To anyone who is struggling, here's how to get started...
Suppose the teeth counts are a, b and c. We know that b=a+c.
So that's tooth ratios of a : (a+c) : c.
Those are the only two variables you need; you don't need to introduce any more.
Once you get the logic set up, it unfolds beautifully to give the solution, and it's all elementary algebra.
You're right. I definitely did it the hard way.
: It's mindless now, clicking towards oblivion. ... How long, K-9?
: INSUFFICIENT DATA.
: Yeah, you never fucking know the answer when it's important.