inverting a percentage (basically inverting the percentages in pie chart)

Question

This is a weird question and I am not sure if it makes sense. Say we have 4 distances (where these are 1 dimensional distances):

5
3
8
6

so the total distance => SUM(5,3,8,6) = 22.

So for the first distance, it represents 3/22 of the total distance, which could be converted to a percent. But in my case, the shortest distance actually means a higher percentage, and a longer distance means a lower percentage.

Originally percentages would be:

(5/22)*100
(3/22)*100
(8/22)*100
(6/22)*100 

Is there a way to invert this somehow mathematically, so that a smaller fraction means a bigger percent, something like:

(22/5)/100
(22/3)/100
(22/8)/100
(22/6)/100

however, this is not right, since there is an obvious smell, in the case of 22/x being bigger than 100. How is this done?

Answer 1

I do not think there is a unique way to do this. For instance, you can consider the "remaining" distance: $$5 \quad\longrightarrow\quad 22-5=17 \quad\longrightarrow\quad \frac{17}{66}\cdot 100 = 25.76\%,$$ $$3 \quad\longrightarrow\quad 22-3=19 \quad\longrightarrow\quad \frac{19}{66}\cdot 100 = 28.78\%,$$ $$8 \quad\longrightarrow\quad 22-8=14 \quad\longrightarrow\quad \frac{14}{66}\cdot 100 = 21.21\%,$$ $$6 \quad\longrightarrow\quad 22-6=16 \quad\longrightarrow\quad \frac{16}{66}\cdot 100 = 24.24\%,$$ where $66=17+19+14+16$.