Problems on clocks are very common in all the aptitude exams. Be it a campus placement paper or other competitive exams like CSAT, CAT, GRE, MAT etc; aptitude questions form a very important part of the paper.
These problems are slightly difficult and time-consuming if you don’t understand the underlying concepts well. Mastery on such topics is important and is required to be ahead in the competition as the other problems on series, percentage, speed and time etc are slightly more comfortable for the larger crowd. So, this is one of the areas where your preparedness can help you get ahead in the race.
Let’s get started:
Clock angle problems relate two different measurements: Angle and Time.
The angle is typically measured in degrees from the mark of no 12 clockwise.(So our reference line is vertical line from centre of the clock to the 12’o clock mark)
It is important to unserstand the rate of change of angle in degrees per minute.
Hour Hand – (360º in 12 hours) – 0.5º/minute
Minute Hand – (360º in 60 min) – 6º/min
Note1: Thus, an important point to note here: Rate of change of angle for minute hand = 12 x Rate of change of angle of hour hand
Some Quick formulae:
1. Angle made by the hour hand at H:M am/pm = 1/2*(60H+M)
2. Angle made by the minute hand at H:M am/pm = 6M
For example: At 3:30 pm
Angle made by the hour hand = 1/2*(60×3 + 30) = 1/2*(210) = 105º
Angle made by the minute hand = 6M = 6*30 = 180º
3. Angle between the minute and the hour hand =1/2*(60H+M) – 6M = 1/2*(60H -11M)
For example: At 3:30 pm
Angle between the Min and Hour hand = 1/2*(60×3 – 11×30) = 75º
4. When are the hour and minute hands of a clock superimposed?
Angle made by Hr hand = Angle made by Min hand
1/2*(60H + M) = 6M
11H = 60H
M = (60/11)*H
M = 5.45*H
To make it simple, Everytime the minute value is a 5.45 times of the hour value, both the hands are overlapped.(e.g 1:05, 2:10, 3:15 etc)
5. How often the hour and minute hands meet?
The hour and the minute hand meet 11 times in 12 hours i.e. 11 times in 720 minutes. Hence, they meet every 720/11 = 65(5/11)minutes or 65.45 minutes
In terms of angle they meet every 360/11 = 32(8/11)º or 32.72º
6. At n’o clock , the angle of the hour hand from the vertical is 30nº. (For example: at 3 o’ clock the angle formed by hour hand is 3*30 = 90º)
7. Hour and the minute hand meet(0º) 11 times in 12 hours and 22 times a day.
8. Hour and the minute hand are in opposite direction(180º) 11 times in 12 hours and 22 times a day.
9. Hour and the minute hand are at any other angle(other than 0º and 180º) 22 times in 12 hours and 44 times per day.
10. Very Important: If the time in the clock is known and asked what will it show if seen in a mirror or vice-versa, simply subtract the given time from 12:00. (For example the time is 8:40 in the mirror then subtracting it from 12 we get 3:20 which will be the time seen when we see the clock in the mirror) [This is a very useful and time saving shortcut for exam and rarely available in the books which increases it’s importance]
11. The hour hand covers one complete rotation(360º) in 12 hours i.e 360º in 720 minutes or 1º per 2 minutes. This result can be used to solve some questions like.
What is the angle covered by the hour hand by the time it shows H:M am/pm? It can be calculated by converting the time into minutes and dividing by 2.(e.g angle covered by hour hand for showing 2:10 is 120+10 min = 130/2 = 65º)
Now that you are familiar with the concept and the formulae, let’s solve some questions.
Q1. If the minute hand of a clock has moved 300º, how many degrees has the hour hand moved?
Ans1. Remember that the speed of hour hand is 1/12 that of the minute hand(Refer to Note1 above). Thus, angle moved by the hour hand = 300/12 = 25º
Q2. A clock when seen in a mirror shows 4:40. What is the correct time?
Ans2: Remember the shortcut to subtract the given time from 12:00(refer to point no 10 above). Thus the correct time is 7:20
Q3.Find the angle between the minute hand and the hour hand when the time in the clock is 6:10.
Ans3. Using the formula, Angle between hour and the minute hand = 1/2*(60H – 11M) = 1/2(60×6 – 10×11)= 1/2*250 = 125º
Q4. A clock started at noon. By 10 minutes past 5, the hour hand has turned through?
Ans4. Remember this simple shortcut for such questions. Time in minutes till 5:10 = 310 minutes. Thus, the angle covered is 310/2 = 155º.
Q5. 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?
Ans. The time after n o’ clock after which the hands of the clock coincide is n+n/11. Therefore, in this case it is 6+6/11 = 72/11 min
We’ll update some more solved questions in this section soon. Stay in touch.
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