Calculating the positions of the hands?

Question

Due to a defect the little hand begins to go the wrong way in the middle of the night at 0 o'clock (midnight). The big hand, however, continues movig in the right direction. Calculate now what times the hands of this clock will lie exactly on top of each other.

Answer 1

Consider what happens every minute. In 60 minutes the big hand moves 360 degrees, so in 1 minute it moves 6 degrees clockwise. In 60 minutes the small hand moves 360/12 = 30 degrees, so in 1 minute it moves 0.5 degrees anticlockwise.
This means the hands are getting closer together at a rate of 6+0.5 = 6.5 degrees per minute.
So they will meet every 360/6.5 = 55.3846 minutes, approximately. So thats 00:55:23, 01:50:46, etc.

[edit] nwordjim has it almost right below me, but for some reason he thinks a clock has 24 hours on it :) If you replace '24' by '12' everywhere, you end up with the same answer as me.

Answer 2

They will lie exactly on top of each other 11 times every 12 hours

But the small hand going backwards changes the times showing on the clock

So time showing (in order) are:

(12 - 1 1/11)h = 10:54:33+,

(12 - 2 2/11)h = 9:49:06+

(12 - 3 3/11h = 8:43:39+

(12 - 4 4/11h = 7:38:11+

(12 - 5 5/11)h = 6:32:44+

(12 - 6 6/11)h = 5:27:17+

(12 - 7 7/11)h = 4:21:50+

(12 - 8 8/11)h = 3:16:23+

(12 - 9 9/11)h = 2:10:55+

(12 - 10 10/11)h = 1:05:28+

(12 - 11 11/11h = 12:00:00

Answer 3

Simplest way of doing it: Say the circle has a circumference of 1 (in arbitrary units), and the hour hand moves a distance x before they meet. Then the minute hand moves distance 1-x at 24 times the speed.
So (1-x)/24 = x
So 1-x = 24x
x= 1/25.

Since the hour hand moves one revolution in 24 hours, 1/25 of a revolution means they will meet after 1/25 of 24 hours = 57.6 minutes.

The next time they meet will be be 57.6 minutes later (because it's the same situation if you don't look at the numbers and you turn your neck. So they will meet at:
00 57 and 36 seconds
01 55 and 12 seconds
02 52 and 48 seconds
03 50 and 24 seconds
04 48 precisely
05 45 and 36 seconds
06 43 and 12 seconds