# Hard word problem - finding how many rounds

Two runners are running for $1$ hour in the Olympic stadium.

One of them $A$ is faster than the other one $B$.

Runner $A$ completes $50$ rounds in $1$ hour and runner $B$ completes $35$ rounds in $1$ hour.

The runners are switching between them a torch while $A$ is holding the torch at the beginning.

Every time $A$ passes $B$ they are switching the torch between them.

How many rounds the torch completed the stadium in $1$ hour?

I tried to build equations but didn't succeed.

Thank you.

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How much time has past between start and the first join of $A$ and $B$?
Let's say after $t$ hours.
After $t$ hours $50t$ rounds are done by $A$ and $35t$ rounds by $B$. So $50t=35t+1$ and $t=\frac{1}{15}$. Then after $\frac{2}{15}$ hours they join for the second time. The torch was for $\frac{1}{15}$ hours in the hands of $A$ and for $\frac{1}{15}$ hours in the hands of $B$ so there were $50\times\frac{1}{15}+35\times\frac{1}{15}=\frac{85}{15}=\frac{17}{3}$ rounds for the torch in $\frac{2}{15}$ hours. That gives $\frac{7\times17}{3}$ rounds in $\frac{7\times 2}{15}=\frac{14}{15}$ hours. Then they join for the $14$-time and the torch now goes to $A$. That results in $50\times\frac{1}{15}=\frac{10}{3}$ rounds for the torch in the last $\frac{1}{15}$ hours and we end up with $\frac{7\times17}{3}+\frac{10}{3}=43$ rounds in $1$ hour.