A Difficult Clock Question?

Question

Ok, just something that I have wondered for a while now. At what times of the day are the minute and hour hands EXACTLY across from each other?? The obvious one is 6 o'clock, but what are the other 11?? And how would you mathematically go about figuring this out without the use of Trial and Error??

Answer 1

i guess: 1:35 , 10:20, 5:55 , 11:25, 7:05 , 6:00 , 12:30 and the others

ooooo soooo confusing?!?

ahahaha i hope you know i drew the clock out so maybe i could be wrong = [

Answer 2

Six o'clock is really the only time the hands are "exactly opposite each other", if you think about it; the way the mechanism of a mechanical clock works, the only hand that really "sweeps" and actually has an infinite number of positions (or angles) is the second hand.  All the other hand positions are governed by gears that "step" rather than sweep....

Answer 3

First, notice that there are 11 such times: 6 o'clock; about 7:05; about 8:11; about 9:16; about 10:22; about 11:27; about 12:33; about 1:38; about 2:44; about 3:49; and about 4:55. (You might want to run through them on a watch or clock to check that these times are the only ones.)

There are only 11 such times, not 12, because every 12 hours, the hour hand also makes a revolution, so the minute hand only passes the hour hand 11 times. (Again, check if you don't believe me.)

Whatever the times are, they are evenly spaced, by the symmetry of the clock face.

In 12 hours, there are 12 * 60 = 720 minutes.

So the times will be every 720 / 11 minutes, or every 65 5/11 minutes.

That is, between any two such times there is an interval of 1 hour, 5 minutes, and 5/11 of a minute.

So the times are exactly:

6:00;

7:05 and 5/11 minute;

8:10 and 10/11 minute;

9:16 and 4/11 minute;

10:21 and 9/11 minute;

11:27 and 3/11 minute;

12:32 and 8/11 minute;

1:38 and 2/11 minute;

2:43 and 7/11 minute;

3:49 and 1/11 minute;

4:54 and 6/11 minute.