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 answer is 20.

Let's break down the solution step-by-step:

Step 1: Understand the speeds and total race distance.
The total distance of the race is 500 m.
The speeds of P and Q are in the ratio of 3 : 4.
Let the speed of P be $3x$ m/s and the speed of Q be $4x$ m/s.

Step 2: Determine how much distance P has left when Q starts.
Q starts running when P has already covered 140 m.
Since the total race is 500 m, the remaining distance P needs to cover to win the race is:
$\mathrm{Remaining\; distance\; for\; P}=500-140=360\text{m}$

Step 3: Calculate the time taken by P to finish the race.
Let $t$ be the time (in seconds) taken by P to cover this remaining 360 m at its speed of $3x$:
$t=\frac{360}{3x}=\frac{120}{x}\text{seconds}$

Step 4: Find the distance covered by Q in the same time.
Q starts running from the starting line at the same time P starts covering the remaining 360 m. In time $t$, the distance covered by Q at its speed of $4x$ is:
$\mathrm{Distance\; covered\; by\; Q}=\text{Speed of Q}\times t=4x\times \frac{120}{x}=480\text{m}$

Step 5: Find the distance between P and Q when P wins.
When P reaches the finish line (completing 500 m), Q has covered 480 m from the starting point.
Therefore, the distance between P and Q at this moment is:
$\mathrm{Distance}=500-480=20\text{m}$