"I mean, there's the problem of figuring out that taking out 0.17 twos from a bag works out to multiplying the contents by 8/9, but you can get that fairly precisely off nineteen twos being a bit less than twelve threes.  Possible self-study problem: rederive that yourself, convince yourself of it, prove it, without looking back at the whiteboard."

"I mean, there's the problem of figuring out that taking out 0.17 twos from a bag works out to multiplying the contents by roughly 8/9, but you can get that fairly precisely off nineteen twos being a bit less than twelve threes.  Possible self-study problem: rederive that yourself, convince yourself of it, prove it, without looking back at the whiteboard."