Two runners start running laps at the same time, from the starting position. George runs a lap in $50$ seconds; Sue runs a lap in $30$ seconds. When will the runners next be side by side?
George's speed: $\dfrac{1}{50}$ lap per second
Sue's speed: $\dfrac{1}{30}$ lap per second
We are looking for the time $t$ when they next meet.
How many laps George has run after time $t$: $\dfrac{t}{50}$ laps
How many laps Sue has run after time $t$: $\dfrac{t}{30}$ laps
This is where I get confused. The solution says: They will next be even when Sue has run exactly one more lap, that is, when $\dfrac{t}{30}-\dfrac{t}{50}=1$.
What is the justification to say they will be next to each other when Sue has run exactly one more lap? To me this just came out of the blue.