Runners A and B run around a circular track. In a four-lap race, A finishes one full lap ahead of B, and this one-lap lead is also equal to a time advantage of 10 minutes. If the race had been only one lap, what time (in minutes) would A take to complete that single lap?

Introduction / Context:
This problem is a race and relative speed question on a circular track. Two runners complete four laps, and A finishes exactly one lap ahead of B. The problem also states that this one lap lead corresponds to a time difference of 10 minutes. We are asked to find how long A would take to complete a single lap if the race were only one lap long.

Given Data / Assumptions:

- Total race distance = 4 laps of the circular track.
- Runner A completes 4 laps in time TA hours (or minutes).
- Runner B completes 4 laps in time TB hours, with TB - TA = 10 minutes.
- At the moment A finishes 4 laps, B has completed only 3 laps, so A beats B by one lap.
- Speeds of both runners are constant.

Concept / Approach:
If A finishes 4 laps while B finishes only 3 laps in the same time, the ratio of their speeds equals the ratio of distances covered, that is 4 : 3. Then we express their times for 4 laps in terms of a common lap length and these speeds. The given information that the difference between their total times for 4 laps is 10 minutes allows us to find A's time for one lap.

Step-by-Step Solution:
Let length of one lap be L units. Let A's speed be vA and B's speed be vB. When A finishes 4 laps, A covers 4L and B covers 3L in the same time. So vA : vB = 4 : 3. Time taken by A for 4 laps = TA = 4L / vA. Time taken by B for 4 laps = TB = 4L / vB. Since vB = 3 vA / 4 from the ratio, TB = 4L / (3 vA / 4) = 16L / (3 vA). Difference in their times for 4 laps is TB - TA = 10 minutes. Compute TA first: TA = 4L / vA. Now TB - TA = 16L / (3 vA) - 4L / vA. Take common denominator 3 vA: TB - TA = (16L - 12L) / (3 vA) = 4L / (3 vA). This 4L / (3 vA) is given as 10 minutes. So 4L / (3 vA) = 10 minutes. Time taken by A for one lap = tA1 = L / vA. Multiply both sides of 4L / (3 vA) = 10 by 3 / 4: L / vA = 10 * 3 / 4 = 7.5 minutes.

Verification / Alternative check:
If A takes 7.5 minutes per lap, then A needs 4 * 7.5 = 30 minutes for 4 laps. The time difference of 10 minutes implies B needs 40 minutes for 4 laps, so B's per lap time is 10 minutes. In 30 minutes (when A has run 4 laps), B would have run 3 laps (3 * 10 = 30 minutes), matching the statement that A beats B by one lap. This confirms that 7.5 minutes for one lap is consistent.

Why Other Options Are Wrong:
8 min and 12 min: These values do not produce the correct combination of a one lap lead and a 10 minute time difference when tested over 4 laps.
12.5 min: This would make A significantly slower, which contradicts the condition that A beats B by one lap and by 10 minutes in a 4 lap race.

Common Pitfalls:
Learners sometimes use the 10 minute difference as the time for one lap directly without relating it to the speed ratio. Another error is to assume that the time difference for one lap is constant instead of being derived from the total 4 lap race. Always start from distance and speed ratios, then use the time difference over the full race to compute the per lap time.

Final Answer:
If the race were only one lap, runner A would take 7.5 minutes to complete the course.