The question is: There is a race on a circular track of length $0.5$ km. The race is $4$ km. Two people start at the same point with speeds $20$ m/s and $10$ m/s respectively. Find the distance covered by the first person when they meet the second person for the third time.
My formulation of the question was, $20t\equiv15t \pmod{0.5}$, where $t$ is time elapsed. However, I have no idea how to solve these type of equations.
Once you correct $.5 km = 500m$
We have $20t \equiv 15t \pmod {500}$
So $5t \equiv 0 \pmod {500}$
When confused. Go back to something less confusing.
$5t\equiv 0 \pmod {500}$
so there is an integer $k$ so that
$5t = 500k$
$t = 100k$
$t$ being any multiple of $100$ seconds will do.
For example: After $100$ seconds, runner 1 will have ran $2000 m$ or or $2 km$ or $4$ laps. Will runner 2 will have run $1500 m$ or $1.5 km$ or $3$ laps.