A Clock is a circular device provided with three hands viz. an hour hand, minute and second hand.
All hands move around the circular dial. The hour hand is smaller than the minute hand and the hour hand is slower than the minute hand. The hour hand shows time in hours and the minute hand shows time in minutes.
The clock is separated into 12 equal pieces with numbers ranging from 1 to 12.
360/12 = 30° for each section.
One hour is when the minute hand completes a full round, i.e., covers 360°.
12 Equal Divisions of a Clock
The angle between any two consecutive divisions = (360°)/12= 30°
Angle Divisions of a clock
Few formula’s that you should know to solve questions related to clocks are as follow:
The difference in the speed = 6°– (1/2°) = 5.5° per minute
Comparing the speed of the minute hand and an hour hand, one can conclude that the minute hand is always faster than the hour hand by 5.5° or an hour hand is always slower than the minute hand by 5.5°
Note: The evaluation of the speed of second hands is not necessary as it travels a corresponding distance of 1 second in a second.
Frequency of coincidence and collision of hands of a clock:
As we know the hands of clock moves at different speeds, they coincide and collide and also make different angle formations among themselves at various times in a day.
Q: How many times in a day do the minute and hour hands of a clock coincide (Angle between them is zero) with each other?
Q: How many times in a day do the minute and hour hands of clock form a 180° straight line in a day?
When the angle between the hands are not perfect angles like 180°, 90° or 270°, the solving of the questions becomes difficult and timeconsuming at the same time. The logic below provides a trick to address problems involving angles of hands for other than standard aspects.
T = 2/11 [H*30±A]
Where:
1. T stands for the time at which the angle formed.
2. H stands for an hour, which is running.
(If the question is for the duration between 4 o’clock and 5 o’clock, it’s the 4^{th} hour which is running hence the value of H will be ‘4’.)
3. A stands for the angle at which the hands are at present.
(The value of A is provided in the question generally)
The clock is divided into two parts: 1^{st} and 2^{nd} half as shown above
If the time given in the question lies in the first half, then the positive sign is considered while evaluating the time else, then the negative sign is used.
Q: At what time between 3 and 4 o’clock, the hands makes an angle of 10 degrees?
Solution:
Given: H = 3 , A = 10
Since both three and four lies in the first half considered a positive sign.
Calculations:
T = 2/11 [H*30±A]
T = 2/11 [3*30+10]
T = 2/11 [90+10]
T = 2/11 [100]
T = 200/11
T =18 2/11
The answer indicates that the hands of a clock will make an angle of 10 between 3 and 4 o’clock at exactly 3:18:2/11 ( 3’ o clock 18 minutes and 2/11 of minutes = 2/11*60 = 10.9 seconds)
This section involves the comparison of time in the accurate clock with the wrong watch. The wrong time indicates that a clock is either slow or fast compared to the correct time. The wrong clock can either be fast or delayed by a few seconds/minutes/ hours or sometimes by a few days and weeks.
Q: A clock gains 5 seconds for every 3 minutes. If the clock started working at 7 a.m. in the morning, then what will be the time in the wrong clock at 4 p.m. on the same day?
Solution:
(a) Case1: When the minute hand is behind the hour hand, the angle between the two hands at M mins past H O’clock:
(b) Case2: When the minute hand is ahead of the hour hand, the angle between the two hands at M mins past H O’clock:
The two hands of the clock will be at right angle between H and (H + 1) O’clock at :
(a) Case 1: When the minute hand is 15 mins spaces behind the hour hand, to be in this position, the minute hand will have to gain (5H – 15) mins spaces over the hour hand.
(b) Case 2: When the minute hand is 15 mins spaces ahead of the hour hand, to be in this position, the minute hand will have to gain (5H + 15) mins spaces over the how hand.
The two hands of the clock will be in the same straight line but not together between H and (H + 1) O’clock at :
The two hands of the clock will be together between H and (H + 1) O’clock at:
Example 1: At what time between 4 and 5, the minute hand will be 2 minutes spaces ahead of the hour hand?
Ans: At 4 O'clock, the two hands are 20 min spaces apart.
In this case, the min hand will have to gain (20 + 2) i.e. 22 – minute spaces.
So, 22 – minute spaces will be gained in (60/55) × 22 = 24 min.
Example 2: The minute hand of a clock overtakes the hour hand at intervals of 65 minutes of correct time. How much does the clock lose or gain in 12 hours?
Ans: In a correct clock, the minute hand and hour hand should meet after every 65 5/11 min.
But we know that they are meeting after every 65 minutes.
So, the gain in 65 minutes is 5/11 minutes.
Gain in 12 hours = (120 × 60/65) × (5/11) = 720/143 = 5 5/143 min.
Example 3: A clock is set right at 10 am. The clock gains 5 minutes in 12 hours. What will be the true time when the clock indicates 3 pm on the next day?
Ans: Time from 10 am to 3 pm on the following day is 29 hrs.
Now, 12 hrs 5 min i.e. 145/12 hrs of this clock = 12 hours of correct clock.
So, 29 hours of this clock is (29 × 12 × 12)/145 = 144/5 = 28 hours 48 minutes.
So, the time is 12 minutes before 3 pm.
Example 4: The minute hand of a clock overtakes the hour band at interval of 60 mins. How much a day does the clock gain or lose?
Ans: In a correct clock, minute hand gains 55 minute spaces over the hour hand in 60 minutes.
Therefore, 60 min spaces are gained in (60 × 60)/55 = 720/11 minutes.
In other words, minute hand overtakes hour hand in every 720/11 minutes.
In the example given, minute hand overtakes hour band in 60 minutes.
Therefore, gain in 60 minute = (720/11)  60 = 60/11 minutes.
Gain in 24 hours = (60/11) × 24 = 1440/11 min
gain/loss =
Since the sign is +ve, clock gains 1440/11 min in a day.
Example 5: The minute hand of a clock overtakes the hour band at an interval of 66 mins. How much a day does the clock gain or lose?
Ans: As we have seen previously, minute hand overtakes hour hand in every 720/11 minute.
In this example, minute hand overtakes hour band in 66 minutes
Therefore, loss in 66 minute = 66  720/11 = 6/11 minutes
Loss in 24 hours = 6/11 × 1/66 × 66 × 24 = 1440/121 min
gain/loss
Since the sign is ve, the clock loses 1440/121 min a day.
Example 6: What is the angle between the hands of the clock at 7:20. At 7 o’ clock, the hour hand is at 210 degrees from the vertical.
Ans: In 20 minutes, Hour hand = 210 + 20*(0.5) = 210 + 10 = 220 {The hour hand moves at 0.5 dpm}
Minute hand = 20*(6) = 120 {The minute hand moves at 6 dpm}. Difference or angle between the hands = 220 – 120 = 100 degrees.
Example 7: 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?
Ans: The watch gains 5 seconds in 3 minutes => 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 or 23 minutes 20 seconds. So, when the correct time is 10 PM, the watch would show 10 : 23 : 20 PM
214 videos139 docs151 tests

1. How can the structure of a clock help in solving clockrelated problems? 
2. What are some important conversions to keep in mind when dealing with clocks? 
3. How can one compare the speed of the hour and minute hands on a clock? 
4. How can one find the time when the angle between the hour and minute hands is known? 
5. What are some key differences between a correct clock and a wrong clock in clockrelated problems? 

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