The formula A/BA

Hi! Professor PoShenLoh talked about how the formula A/BA can apply to every race and catchup question, could you explain why this formula works in every question? And what are some other questions that might use this formula?

Hey @victorioussheep!
Here's the thing: try solving the exact same problem, but instead of the first runner running 1 lap in 57 seconds, let's say he runs it in A seconds. Then, the second runner will run a lap in B seconds. Try to see if you can figure out (in terms of A & B) how many laps the first runner has run when he catches up to the second runner.
If you can figure it out, then great job!! this is great because it means you truly understand what's going on! (answer should be A/(BA))
Here's the thing: memorizing formulas isn't really that useful a lot of the time, unless you REALLY need the time. To be honest, once you actually understand how to solve something, most of the time, the difference in time between memorizing a formula and actually solving it again isn't more than ~1 minute. Although I do agree, sometimes it is more convenient to memorize a formula (like quadratic formula & others). Also, once you know how to solve it, you'll know when to use the formula. From what I see, you can only use it on a circular track with constant speeds because the first runner needs to loop back in order to catch up to the second person.Phew, that was a mouthful. Hope some of that is useful to you!

@quacker88 OMG, thank you for taking the time to answer my question, I truly appreciate it!

@victorioussheep @quacker88 Great answer!!

@quacker88 Hi! Thanks for the helpful explanation because I had the same question victorioussheep did. It's just when I tried to solve the problem you gave, I got the answer B/(BA).
 (1/A1/B)t=1
 Simplifying (1/A1/B), I get (BA/AB)t=1 and so t= AB/BA.
 t is the time it takes for the first runner to catch up with the second runner. So the number of laps the first runner runs is AB/BA divided by A which is B/BA.
Also, I think how Prof. Loh got A/AB was by solving for the number of laps the trainer or the second runner was running.
I don't know if this is correct, but I hope you can look over my work. Thanks!