Ratio of Time reuqired

nikhilgmat31 wrote:A runner starts running on a circular path of radius 'r' meters. His average speed (in meters/min) is �r during the first 30sec, �r/2 during the next one minute, �r/4 during the next 2 min, �r/8 during the next 4 min and so on... What is the ratio of time for the nth round to that for the n-1 round?

a) 4
b) 8
c) 16
d) 32
e) 64

One lap = 2πr meters.
Rate for the first 30 seconds = πr meters per minute = 1/2 lap per minute.

We can plug in ANY DISTANCE AND RATE such that the runner travels at a rate of 1/2 lap per minute for the first 30 seconds.
Let one lap = 16 meters and the rate for the first 30 seconds = 8 meters per minute.
After the first 30 seconds, the time for each interval DOUBLES, while the rate is REDUCED BY 1/2.

First 30 seconds:
At a rate of 8 meters per minute, the distance traveled in the first 1/2 minute = (8)(1/2) = 4 meters.
Next minute:
At a rate of 4 meters per minute, the distance traveled over the next minute = (4)(1) = 4 meters.
Next 2 minutes:
At a rate of 2 meters per minute, the distance traveled over the next 2 minutes = (2)(2) = 4 meters.
Next 4 minutes:
At a rate of 1 meter per minute, the distance traveled over the next 4 minutes = (1)(4) = 4 meters.
After 4 intervals, the total distance traveled = 4+4+4+4 = 16 meters = 1 lap.
Total time to complete the first lap over the first 4 intervals = 1/2 + 1 + 2 + 4 = 7.5 minutes.

Notice that the distance traveled in each interval is THE SAME:
4 meters.
Since 4 meters = 1/4 lap -- implying that 1/4 lap is completed over the course of each interval -- another lap will be completed after 4 more intervals.
Since the time for each interval keeps doubling, the total time to complete the second lap over the next 4 intervals = 8+16+32+64 = 120 minutes.

Resulting ratio:
(total time for the second lap)/(total time for the first lap) = 120/7.5 = 240/15 = 16.

The correct answer is C.