How much does a group cost
For a group of n people, each person must pay
dollars per hour. See the following table:
| Number of Participants | Hourly cost per Person |
| 2 | $12 |
| 3 | $9.17 |
| 4 | $7.50 |
| 5 | $6.39 |
| 6 | $5.58 |
| 7 | $4.98 |
| 8 | $4.50 |
| ⋮ | ⋮ |
| 16 | $2.63 |
| ⋮ | ⋮ |
Here's a graph of h(x) as well:
Why this function?
In short: it's good for you and it's good for me.When looking for a formula to decide on the group prices, I had these key principles in mind: group rates should be much lower for students (this is all about them anyway!), and much smaller for larger groups, but I should still be able to benefit from tutoring larger groups. Naturally, each student is going to pay the same amount, so d(n) = g(n)/n for some function g(n). In the end, g(n) is the total amount of money I receive. Since I've decided that d(n) should be small for large groups, it's natural to require that
. In math-speak, this means that g(n) is sublinear, or in other words that as n gets really big, g(n) becomes "much" smaller than some increasing linear function. Some good examples of sublinear functions include constants and logarithms. Also, any two sublinear functions sum together to form a sublinear function. So a good guess would be to set 
. I chose 2 for the base of the logarithm because it's easy to work with when dealing with small integral values of n. Since the cost of a group of one person should be the same as an individual tutoring session, I derived 
. Because g(n) is divided by n in d(n) and people prefer not to pay in fractions of pennies, it makes sense to try and have n be a divisor of g(n) for some values of n. 18 is divisible by 6, which is divisible by 2 and 3, it makes sense for b to be a multiple of 6. I don't need to charge much for extra people, so b = 6 seems very reasonable. After graphing the results, I found I acheived both of my goals successfully. In short, math saves the day again!
Copyright 2010 Berkeley Churchill.