I was thinking of the following problem:
Suppose I have an agency that offers workshops to companies. There are four different "type/rank" of professionals in my agency that could give these workshops. Call these four types R1,R2,R3 and R4. For example, a workshop could be given by two people of rank R1 and one of rank R4, or maybe there is also a workshop just given by three persons of rank R3. Now, suppose the following:
A workshop was given by four persons of the agency of rank R1,R2,R3 and R4 respectively. Suppose the money earned was $T$ dollars and the money is distributed among these for employees according to this criteria:
R1 takes 15%, R2 takes 20%, R3 takes 30% and R4 takes 35%.
So my problem is: for an arbitrary workshop where there is another number of employees of each rank, could I distribute the money based on the criteria adopted in the example described?
For example, suppose another workshop is given by one person of rank R2 and two persons of rank R3, couldn't I distribute the money among these people trying to follow the criteria of the other case?
I don't know if this way of reasoning is correct: if of 100% R3 took 30% then of 80% R3 takes 37,5%, and if R2 took 20% of 100%, of 80% he/she takes 25%. So in this workshop, the money is distributed with two R3's taking 37,5% of the total and one R2 taking 25%.
If this method is correct, could anyone explain me why it works?
If $n_i$ persons of rank$R_i$ participate, then it seems that each participating person of rank $i$ should get $\frac{p_iT}{p_1n_1+p_2n_2+p_3n_3+p_4n_4}$, where $p_1=15$, $p_2=20$, $p_3=30$, $p_4=45$.