Figure Graduated Intervals Between Two Percentages

Question

Given 100%, I need an equation to figure the equal steps that will occur between two percentages, where the first known percentage starts at 0% and the last known percentage ends at 100%.

For example, if the initial percentage is 17% and the final percentage is 3%, what will the graduated ( maybe linear? ) intervals be?

The simplest example that I can lay out without an equation would be if the first value is 40% and the end value is 10%. The final full set of percentages would be 40%, 30%, 20%, 10%. ( Total of 100% )

Answer 1

It sounds like you are looking for an arithmetic progression with the specified starting and ending values and summing to $100$. Let the starting number be $s$ and the ending number be $e$. The number of pairs of terms is then $\frac {100}{s+e}$. If this doesn't come out an integer or half integer you are stuck. $n=2\frac {100}{s+e}$ is the number of terms. Then the difference between the terms is $\frac {e-s}{n-1}$ as there are $n-1$ spaces between the terms. In your example, $s=17, e=3, n=10$ and the term difference is $-\frac {14}{9}$

Answer 2

Call $M$ the max of your list and $m$ the min, say $n=100/(M+m)$. Your step size will then be $p=(M-m)/(2n-1)$. This will only work if $2n$ is integer.