In a 1 kilometre race, runner A beats runner B by 28 metres, and B takes 7 seconds to cover this remaining 28 metres. What is A's total time to complete the 1 km race?

Introduction / Context:
This problem involves a race where one runner finishes ahead of another by a certain distance, and you know how long the slower runner takes to cover that remaining distance. From this information, you can determine the slower runner's speed and then find the faster runner's time for the entire race. The question tests your understanding of speed, distance, and time relationships and how to interpret race conditions carefully.

Given Data / Assumptions:

• Total race distance = 1 km = 1000 metres.
• A beats B by 28 metres.
• This means when A finishes 1000 m, B has covered 972 m.
• B takes 7 seconds to cover the remaining 28 m.
• Speeds are assumed constant throughout the race.
• We are asked for A's time to complete 1000 m.

Concept / Approach:
Key ideas:

• First find B's speed from the information about the last 28 metres.
• Use B's speed to find the time B would take to run the full 1000 metres.
• Use the ratio of distances at the instant A finishes to relate A's and B's times.
• At the moment A finishes, B has covered only 972 m, so B's time to run 972 m equals A's time to run 1000 m.

Step-by-Step Solution:
Step 1: B covers 28 m in 7 seconds. Step 2: So B's speed v_B = distance / time = 28 / 7 = 4 m/s. Step 3: Time taken by B to run the entire 1000 m is T_B_total = 1000 / 4 = 250 seconds. Step 4: At the instant A finishes the race, B has covered only 972 m. Step 5: Time taken by B to cover 972 m is T_B_972 = 972 / 4 = 243 seconds. Step 6: This time T_B_972 is equal to A's time to cover 1000 m, since they started together and at that instant A finishes. Step 7: Therefore, A's time T_A = 243 seconds. Step 8: Convert 243 seconds to minutes and seconds: 243 / 60 = 4 minutes with a remainder of 3 seconds. Step 9: So A's time is 4 minutes 3 seconds.

Verification / Alternative check:
We can confirm consistency:

• A covers 1000 m in 243 seconds, so A's speed v_A = 1000 / 243 ≈ 4.115 m/s.
• B's speed is 4 m/s.
• In 243 seconds, B runs distance = v_B * time = 4 * 243 = 972 m.
• Difference between A and B at that instant = 1000 - 972 = 28 m.

This matches the given condition that A beats B by 28 m, confirming that the computed time is correct.

Why Other Options Are Wrong:
5min,4sec, 2min,3sec, and 3min,4sec do not satisfy the relationship between speeds and distances. If you plug these times into speed calculations, B's position at A's finish does not differ by exactly 28 m, so these times cannot be correct. Only 4min,3sec yields the correct 28 m lead at the finish line.

Common Pitfalls:
Some students misinterpret "A beats B by 28 metres in 7 seconds" as meaning A takes 7 seconds to run 28 metres more, which is not what the question says. The phrase actually means B needs 7 more seconds to cover the 28 metre gap after A has already finished. Another common error is to mis-handle the conversion between metres and kilometres or between seconds and minutes. Always convert units properly and carefully interpret who reaches which point when.

Final Answer:
Runner A completes the 1 kilometre race in 4 minutes 3 seconds.