- Few Basic things to understand -
The angle covered in a circle (or a dial of a clock) is 360°
As we know, the Hour hand completes one round of the dial in 12 hours. Thus, angle covered by Hour Hand in each hour is 30°
As an hour has 60 minutes, angle covered by Hour hand in a minute is 0.5°
Similarly,the Minute Hand completes one round of the dial in 1 hour (i.e. in 60 minutes).
Thus, angle covered by Minute hand in a minute is 6°
Ratio of speed of Hour Hand and Minute Hand -
Relative speed of minute hand over hour hand is (6 - 0.5) = 5.5° per minute
- Concept of Minute Spaces
As you can see in the image above, each digit on the dial of a clock corresponds to an hour. The dial of a clock is divided into 12 such hours.
Angle between each digit is 30° (360° / 12 = 30°) which is obviously same as the angle traced by Hour Hand in 1 Hour.
Angle between two consecutive digits is divided in 5 equal parts. These can be called as "Minute Spaces".
The whole face or dial of a clock is divided into 60 Minute Spaces.
Angle between consecutive Minute Spaces is 6° (30° / 5 = 6°) which is obviously same as the angle traced by Minute Hand in 1 Minute.
- How to calculate angle between Hour Hand and Minute Hand at a given time -
The generic formula to calculate angle between Hour hand and Minute Hand at any given time is -
where, M is Minutes and H is HourReflex angle - Reflex angle between hands of a clock is the larger angle. It is more than 180° but less than 360°
- Frequency on coinciding of hands -
Thus, two hands of a clock coincide 11 times in 12 hours (as between 11 and 1, hands coincide only one time i.e. at 12 o'clock).
Therefore, two hands of a clock coincide 22 times in a day.
Similarly, two hands of a clock are opposite 22 times in a day. [Opposite means angle = 180°]
Thus, two hands of a clock are in a straight line (opposite or coincide) 44 times in a day
Two hands of a clock are at any other angle (30°, 45° etc.) 44 times in a day
- Time displayed by a clock in a Mirror -
(As you can see above, the actual time is 07:41. Now, subtracting it from 12:00 we get 04:19 which is the time displayed when we see the clock in the mirror)
- Fast and Slow Clocks -
If a clock indicates 11:45 when actual time is 12:00, its said to be 15 minutes too slow
NOTE - If a clock is constantly gaining or loosing time, it'll show the correct time only when it has gained or lost exactly 12 Hours
Two hands of a clock coincide every 65(5/11) minutes [This is explained in Example-8 below]
If two hands of a clock coincide in a time interval less than 65(5/11) minutes, the clock will loose the time.
If two hands of a clock coincide in a time interval more than 65(5/11) minutes, the clock will gain the time.
The time lost/gained by a clock in a day whose hands coincide after every M minutes is given by -
Time Lost/Gained (in minutes) in a day -
Example 1 -
Calculate the angle between Hour Hand and Minute Hand of a clock at 03:00
Solution -
At 03:00, the Hour Hand is at 3 and the minute Hand is at 12. So, both are separated by 15 Minute Spaces.
Thus, angle between them is - 15 * 6° = 90°
[Angle between consecutive minute spaces is 6°]
NOTE - The angle at a time will be same irrespective of am/pm
Example 2 -
Calculate the angle between Hour Hand and Minute Hand of a clock at 2:20
At 2:20, the minute hand of a clock is at digit 4 and the hour hand is slightly ahead of digit 2.
Now, the angle between digits 2 and 4 is 60°.
At 2:00, the hour hand will be pointing exactly at digit 2. In 20 minutes, the hour hand will move ahead of digit 2 by 10° (as it traces 0.5° per minute).
Thus, the angle between Hour Hand and Minute Hand at 2:20 is (60° - 10° = 50°).
One more way to find out the solution is to find out the angles of Hour Hand and Minute Hand with respect to Vertical line i.e. digit 12. [Refer the image shown]
At 2:20, minute hand is 120° and hour hand is 70° away from digit 12.
Thus, the angle between hour hand and minute hand is 120° - 70° = 50°.
Lets try to calculate the angle using the generic formula. Here, M is 20 and H is 2. So, Angle between Hour Hand and Minute Hand is -
(11/2)*20 - (30)*2 = 110 - 60 = 50°
Example 3 -
At what time, in minutes between 6 o’clock and 7 o’clock, do the hour hand and the minute hand of the clock coincide?
Solution -
When hands of a clock coincide, angle between them is Zero (0°).
Thus, we have 0 = (11/2)*M - (30)*6
M = 360/11 = 32(8/11)
Thus, between 6 & 7, hands of clock will coincide at 32(8/11) minutes after 6.
Example 4 -
At what time between 7 and 8 o'clock will the hands of a clock be at right angle ?
Solution -
At 7 o'clock hour hand is 35 minutes space ahead of the hour hand.
When both hands are at right angled, they are 15 minutes space apart.
Case 1: When minute hand is 15 min. space ahead the hour hand
In this case minute hand has to cover (35+15)= 50 min spaces
now 55 min space is gained in 60 min
So, 50 min spaces will be gained in (60 * 50) / 55 = 54(6/11) minutes
Case 2: When minute hand is 15 min. space behind the hour hand
In this case minute hand has to cover (35-15)= 20 min spaces
now 55 min space is gained in 60 min
So, 20 min spaces will be gained in (60 * 20) / 55 = 21(9/11) minutes
Thus, between 7 and 8 o'clock, the hours of the clock will be at right angles twice, 21(9/11) minutes and 54(6/11) minutes past 7.
Example 5 -
A clock is set right at 8 a.m. The clock looses 15 minutes in 24 hours. What will be the true time when clock indicates 8 p.m on the third day ?
Solution -
As clock looses 15 minutes in 24 hours, it covers 23 hrs 45 minutes in a day.
i.e. 23(45/60) = 95/4 hrs
i.e. 95/4 hrs of this clock = 24 hrs of correct clock
Now, Time covered in this clock from 8 a.m. on first day to 8 p.m of third day = 60 hrs
Thus, time covered in actual clock = 60 * 24 * (4/95) = 61(1/95) ~ 61 hours
Therefore, true time will be ~ 9 pm
Example 6 -
If a clock shows a time of 13 minutes past 3 in a mirror, what is the actual time ?
Solution -
Time shown in the mirror - 03:13
Thus, actual time will be -
12:00 - 03:13 = 08:47
Example 7 -
A clock started at noon. By 10 minutes past 5, the hour hand has turned through ?
Solution -
Time in minutes till 5:10 from noon = 310 minutes.
Hour hand traces an angle of 0.5° in a minute.
Thus, the angle covered till 05:10 is 310/2 = 155ยบ
Example 8 -
What are the times when the hour hand and minute hand of a clock coincide ?
Solution -
We know that, Relative speed of minute hand over hour hand is (6 - 0.5) = 5.5° per minute
Thus, in 60 minutes, minute hand covers 5.5°*60 = 330° more than hour hand.
Now, hands of clock coincide at 12:00. To coincide again, minute hand will need to trace 360° more than hour hand.
As minute hand traces 330° more in 60 minutes, it will require 65(5/11) minutes to trace 360° more.
Thus hands of clock will coincide after the interval of 65(5/11) minutes.
Thus, the times at which hands will coincide will be -
> 12:00
> 12:00 + 65(5/11) = 5(5/11) minutes past 1 [01:05(5/11)]
> 12:00 + 65(5/11) + 65(5/11) = 10(10/11) minutes past 2 [02:10(10/11)]
> 12:00 + 65(5/11) + 65(5/11) + 65(5/11) = 15(15/11) minutes past 3 [03:16(4/11)]
and so on.....
Example 9 -
How much does a watch loose per day, if its hands coincide every 64 minutes?
Solution -
Time Lost/Gained (in minutes) in a day = ((720/11) - M) * (24 * 60 / M)
Here M = 64.
Thus, time lost in 24 hours = ((720/11) - 64) * (24 * 60 / 64) = 32(8/11) minutes
Example 10 -
What is the reflex angle between hands of a clock at 10:20 ?
Solution -
Here H = 10 & M = 20
Angle between hands of a clock at 10:20 is -
| (11/2)* 20 - (30)10 | = |110 - 300 | = 190°
Note : As the angle value calculated above is already greater than 180°, it will be termed as Reflex Angle value.
So, the acute angle between hands of a clock will be 170°.
Example 11 -
A watch gains 5 seconds in 3 minutes and was set right at 8 AM. What time will it show at 10 PM on the same day?
Solution -
The watch gains 5 seconds in 3 minutes i.e. 100 seconds in 1 hour.
From 8 AM to 10 PM on the same day, time passed is 14 hours.
In 14 hours, the watch would have gained 1400 seconds i.e. 23 minutes 20 seconds.
So, when the correct time is 10 PM, the watch would show 10 : 23 : 20 PM
