How many times in a day do the hour hand and minute hand point in the same direction?

Answer 1

I just did some tallying in my head, with pencil and paper handy. In a 24 hour day, I count 44 times.

Answer

The continuous movement of the hour hand brings about something analogous to the "sidereal day" problem. Most of the time, there are 2 occurances per hour of 90 degrees between the hands. But after 2 PM, for example, there is an occurance before 2:30, and the next occurance is 3 PM. There is then one occurance after 3 PM, the next one being after 4 PM. The same thing happens again at 8 and 9. So it seems that a total of 44 is probably right. Another way to 'visualize' that 'twice per hour' can't always work is to see that occurances of 90 degrees must be farther apart than 30 minutes, because both hands are advancing, not just the minute hand. For some hours, the first occurance will be late enough in the hour that the next occurance is in the next hour.

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Let me show you a mathematical approach.
Common sense dictates that the minute hand moves at a faster rate of 5.5 degrees a minute (because the hour hand moves 0.5 degrees a min and the minute hand moves 6 degrees a minute).

We start at 12 midnight. The hands are together. For subsequent 90 degree angles to occur, the minute hand must "overtake" the hour hand by
90 degrees, then 270 degrees, then 360 + 90 degrees, then 360 + 270 degrees, then 360 + 360 +90 degrees.. and so on.
This can be re-expressed as: (1)90, 3(90), 5(90), 7(90), 9(90), 11(90)... n(90).
The number of minutes this takes to happen can be expressed as (1)90/5.5, 3(90)/5.5, 5(90)/5.5, 7(90)/5.5, 9(90)/5.5, 11(90)/5.5... n(90)/5.5.

In one day, there are 24 hr * 60 mins = 1440mins
To find the maximum value of n,
n(90)/5.5 = 1440
n = 88

but as seen from above, n must be an odd number (by pattern recognition and logic)
hence n must be the next smallest odd number (87)
counting 1,3,5,7,9,11......87, we see that the number of terms = (87-1)/2 +1 = 44.

In other words, the minute hand "overtakes" the hour hand on 44 occasions in 24 hours in order to give a 90 degree angle.

Therefore the answer to your question is 44.

Answer 2

22 times in a 24 hour day. At midnight (the start of the day) the minute hand and hour hand are both pointing at 12. It will happen every 65 minutes 27.27 seconds. Here are the times:

1. 12:00:00 midnight
2. 1:05:27
3. 2:10:55
4. 3:16:22
5. 4:21:49
6. 5:27:16
7. 6:32:44
8. 7:38:11
9. 8:43:38
10. 9:49:05
11. 10:54:33

That's 11 times in the morning, then at 12 noon, the cycle repeats, so 22 times.

Answer 3

This happens every 32 minutes, 43.64 seconds, starting at 12:16:22.

Here are the times:

12:16:22, 12:49:05, 01:21:49, 01:54:33, 02:27:16, 03:00:00, 03:32:44, 04:05:27, 04:38:11, 05:10:55, 05:43:38, 06:16:22, 06:49:05, 07:21:49, 07:54:33, 08:27:16, 09:00:00, 09:32:44, 10:05:27, 10:38:11, 11:10:55, 11:43:38. This is for a 12 hour clock, then it would repeat for the PM times, so 44 times each day.

See the question, "What are the times at which the hour and minute hands of a clock are coincident at right angles or opposite?"

Answer 4

On a normal 12-hour clock, the minute hand moves thru 360° in 1 hour, 360° in 60 minutes, or 6° every minute.

On a normal 12-hour clock, the hour hand moves thru 360° in 12 hours, 360° in 720 minutes, or ½° every minute. Beginning at 0:00:00, when the hands are coincident (!), the number of minutes (x) needed for the minute hand to get 90 degrees ahead of the hour hand is found in:

6x - 0.5x = 90

x = 16.363636 minutes or 16:21.818181 minutes:seconds Most importantly, every 90 degree change in the relationship between the two hands will require this same amount of time to transpire.

Beginning with any coincidence (!), it takes 16.36 minutes for the minute hand to get 90 degrees ahead of the hour hand (>), in another 16.36 minutes the minute hand will be 180 degrees ahead (or behind) of the hour hand or directly opposite the hour hand (-). In another 16.36 minutes, the minute hand will be 90 degrees behind the hour hand (<) and yet another 16.36 minutes will find the hands coincident (!) again. There are exactly 88 such 16.36 minute intervals in 24 hours and the times are listed below.

0:00:00 !... ..12:00:00 !

0:16:22 >.. ..12:16:22 >

0:32:44 - .....12:32:44 -

0:49:05 < ....12:49:05 <

1:05:27 ! .....13:05:27 !

1:21:49 > ....13:21:49 >

1:38:11 - .....13:38:11 -

1:54:33 < ....13:54:33 <

2:10:55 ! .....14:10:55 !

2:27:16 > ....14:27:16 >

2:43:38 - .....14:43:38 -

3:00:00 < ....15:00:00 <

3:16:22 ! .....15:16:22 !

3:32:44 > ....15:32:44 >

3:49:05 - .....15:49:05 -

4:05:27 < ....16:05:27 <

4:21:49 ! .....16:21:49 !

4:38:11 > ....16:38:11 >

4:54:33 - .....16:54:33 -

5:10:55 < ....17:10:55 <

5:27:16 ! .....17:27:16 !

5:43:38 > ....17:43:38 >

6:00:00 - .....18:00:00 -

6:16:22 < ....18:16:22 <

6:32:44 ! .....18:32:44 !

6:49:05 > ....18:49:05 >

7:05:27 - .....19:05:27 -

7:21:49 < ....19:21:49 <

7:38:11 ! .....19:38:11 !

7:54:33 > ....19:54:33 >

8:10:55 - .....20:10:55 -

8:27:16 < ....20:27:16 <

8:43:38 ! .....20:43:38 !

9:00:00 > ....21:00:00 >

9:16:22 - .....21:16:22 -

9:32:44 < ....21:32:44 <

9:49:05 ! .....21:49:05 !

10:05:27 > ..22:05:27 >

10:21:49 - ...22:21:49 -

10:38:11 < ..22:38:11 <

10:54:33 ! ...22:54:33 !

11:10:55 > ..23:10:55 >

11:27:16 - ...23:27:16 -

11:43:38 < ..23:43:38 <

Answer 5

12.30; 1.35; 2.40; 3.45; 4.50; 5.55; 6.00; 7.05; 8.10; 9.15; 10. 20; 11.25 - AM

12.30; 1.35; 2.40; 3.45; 4.50; 5.55; 6.00; 7.05; 8.10; 9.15; 10. 20; 11.25 - PM

Answer: 24

The answer, 24, is correct; however, the times stated are not correct, they are only approximate. For example, 12.30 is close, but the correct time is 12:32.44. The hour hand moves while the minute hand moves. In the list above, 6:00.00 is the only exactly correct time stated. The rest are approximate.

Answer 6

I'm pretty sure the answer is 44 times (22 times per revolution of the hour hand).

1. 00:16
2. 00:49
3. 01:21
4. 01:54
5. 02:27
6. 03:00
7. 03:32
8. 04:05
9. 04:38
10. 05:11
11. 05:43
12. 06:16
13. 06:49
14. 07:21
15. 07:54
16. 08:27
17. 09:00
18. 09:32
19. 10:05
20. 10:38
21. 11:11
22. 11:43
23. 12:16
24. 12:49
25. 13:21
26. 13:54
27. 14:27
28. 15:00
29. 15:32
30. 16:05
31. 16:38
32. 17:11
33. 17:43
34. 18:16
35. 18:49
36. 19:21
37. 19:54
38. 20:27
39. 21:00
40. 21:32
41. 22:05
42. 22:38
43. 23:11
44. 23:43

(confirmed)

Answer 7

44 times in one day. At first, it seems like 48 times would be the answer. Each time where the minute hand is before the hour hand, then after the hour hand. Starting at approx. 12:05 AM, 1:00 AM, approx 1:11 AM, approx 2:06 AM etc.

But the hour and minute hand are moving relative to one another at a speed of 5.5° per minute, because the minute hand moves at 6° per minute (360° / 60 minutes per full circle), while the hour hand moves 0.5° per minute (360° / 720 minutes per full circle). So it happens almost twice per hour. See the related link for more information.

Answer 8

a right angle is 90 degrees so it would form twice between one and two oclock... once at 1:15 and once at 1:45.

Answer 9

18

Answer 10

22