In this problem, we are given that:

• P beats Q by 120 meters or 30 seconds in a 1 km (1000 meters) race.
• We need to find the time taken by P to cover the entire 1 km race.

Let's assume the time taken by P to cover the race is $t_P$ seconds.

Step-by-Step Solution:

1. Let the speed of P be $v_P$ meters per second. $v_P = \frac{1000}{t_P}$

2. P beats Q by 120 meters or 30 seconds. This means that when P finishes the 1000 meters, Q is 120 meters behind, or equivalently, Q takes 30 more seconds to finish the race.

3. The total distance for the race is 1000 meters, and Q would cover the remaining 120 meters in 30 seconds. So, Q's speed $v_Q$ is: $v_Q = \frac{120}{30} = 4 \text{ m/s}$

4. Using Q's speed, we know that Q would take $t_Q$ seconds to finish the full 1000 meters: $t_Q = \frac{1000}{v_Q} = \frac{1000}{4} = 250 \text{ seconds}$

5. Since P beats Q by 30 seconds, the time taken by P to complete the race is: $t_P = t_Q - 30 = 250 - 30 = 220 \text{ seconds}$

Answer:

The time taken by P to cover the 1 km race is 220 seconds.

Would you like further details or clarification on any step? Here are some related questions to explore:

1. How can we calculate the speed of Q in this problem?
2. What would happen if P and Q were closer in their speeds?
3. How would the solution change if the distance of the race were different?
4. Can you derive a formula for the time difference between P and Q for any race distance?
5. What other methods could be used to solve this type of problem?

Tip: Always keep units consistent when solving problems involving time, distance, and speed!