# How many times will minute hand and hour handa) coincide in one dayb) perpendicular in one dayc) opposite in one day

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### 2 Answers

**The hands will be opposite to each other 22 times in 24 hours actually** (kindly ignore the typing mistake in the last line of my earlier answer).

Here is another method to solve the problem: The minute hand travels an angular distance of 360 degrees in 60 minutes. So, it covers 6 degrees in one minute.

The hour hand covers 2*360 degrees in 24 hours, i.e 1440 minutes. So, in one minute, it covers 1440/720 = 1/2 degree. Difference in angular distance travelled by the minute hand and the hour hand in one minute is thus 6-1/2 = 11/2 degrees. So, on a full rotation (360ᵒ), any similar event between them will be repeated every `360/(11/2)` = `65 5/11` minutes.

In twenty four hours, i.e. 24*60 = 1440 minutes, the number of times angle between the minute hand and hour hand will be same, can be calculated by dividing 1440 minutes with this value, thus n = `1440/(720/11)` = 22.

**So, a) 0ᵒ angle (event of coincidence) will be obtained = 22 times. **

**and similarly, c) 180ᵒ angle (event of opposition) will be obtained = 22 times. **

**b) 90ᵒ angle (event of perpendicular disposition) will be obtained = 22*2 = 44 times (as this situation arises twice on a complete rotation).**

a) The hands of a clock coincide 11 times in every 12 hours (Since between 11 and 1, they coincide only once, *i.e.,* at 12 o'clock).**AM**

12:00

1:05

2:11

3:16

4:22

5:27

6:33

7:38

8:44

9:49

10:55 **PM**

12:00

1:05

2:11

3:16

4:22

5:27

6:33

7:38

8:44

9:49

10:55

The hands overlap about every 65 minutes, not every 60 minutes.**Thus the minute hand and the hour hand coincide 22 times in a day.**

(b) In 12 hrs. the hands of the clock will be perpendicular to each other for 22 times as in 1 hr. time the hands of the clock wil be at rt. angles for 2 times but between 1` `1'o clock and 1'o clock they will be at rt. angles only twice.

**Therefore, in 24 hrs. they will be at rt. angles 44 times.**

(c) Eleven times in 12 hrs.. The hands will be directly opposite each other at the following times:

12:32 and 8/11 of a minute

1:38 and 2/11 of a minute

2:43 and 7/11 of a minute

3:49 and 1/11 of a minute

4:54 and 6/11 of a minute

6:00

7:05 and 5/11 of a minute

8:10 and 10/11 of a minute

9:16 anfd 4/11 of a minute

10:21 and 9/11 of a minute

11:27 and 3/11 of a minute.

**Therefore, the hands of the clock will be opposite to each other 44 times.**