1+p_2}{q_1+q_2} = \frac{q_1}{q_1+q_2}\frac{p_1}{q_1} + \frac{q_2}{q_1+q_2}\frac{p_2}{q_2}.$$
This shows, for instance, that $(p_1+p_2)/(q_1+q_2)$ always lies between $p_1/q_1$ and $p_2/q_2$.
In fact a simple argument shows that the function $\psi:\mathbb{Q}^2\to\mathbb{Q}$ so defined is discontinuous (with respect to the usual order topology) everywhere except the diagonal. Indeed, consider replacing $p_2/q_2$ with a nearby rational but with much larger denominator. The value of $(p_1+p_2)/(q_1+q_2)$ is then changed to pretty much $p_2/q_2$. This puts a lower bound on the simplicity of a formula for $\psi$.