A and B run a 5 km race on a 400 m circular course. Their speeds are in the ratio 5 : 4. How many times does the winner pass the other during the race?

Introduction / Context:
Laps gained on a circular track equal the relative distance advantage divided by the track length. The number of passes equals the count of complete 400 m gains; the fractional part indicates a partial, not complete, pass near the end.

Given Data / Assumptions:

• Total winner distance = 5,000 m.
• Speed ratio 5:4 ⇒ loser’s distance at winner’s finish = 5,000 * (4/5) = 4,000 m.
• Track length = 400 m.

Concept / Approach:
Relative lead at finish = 5,000 − 4,000 = 1,000 m. Full passes = 1,000 / 400 = 2.5 times (two complete passes and half a lap extra).

Step-by-Step Solution:

Verification / Alternative check:
Laps by winner = 5,000/400 = 12.5; loser laps = 4,000/400 = 10; difference = 2.5 passes.

Why Other Options Are Wrong:
Other fractional counts do not match the 1,000 m relative lead.

Common Pitfalls:
Rounding 2.5 down to 2; the question allows fractional counts of passes.

Final Answer:
2 1/2 times