Two people meeting on a track

Question

We have two people on a circular track where one lap is $d$ meters. Two runners, who run at rates $r_1$ and $r_2$, start running. At what time will they meet up again? These numbers may be decimal.

I tried representing this as $r_1 \cdot t \equiv r_2 \cdot t \pmod{d}$ but could not get anywhere from here. Since the numbers may be decimal I don't know if we can use modular arithmetic methods, or if it even makes sense if $t,d$ aren't coprime.

I tried re-representing it as $r_1 \cdot t = r_2 \cdot t + jd$ but now this simply introduces another unknown variable.

Solving for $t$ this is $t = \frac{jd}{r_1 - r_2}$. Like is it enough to just pick $j=0, 1, 2, 3, 4,...$ and these are the meeting points in time?

Answer 1

Yes. They will meet for each of your values of $j$. The first time they will meet after starting will be when $j=1$ (so the faster has exceeded the slower by $1$ full lap).

Note: You have set things up so that $r_1$ should be greater than $r_2$. [You could also say they will meet after time $\frac{d}{\mid r_1-r_2\mid}$, and then either one could be faster. But they still had better not go at the exact same speed.]

Answer 2

Another way to interpret your answer and see that it is correct is that the distance between the two runners increases at a rate of $r_1-r_2$ (or possibly $|r_1-r_2|$, if you're so inclined). Thus the time it takes for this distance to correspond to exactly $j$ laps is $\frac{jd}{r_1-r_2}$