In a 500 m race, P and Q have speeds in the ratio of 3 : 4. Q starts the race when P has already covered 140 m. What is the distance between P and Q (in m) when P wins the race?

The correct option is 20.

Let the speed of P be $3v$ and the speed of Q be $4v$ (where $v$ is a constant multiplier).
The total distance of the race is $500\text{m}$.

Q starts running when P has already covered $140\text{m}$. Therefore, the remaining distance P needs to run to finish and win the race is:
$$\text{Remaining distance of P}=500-140=360\text{m}$$

The time ($t$) taken by P to run this remaining $360\text{m}$ is:
$$t=\frac{360}{3v}=\frac{120}{v}$$

During this same time interval $t$, Q runs from the starting point. The distance covered by Q is:
$$\text{Distance covered by Q}=\text{Speed of Q}\times t$$
$$\text{Distance covered by Q}=4v\times \frac{120}{v}=480\text{m}$$

When P wins (reaching the $500\text{m}$ mark), Q has covered $480\text{m}$ from the start.
The distance between P and Q at this moment is:
$$\text{Distance}=500-480=20\text{m}$$