Mistaken Time
Geometry
Level
4
Between 5 and 6 o'clock, a lady looked at her watch. She mistook the hour hand for the minute hand and vice versa. As a result, she thought the time was approximately 55 minutes earlier. Exactly how many minutes earlier was the mistaken time?
The answer is 55.385.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Suppose the current time is 5 + x o'clock, with 0 < x < 1 ( x is measured in hours). Suppose the mistaken time is 5 + x + a o'clock; we know that a ≈ − 1 1 / 1 2 .
The position of the hour and minute hand (in natural "clock" units, where one revolution is 12 units) are h = 5 + x ; m = 1 2 x . Replacing x by x + a and inverting, we find that also h = 1 2 ( x + a ) − 1 2 n ; m = 5 + x + a . Here n is an integer, representing the number of full revolutions the minute hand makes between the mistaken and correct time.
Equating, we get the equations { 1 2 x = 5 + x + a 1 2 ( x + a ) − 1 2 n = 5 + x . Subtract the top from the bottom equation to eliminate x : 1 2 a − 1 2 n = − a ∴ 1 3 a = 1 2 n ∴ a = 1 3 1 2 n . Since a is close to − 1 1 / 1 2 , we must have n = − 1 .
Thus the actual time difference is 6 0 ∣ a ∣ = 6 0 × 1 2 / 1 3 = 5 5 1 3 5 ≈ 5 5 . 3 8 5 minutes.