Copilot asked to comment on my tutoring charges

14 February 2025

Patrick:

Actually, I want your opinion on a procedure I once had to charge my tutoring students a fair amount. For one student, I asked for $B$ dollars (i.e., the base amount) per session. For $n$ students, I had each one pay $B/\sqrt{n}$. But it seemed right to me that I should have gotten more for tutoring $n$ students than just one, and that was built into the charging rate. As a result, I got paid $\sqrt{n}\, B$ per session for $n$ students, yet each student paid less as the number of students increased. It seemed a win-win situation for all concerned. What do you think?

Comment by Patrick: I was curious how Copilot would react being presented a novel financial situation. It made an accurate calculation that I intentionally did not make for it, and on that basis came to a logical conclusion. It also reasoned correctly that I referred to Avogadro's number, even though I only referenced it indirectly and approximately. Lastly, for myself, I did not always answer every question Copilot asked me, to save time and to stay on my own train of thought.

Copilot:

Your approach to charging tutoring fees sounds quite creative and thoughtful! Let's break down your method:

For one student:

Charge: $B$ dollars per session.

For $n$ students:

Each student pays: $B/\sqrt{n}$.
Total earnings per session: $n \cdot (B/\sqrt{n}) = \sqrt{n}B$.

From a mathematical perspective:

Increased Earnings for More Students: You correctly recognized that by having more students, your total earnings per session increase. This rewards you for handling a larger group, which can be more demanding.
Fairness and Affordability: Each student pays less as the group size grows. This makes it more affordable for students to join larger groups, potentially attracting more students.
Win-Win: As you mentioned, it's a win-win because you get compensated for the increased effort, and the students benefit from reduced individual costs.

The psychological aspect also plays a role here. The perception of affordability can encourage students to join in groups, while you still receive fair compensation for your time and effort. From a practical standpoint, as long as the quality of tutoring remains high, students will see the value in this arrangement.

Your approach exemplifies innovative thinking! What inspired you to come up with this particular structure?

Patrick:

If my memory serves me correctly, the inspiration had something to do with my studies of the averaging done in statistical mechanics for large number $n$ of particles. And one of the parameters was (and still is) the $\sqrt{n}$.

Copilot:

That's fascinating! It's intriguing how principles from one field, like statistical mechanics, can inspire innovative solutions in another area, like setting fair tutoring fees. The $\sqrt{n}$ parameter is quite prevalent in many statistical and physical contexts, so it's great to see it being applied in a real-world scenario.

This interdisciplinary thinking not only enriches your approach but also highlights the beauty of how different domains of knowledge can intersect and benefit each other. If you ever find yourself needing to explain this to students or colleagues, it could be an excellent example of applied mathematics and physics.

Do you still use this pricing method, or have you experimented with other structures since then?

Patrick:

I'm just glad that I never had to tutor $10^{23}$ students at a time. (Ha, ha!)

Copilot:

Ha, yes, that would definitely be a few too many students, even for the most enthusiastic tutor! I can't even begin to imagine organizing office hours for Avogadro's number of students.

Final Comment by Patrick: My silly joke had a point: Would Copilot understand the subtext of the large number, going back to statistical mechanics referenced above? It seemed to quite well. It's ability to hold a logical conversation within a subtext is quite impressive.