over the course of 18 months, you want his pay to increase to 50,000 per year.
he gets paid every other week.
18 months is 1 and 1/2 years.
there are 52 weeks in a year.
therefore there are 52 + 26 = 78 weeks in 1 and 1/2 years.
you pay him every two weeks, so there are 78 / 2 = 39 bi-weekly pay periods in a year and a half.
the increase is 150,000 - 115,000 = 35,000.
divide that by 39 to get an increase of 897.4358974 every payday.
add a payday to make it 40 paydays and the amount you give him every payday will be increased by 875 dollars.
you won't be right on, but you'll be close.
in effect, this is an arithmetic progression.
the formula is An = A1 + (n-1) * d
in this formula, A1 = 115,000, d = 875, n = 41.
the formula becomes A40 = 115,000 + 40 * 875 = 150,000
you can also make a straight line equation out of it.
the equation will be y = 875 * x + 115,000
in this particular formula, x will be equal to 40.
when that happens, the formula becomes 875 * 40 + 115,000 = 150,000
his increase should occur sometime in the 18th month.
i used excel to model the situation to see what happens.
i assumed the start date was october 1, 2020.
the results are shown below:
the problem with bi-weekly paydays is that they don't correlate to months very well.
if the payday was once a month, then you could figure 18 months exactly.
since the payday is bi-weekly, it makes more sense to count paydays that will fall somewhere between the 18th month and the 19th month.
i could have used 39 months, but then the figures per payday wouldn't be integer, so the increase would be dollars and cents with some small error at the end.
it's not terrible, but 40 paydays made the increase an integer which was much cleaner.
the actual number of months could be counted as shown below:
40 paydays is equal to 80 weeks.
since there are 52 weeks in a year, then the number of years is 80 / 52 = 1.538461538.
multiply that by 12 months in a year to get 18.46153246 months.
that's some time between the 18th month and the 19th month, april 14t5h to be exact, assuming the start date was october 1, 2020.
not sure if this is what you're looking for.
my attempt was to make the problem realistic and to make the increase each payday an integer.
hopefully this will satisfy your requirements.
if not, let me know what you would really want and i'll modify my response accordingly.