Division of number with fixed decimal (sharing monetary fee)

Question

I'm possibly writing a lot of terms incorrectly in this question since I'm kinda lost in how to look for this.

The problem I have is, given a monetary value that must be paid (a fee of some sort), and also given that this value comes from multiple sources, and I would like to correclty alocate them. How to correctly do the division. For instance, if I bought itens A, B and C, together they have a fee of $2.45. What I would do is:

fee of A alone = \$A/(\$A + \$B + \$C)*TotalFee

(and so on)

However, rounding the fees to two decimal places, and summing them at the end, sometimes gives a different value than the original fee.

I usually just adjust by hand, but I imagine there is a correct way to adjust that, possibly using the remaiders of division.

Thank you!

Answer 1

This is surprisingly tricky to get right! Here is what I would do:

Call your three partial fees (expressed as a whole number of cents) $F_A,F_B,$ and $F_C$.

Suppose first that the sum of these is too small: $$F_A+F_B+F_C<245$$ First, get the partial fees $X_A,X_B,X_C$ to more decimal places, using standard floating-point arithmetic. Then calculate the three discrepancies $$D_A=X_A-F_A,D_B=X_B-F_B,D_C=X_C-F_C$$

Find the largest of these; say this is $D_B$ (if two or three are equally large, choose one at random). Now just increase $F_B$ by one cent.

This will usually be enough to make the total come out right. If not, just repeat the procedure; this will certainly be enough (for three items).

If $F_A+F_B+F_C$ is too large instead of too small, you should be able to work out how to modify this procedure accordingly.

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Depending on your requirements, it might be more equitable to express those discrepancies as ratios: $$D_A=(X_A-F_A)/X_A,D_B=(X_B-F_B)/X_B,D_C=(X_C-F_C)/X_C$$ This is because, for instance, a 1-cent markup on a 10-cent item could be seen as more significant than a 2-cent markup on a 60-cent item. But this is your call.