PROBLEMS ON CLOCKS

IMPORTANT FORMULATE

>>Minute Spaces

The face or dial of clock is a circle whose circumference is divided into 60 equal parts, named minute spaces

>>Hour hand and minute hand

A clock has two hands. The smaller hand is called the hour hand or short hand and the larger one is called minute hand or long hand.

>>55 min spaces are gained by minute hand (with respect to hour hand) in 60 min.

(In 60 minutes, hour hand will move 5 min spaces while the minute hand will move 60 min spaces. In effect the space gain of minute hand with respect to hour hand will be 60 - 5 = 55 minutes.)

>>Both the hands of a clock coincide once in every hour.

>>The hands of a clock are in the same straight line when they are coincident or opposite to each other.

>>When the two hands of a clock are at right angles, they are 15 minute spaces apart.

>>When the hands of a clock are in opposite directions, they are 30 minute spaces apart.

>>Angle traced by hour hand in 12 hrs = 360°

 1 hour=360/12= 30°

 Degrees turned by hour hand in 1 minute=0.5°

Angle made by the  hour hand at H:M am/pm = 1/2*(60H+M)

Note: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.

Ex: angle covered by hour hand for showing 2:10 is 120+10 min = 130/2 = 65º)

>>Angle traced by minute hand in 60 min. = 360°.

 Degrees turned by minute hand in 1 minute=6°

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

>>Angle between the minute and the hour hand

=1/2*(60H+M) – 6M

 = 1/2*(60H -11M)       

Note:

Rate of change of angle for minute hand = 12 x Rate of change of angle of hour hand

>> 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

Thus 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)

>>How often the hour and minute hands meet?

The hour and the minute hand meet 11 times in 12 hours i.e. 11times 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

>>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)

>> Note: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)

                                                                                                     

>>If a watch or a clock indicates 9.15, when the correct time is 9, it is said to be 15 minutes too fast.

>>If a watch or a clock indicates 8.45, when the correct time is 9, it is said to be 15 minutes too slow.

>>The hands of a clock will be in straight line but opposite in direction, 22 times in a day

>>The hands of a clock coincide 22 times in a day

>>The hands of a clock are straight 44 times in a day

>>The hands of a clock are at right angles 44 times in a day.

SOLVED EXAMPLES

Ex 1:Find the angle between the hour hand and the minute hand of a clock when  3.25. Solution:angle  traced by the hour hand in 12 hours = 360° Angle  traced  by  it  in  three  hours  25  min  (ie)  41/12  hrs=(360*41/12*12)°  =102*1/2° angle traced by minute hand in 60 min. = 360°. Angle traced by it in 25 min. = (360 X 25 )/60= 150° Required angle = 1500 – 102*1/2°= 47*1/2°

Alternate Method:

Hour hand:

Angle traced by hour hand in 1 min=0.5°

3 hours is=3*60*0.5°=90°

Minute hand:Angle traced by minute hand in 1 min=6°

In 25 minutes=25*6°=150°

For 25 minutes hour hand will move additional 12.5°.so total hour hand angle traced=90°+12.5°=102.5°

Total minute hand traced angle is=150°

Required angle=150°-102.5°=47.5°.

Ex 2:At what time between 2 and 3 o'clock will the hands of a clock be together?

Solution: At 2 o'clock, the hour hand is at 2 and the minute hand is at 12, i.e. they  are 10 min  spaces apart. To be together, the minute hand must gain 10 minutes over the hour hand. Now, 55 minutes are gained by it in 60 min. 10 minutes will be  gained in (60 x 10)/55  min. = 120/11 min. The hands will coincide at 120/11 min. past 2.

Ex. 3. At what time between 4 and 5 o'clock will the hands of a clock be at right  angle?

Sol: At 4 o'clock, the minute hand will be 20 min. spaces behind the hour hand,  Now, when the two hands are at right angles, they are 15 min. spaces apart. So,  they are at right angles in following two cases.        Case I. When minute hand is 15 min. spaces behind the hour hand: In this case min. hand will have to gain (20 ­ 15) = 5 minute spaces. 55 min. spaces  are gained by it in 60 min.

 5 min spaces will be gained by it in 60*5/55  min=60/11min.

:. They are at right angles at 60/11min. past 4. Case II. When the minute hand is 15 min. spaces ahead of the hour hand: To be in this position, the minute hand will have to gain (20 + 15) = 35 minute spa'  55 min. spaces are gained in 60 min. 35 min spaces are  gained in (60 x 35)/55 min =420/11

:. They are at right angles at 420/11 min. past 4.

Ex. 4. Find at what time between 8 and 9 o'clock will the hands of a clock  being  the same straight line but not together.               

  Sol: At 8 o'clock, the hour hand is at 8 and the minute hand is at 12, i.e. the two  hands_ are 20 min. spaces apart. To be in the same straight line but not together they will be 30 minute spaces apart.  So, the minute hand will have to gain (30 ­ 20) = 10 minute spaces over the hour 

hand. 55 minute spaces are gained. in 60 min. 10 minute spaces will be gained in (60 x 10)/55 min. = 120/11min. :. The hands will be in the same straight line but not together at 120/11 min. 

Ex.  5.  At  what  time  between  5  and  6  o'clock  are  the  hands  of  a  clock  3minapart?

. Sol. At 5 o'clock, the minute hand is 25 min. spaces behind the hour hand.        Case I. Minute hand is 3 min. spaces behind the hour hand. In this case, the minute hand has to gain' (25 ­ 3) = 22 minute spaces. 55 min. are  gained in 60 min. 22 min. are gaineg in (60*22)/55min. = 24 min. :. The hands will be 3 min. apart at 24 min. past 5. Case II. Minute hand is 3 min. spaces ahead of the hour hand. In this case, the minute hand has to gain (25 + 3) = 28 minute spaces. 55 min. are  gained in 60 min.  28 min. are gained in  (60 x 28_)/55=346/11 The hands will be 3 min. apart at 346/11 min. past 5.

Ex 6. Tbe minute hand of a clock overtakes the hour hand at intervals of 65  minutes of the correct time. How much a day does the clock gain or lose?         Sol: In a correct clock, the minute hand gains 55 min. spaces over the hour hand in  60 minutes. To be together again, the minute hand must gain 60 minutes over the hour hand. 55  min. are gained in 60 min. 60 min are gained in  60 x 60 min =720/11 min.                                    55

But, they are together after 65 min. Gain in 65 min =720/11­65 =5/11min.

Gain in 24 hours =(5/11 * (60*24)/65)min =440/43 The clock gains 440/43  minutes in 24 hours.

Ex. 7. A watch which gains uniformly, is 6 min. slow at 8 o'clock in the  morning Sunday and it is 6 min. 48 sec. fast at 8 p.m. on following Sunday.  When was it correct?

Sol. Time from 8 a.m. on Sunday to 8 p.m. on following Sunday = 7 days 12 hours  = 180 hours

The watch gains (5 + 29/5) min. or 54/5 min. in 180 hrs. Now 54/5  min. are gained in 180 hrs. 5 min. are gained in (180 x 5/54 x 5) hrs. = 83 hrs 20 min. = 3 days 11 hrs 20 min. Watch is correct 3 days 11 hrs 20 min. after 8 a.m. of Sunday. It will be correct at 20 min. past 7 p.m. on Wednesday.

Ex 8. A clock is set right at 6 a.m. The clock loses 16 minutes in 24 hours.  What will be the true time when the clock indicates 10 p.m. on 4th day?

Sol. Time from 5 a.m. on a day to 10 p.m. on 4th day = 89 hours. Now 23 hrs 44 min. of this clock = 24 hours of correct clock.

356/15 hrs of this clock = 24 hours of correct clock. 89 hrs of this clock = (24 x 31556 x 89) hrs of correct clock. = 90 hrs of correct clock. So, the correct time is 11 p.m.

Ex. 9. A clock is set right at 8 a.m. The clock gains 10 minutes in 24 hours will  be the true time when the clock indicates 1 p.m. on the following day?

Sol. Time from 8 a.m. on a day  1 p.m. on the following day = 29 hours. 24 hours 10 min. of this clock = 24 hours of the correct clock. 145 /6  hrs of this clock = 24 hrs of the correct clock

29 hrs of this clock = (24 x  6/145 x 29) hrs of the correct clock = 28 hrs 48 min. of correct clock

The correct time is 28 hrs 48 min. after 8 a.m. 

Ex.10. If the minute hand of a clock has moved 300º, how many degrees has the hour hand moved?

Sol. Remember

Rate of change of angle for minute hand = 12 x Rate of change of angle of hour hand

 Thus, angle moved by the hour hand = 300/12 = 25º

Ex.11. A clock when seen in a mirror shows 4:40. What is the correct time?

Sol: Remember the shortcut to subtract the given time from 12:00.

Thus the correct time is 7:20

Ex.12.Find the angle between the minute hand and the hour hand when the time in the clock is 6:10.

Sol. Using the formula, Angle between hour and the minute hand = 1/2*(60H – 11M) = 1/2(60×6 – 10×11)= 1/2*250 = 125º

Ex.13. A clock started at noon. By 10 minutes past 5, the hour hand has turned through?

Sol. Remember this simple shortcut for such questions. Time in minutes till 5:10 = 310 minutes. Thus, the angle covered is 310/2 = 155º.

Ex.14. 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

Ex.15: At what time do the hands of a clock between 7:00 and 8:00 form 90 degrees?

Ans: At 7 o' clock, the hour hand is at 210 degrees from the vertical.

In 't' minutes

Hour hand = 210 + 0.5t

Minute hand = 6t

The difference between them should be 90 degrees. Please note that it can be both before the meeting or after the meeting. You will get two answers in this case, one when hour hand is ahead and the other one when the minute hand is ahead.

Case 1: 210 + 0.5t - 6t = 90

=> 5.5t = 120

=> t = 240/11 = 21 minutes 9/11th of a minute

Case 2: 6t - (210 + 0.5t) = 90

=> 5.5t = 300

=> t = 600/11 = 54 minutes 6/11th of a minute

So, the hands of the clock are at 90 degrees at the following timings:

7 : 21 : 9/11th and 7 : 54 : 6/11th

Ex.16: At what time do the hands of the clock meet between 7:00 and 8:00

Ans: At 7 o' clock, the hour hand is at 210 degrees from the vertical.

In 't' minutes

Hour hand = 210 + 0.5t

Minute hand = 6t

They should be meeting each other, so

210 + 0.5t = 6t

=> t = 210/5.5 = 420/11= 38 minutes 2/11th minute

Hands of the clock meet at 7 : 38 : 2/11th

Ex.17: 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

Ex.18: A watch gains 5 seconds in 3 minutes and was set right at 8 AM. If it shows 5:15 in the afternoon on the same day, what is the correct time?

Ans: The watch gains 5 seconds in 3 minutes => 1 minute in 36 minutes

From 8 AM to 5:15, the incorrect watch has moved 9 hours and 15 minutes = 555 minutes.

When the incorrect watch moves for 37 minutes, correct watch moves for 36 minutes.

=> When the incorrect watch moves for 1 minute, correct watch moves for 36/37 minutes

=> When the incorrect watch moves for 555 minutes, correct watch moves for (36/37)*555 = 36*15 minutes = 9 hours

=> 9 hours from 8 AM is 5 PM.

=> The correct time is 5 PM.

Ex.19: A watch loses 5 minutes every hour and was set right at 8 AM on a Monday. When will it show the correct time again?

Ans: For the watch to show the correct time again, it should lose 12 hours.

It loses 5 minutes in 1 hour

=> It loses 1 minute in 12 minutes

=> It will lose 12 hours (or 720 minutes) in 720*12 minutes = 144 hours = 6 days

=> It will show the correct time again at 8 AM on Sunday.

PRACTICE QUESTIONS

·         A watch which gains uniformly is 2 minutes low at noon on Monday and is 4 min. 48 sec fast at 2 p.m. on the following Monday. When was it correct?

·         At what time, in minutes, between 3 o'clock and 4 o'clock, both the needles will coincide each other?

·         At what time between 4 and 5 o'clock will the hands of a watch point in opposite directions?

·         At what time between 7 and 8 o'clock will the hands of a clock be in the same straight line but, not together?

·         At what time between 5.30 and 6 will the hands of a clock be at right angles?

·         The minute hand of a clock overtakes the hour hand at intervals of 65 minutes of the correct time. How much a day does the clock gain or lose?

·         A watch which gains 5 seconds in 3 minutes was set right at 7 a.m. In the afternoon of the same day, when the watch indicated quarter past 4 o'clock, the true time is:

·         A clock is set right at 5 a.m. The clock loses 16 minutes in 24 hours.What will be the true time when the clock indicates 10 p.m. on 4th day?

·         A watch which gains uniformly ,is 5 min,slow at 8 o'clock in the morning on sunday and it is 5 min 48 sec.fast at 8 p.m on following sunday. when was it correct?

·         How much does a watch lose per day, if its hands coincide ever 64 minutes?

·         A watch which gains uniformly is 2 minutes low at noon on monday and is 4 min.48 sec fast at 2 p.m on the following monday. when was it correct ?

In case of any doubts please write us back,meanwhile we will get back to you with some more examples and tricks.





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