I'm not exactly sure which method to use to solve my problem - pure algebra, probability or something else.
I have a certain set of numbers for multiple years: A = total amount of people working in factories and B = total number of factories. The third number I can calculate pretty fast, it is C = average of workers per factory, rounded.
For one year I'm lucky to have a more complete data set which is a table that differentiates between four different size classes of factories as well as the amount of people and factories in these groups respectively.
Therefore, I know A, B, C, A1, A2, A3, A4, B1, B2, B3, B4, C1, C2, C3 and C4. I want to project this data onto the other years where I only have A, B and C on the assumption that C1 to C4 don't change much through time even though A, B and C do. The upper table is the whole data set as I have it for that year. The lower table shows what it would look like for every other year
Since I've fixed the average, we have A1 = B1 x 77, which gets rid of one of the unknown constants, but I'm not sure now that this is the approach that I need. Instead, I think I need to find out how many possible solutions there are for 77xB1 + 206xB2 + 663xB3 + 1055xB4 = 98200 if B1 + B2 + B3 + B4 = 482 and B1 > B2 > B3 > B4 and A4 > A1 > A3 > A2.
The goal would be to have an Excel sheet where I type in A and B for the different years and it fills in the numbers on the basis of my formula.