A $10\text{km}$ race is organized on an $800\text{m}$ circular racecourse. $P$ and $Q$ are contestants. The ratio of their speeds is $5:4$, and the difference between their speeds is $1\text{m/s}$. How many times will the winner overtake the loser?

Let the speeds be $5k$ and $4k$.Difference $=5k-4k=1⟹k=1$.Speed of $P=5\text{m/s}$, Speed of $Q=4\text{m/s}$.$P$ is the winner. Distance $=10\text{km}=10,000\text{m}$.Time taken by $P$ to finish $=10,000/5=2,000\text{s}$.Relative speed (same direction) $=5-4=1\text{m/s}$.Overtaking occurs when the faster person covers one full track length ($800\text{m}$) more than the slower person.Time to first overtake $=\text{Track Length}/\text{Relative Speed}=800/1=800\text{s}$.Subsequent overtakes occur at $1,600\text{s}$, $2,400\text{s}$, etc.Since the race ends at $2,000\text{s}$, $P$ overtakes $Q$ at $800\text{s}$ and $1,600\text{s}$.Total times $=2$.