rblog

Now that Emily is 9, it is time to start an allowance. Of course, the question is "how much"? And "how much" needs to be an answer that automatically changes as Emily ages. So, figuring that increases should be small initially and larger as she reaches her mid-teens, I proposed the Allowance function A(n) where n is age in years.

My proposal was:
A(n) = (n-8)^2/2 + 2
So, when Emily is 9, this results in $2.50. At 10, it becomes $4.00 and at 11, we are looking at $6.50.

Molly had a different proposal:
Start at $2.50. Then, each year, add one more dollar for each of the years you have been receiving an allowance. So, at age 10, it would be $3.50 (goes up by 1 dollar for the one year of receiving an allowance), then at age 11 it would be $5.50 (2 more dollars, since she had an allowance for two years).
For the mathematically inclined, it is not difficult to see that Molly's proposal works out to:
A'(n) = (n-8)*(n-9)/2 + 2.50

which is nearly identical. Emily (wisely) chose A(n).

So, in case you are looking for an allowance function, we offer this to you. Of course we neglected to mention that A(n) = 0 for n < 9 and n > 18 :).