Formula for Contractor's Hourly Rate

Assume the number of work hours per year is 12 (months) x 4 (weeks per month -- for the purposes of this problem, assume 4 weeks per month) x 40 (hours per week) = 1920 hours per year.

Furthermore, assume a goal of incentivizing companies to hire a contractor for as many possible hours per week for as many possible months.

If a contractor is hired for 12 months at 40 hours a week, a multiplier of 1 is applied to a set hourly rate (e.g. 100 per hour x 1 = $100). The minimum number of months a contractor can be hired is 1 month at 5 hours a week. The multiplier for that is 2 (e.g. 100 per hour x 2 =$200).

What formula can be used to compute this multiplier? What should the multiplier be if the contractor is hired for 6 months at 20 hours a week? Or for 3 months at 15 hours a week?

Trying to find a formula that would apply to 1 (month) and 5 (hours a week) to equal 2 (multiplier) and use that same formula that would apply to 12 (months) and 40 (hours a week) to equal 1 (multiplier).

UPDATE: I think 25 hours (halfway between 10 and 40) at 7 months (halfway between 1 and 12) would probably yield a 1.5 multiplier.

• These multipliers seems rather arbitrary and I cannot immediately see a reason for the multiplier of 2 in the case that you provided ... Are these numbers taken from a real company? Or is this a math book example? – Matti P. Sep 5 '18 at 13:19
• Not from a real company. Not a math book example. – MathNewbie Sep 5 '18 at 13:20
• how many hours would be in a month if contractor is working 5 hours a week? Should it be 20? A month is not exactly 4 weeks. – Vasya Sep 5 '18 at 13:21
• Do you know why this multiplier is added? – Matti P. Sep 5 '18 at 13:21
• The multiplier seems to be chosen at will. Is the goal of the multiplier to assess the time? Since it seems you have a fixed rate of 100 dollars per hour. How do you determine your multiplier of 2? – Jan Sep 5 '18 at 13:21

Assuming you want a linear relationship between hours worked and multiplier, you have two data points: $(20,2)$ and $(480,1)$. The line that goes thru these two points is defined by this equation: $m=-h/460+2\frac{1}{23}$ where $m$ is multiplier and $h$ is the number of hours.
• And how did you come up with $2\frac{1}{23}$? – MathNewbie Sep 5 '18 at 14:48
• @MathNewbie: we cannot mix months and hours, we need to use the same units. What I did is I calculated equation of a line that goes thru 2 data points. See here: mathworld.wolfram.com/Two-PointForm.html Plug in $h=20$ and you'll get $m=2$, plug in $h=480$ and you'll get $m=1$ – Vasya Sep 5 '18 at 14:53