two objects moving in opposite directions.

Question

I don't need a specific answer for this question, and would rather prefer to know how to solve questions like this one. So far I've tried using the $v=d/t$ formula to form equations, but haven't managed to arrive at a solution.

Two joggers each run at their own constant speed and in opposite directions from each other around an oval track. They meet every 36 seconds. The first jogger completes one lap of the track in a time that, when measured in seconds, is a number (not necessarily an integer) between 80 and 100. The second jogger completes one lap in a time, $t$ seconds, where $t$ is a positive integer. What is the product of the smallest and largest possible values of $t$?

Answer 1

What I did was I drew the oval track and then tracked each runner on different directions. It says hey meet every 36 seconds but runner 1 takes from 80s to 100s so first I assume runner 1 does it at 80s, so half of it is 40 seconds. That tells me that by the time they meet (36s) runner 1 hasn't reached the half, so that means runner 2 is faster than runner 1 (since the other end of runner 1's time is 100).

That takes me to say

t1min = 80s and t2min = 80s-x 

then similarly

t1max = 100s and t2max = 100s-x

So the product of t2min and t2max (t2 is t from the problem) is:

-x^2 - 180x + 8000

Answer 2

Say the first jogger takes $t'$ seconds to complete the lap, and the length of the lap is $d$ metres. Then, you get speeds of both the joggers, say $v_1$ and $v_2$. Take relative velocity. The relative velocity in this case would be $v_1+v_2$, since they're both running towards each other.

So, we are assuming that instead of both the joggers running simultaneously, jogger $1$ is stationary and jogger $2$ is running with a speed $v_1+v_2$.

Since they meet after $36$ seconds, it means that jogger $2$ covers a distance equal to one lap, that is, $d$ metres, in $36$ seconds with a speed of $v_1+v_2$. So you can write $v_1+v_2=\cfrac{d}{36}$. The term $d$ cancels out and you get an equation in terms of $t$ and $t'$. Use the values of $t'$ and see for which values, you get integer value for $t$.