Correct me If I am wrong, the question specifically states that "each person eats one piece of each cake." This means that if A brings 3 cakes and B brings 5 cakes then person C eats one piece of each cake, so he must eat 3 pieces from A and 5 pieces from B. Hence if each piece costs 1 coin, 3 coins should go to A while 5 to B.

For summarising the solution to this answer I got the general equation of (3n*(Σp-k))/Σp. the letter 'n' is the number of cakes present, 'Σp' is the total number of people eating cakes. The 'k' is the number of people who provide the cakes. The expression 'Σp-k' is the number of extra people. The equation provides a result for the sum of all coins that need to be paid back this will be 'Σc', to calculate the number of coins that need to be paid back per person the Σc needs to be divided by the number extra people this gives the result (Σc)/(Σp-k). Thus we can state that the number of coins that need to be paid back per person is given by equation (3n*(Σp-k))/((Σp*(Σp-k)) = Σc/Σp-k. This simplifies the left side equation to 3n/Σp.

Thus the formulas I obtained are:
- (3n*(Σp-k))/Σp = Σc : sum of coins needed to be paid back
- (Σc)/(Σp-k) or 3n/Σp : number of coins that need to be given back per person

I think that the problem with the question lies in the wording, the answer provided is correct is the people ate the 8 cakes between each other. I also would like to say that the answer provided would only be one of many possible answers, this is because it assumes that person A eats all her cakes and then gives the others away. However if they were the to randomly eat 8 slices each the number of combinations possible would be 24!/8! (each slice is on its own) where the combination in the answer would be just one of many.

I am not sure if my maths is 100% correct, please do correct me if I have made any mistakes!