P gives Q a head-start of 10 seconds in a 600 m race, but both finish the race at the same time. Find the time taken by P to finish the race if the speed of Q is 4 m/sec.

The correct option is 140 Seconds.

Let's break down the problem step-by-step to understand why this is the correct answer:

Step 1: Understand the race parameters for Q
The total distance of the race is $600\text{m}$.
The speed of runner Q is given as $4\text{m/sec}$.

Step 2: Calculate the time taken by Q to finish the race
Using the formula for time, $\text{Time}=\frac{\text{Distance}}{\text{Speed}}$, we can find the time taken by Q:
${\text{Time}}_Q=\frac{600}{4}=150\text{seconds}$.

Step 3: Analyze the head-start condition
The problem states that P gives Q a head-start of $10\text{seconds}$. This means that Q starts running $10\text{seconds}$ before P starts.
Since both runners finish the race at the same time, the total duration for which P runs must be $10\text{seconds}$ less than the total time Q spent running.

Step 4: Calculate the time taken by P to finish the race
Let ${\text{Time}}_P$ be the time taken by P to finish the race.
${\text{Time}}_P={\text{Time}}_Q-10$
${\text{Time}}_P=150-10=140\text{seconds}$.

Thus, the time taken by P to finish the race is 140 Seconds.