In a $500\text{m}$ race, the ratio of the speeds of $P$ and $Q$ is $3:4$. When the race starts, $P$ is $140\text{m}$ ahead. What is the distance between $P$ and $Q$ when $P$ wins the race?

Total race distance $=500\text{m}$.$P$ starts $140\text{m}$ ahead, so $P$ needs to cover $500-140=360\text{m}$ to reach the finish line and win.The ratio of speeds of $P$ and $Q$ is $V_P:V_Q=3:4$.Since time is constant for both until $P$ wins:
$$\frac{\text{Distance covered by }P}{\text{Distance covered by }Q}=\frac{V_P}{V_Q}=\frac{3}{4}$$
$$\frac{360}{{\text{Distance}}_Q}=\frac{3}{4}$$
$${\text{Distance}}_Q=\frac{360\times 4}{3}=480\text{m}$$
When $P$ reaches the finish line ($500\text{m}$ mark), $Q$ has covered $480\text{m}$ from the starting point.Distance between $P$ and $Q=500-480=20\text{m}$.