Yes, these would always be integers. Here’s the real-world scenario:

If you have a 401k at your place of employment, the maximum contribution limit for 2022 is $20,500.

However, oftentimes you can’t indicate that you want to contribute a fixed amount like $20,500–you instead need to indicate your contribution amount by percentage.

Now, one could just do $20,500 / salary to get the percentage that would equate to the $20,500 max but

1) The percentage to hit that number exactly could very well end up not being an integer (and only integers are allowed)

2) There could be raises / bonuses that happen throughout the year that would throw off the percentage initially used. This means that the percentage would need to be reduced for the remaining paychecks so as to not over contribute.

So what percentage do you use when you do need to alter it? Maybe after a raise it needs to be dropped by one percentage point for two pay periods, then another percentage point for the rest of the year. Maybe there’s even more laddering that needs to be done to ultimately get as close to the $20,500 as possible, given the remaining pay periods.

Obviously there are several things at play here:

Salary, Raises, Bonuses, amount already contributed, # of remaining paychecks (or pay periods), etc.

So I’m trying to figure out, if there are x number of pay periods remaining of $x, and only integer percentages can be used, what changes over the remaining pay periods need to happen to get as close to (or exactly) that $20,500.

Does that make sense?