# 22. Clocks

Introduction:
For solving the problems on clocks the following points will be helpful
1) Minutes hand covers 360o in 1 hour, i.e., in 60 minutes. Hence, minutes hand covers 6° per minute. (360/60 = 6°)
2) Hour hand covers 360o in 12 hours, hence hour hand covers 30o per hour i.e., ½ per minute (30/60 = ½o)

3) Hours hand and minute’s hand coincide with each other:
12 hours 24 hours/1day
a. 0o with each other (i.e.., they coincide with each other) 11 times 22 times
b. 180o with each other (i.e., they lie on the same straight line) 11 times 22 times
c. Remaining all angles except 0o & 180o 22 times 44 times

Note:
1. The time gap between any two coincidences is minutes
Correct time:-
If a clock is said to be a correct time then the time gap between any two coincidences is minutes.
• If a clocks coincidence time is less than min’s then the clock is going fast (or) gaining time.
• If a clock coincidence time is greater than min’s then the clock is going slow (or) losing time

To calculate the angle between two hands of a clock

Ex: 1 Find the angle between the min’s hand and hour’s hand of a clock at 5 hours 20 min’s?
(a) 45o
(b) 40o
(c) 30o
(d) 20o
Sol: Here we have hours = 5 hrs; minute = 20 min
=|110 - 150| = |-40|= 400
: Angle between the two hand is 40o
Ans: b

Ex: 2 At what time between 2 and 3 0’ clock will the hands of a clock be together?
Sol: At 2 0’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
The hand will coincide at past 2.

Ex: 3 At what time between 4 and 5 0’clock will the hands of a clock be at right angle?
Sol: At 4 0’ 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 1.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
They are at right angles at 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 spaces.
55 min. spaces are gained in 60 min.
35 min, spaces are gained in
They are at right angles at past 4.

Ex: 4 Find at what time between 8 and 9 0’ clock will the hands of a clock be in the same straight line but not together.
Sol: At 8 0’ clock, the hour hand is at 8 and the minute hand is at 12, i.e., the two hands are 20 min. space 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
The hands will be in the same straight line but not together at past 8.

Ex: 5 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?
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
But, they are together after 65 min.
Gain in 65 min.
Gain in 24 hours
The clock gains minutes in 24 hours.

Ex: 6 A watch which gains uniformly, is 5 min. slow at 8 0’ 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?
Sol: Time from 8 a.m. on Sunday to 8 p.m on following Sunday = 7 days 12 hours = 180 hours
The watch gains in 180 hrs.
Now min, are gained in 180 hrs.
5 min, are gained in 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

EXERCISE
1. What was the angle covered by the hour hand in 15 minutes?
(a) 45o
(b) 30o
(c) 71/2o
(d) 90o
(e) 15o
2. By how many degrees will the minute hand move, in the same time, in which the hour hand moves by 12o?
(a) 144o
(b) 6o
(c) 24o
(d) 120o
(e) 140o
3. What is the angle between the two hands of a clock, when it shows 30 min past 3 0’ clock?
(a) 110o
(b) 75o
(c) 60o
(d) 105o
(e) 90o
4. What is the angle between the two hands of a clock at 45 min past 6 0’ clock?
(a) 43 ½o
(b) 135o
(c) 75 ½o
(d) 67 ½o
(e) 90o
5. What is the angle between the two hands of a clock 20 min’s past 11 0’ clock?
(a) 145o
(b) 140o
(c) 120o
(d) 130o
(e) 180o
6. At what time between 4 0’ clock & 5 0’ clock are the hands of a clock together?
(a) 4 hr 21 min
(b) 5 hr 21 min
(c) 4 hr 20 20/11 min
(d) 4 hr 15min
(e) None
7. At what time between 8 0’ clock & 9 0’ clock will the hands of a clock coincide?
(a) 8 hr 23 2/11
(b) 8 hr 51 3/11min
(c) 8 hr 43 7/11 min
(d) 8 hr 17 2/11 min
(e) Can’t be determined
8. At what time between 4 0’ clock & 5 0’ clock are the hands of a clock in the opposite directions?
(a) 4 hr 54 6/11 min
(b) 4 hr 11 10/11 min
(c) 4 hr 38 2/11 min
(d) 4 hr 43 7/11 min
(e) Both A & B
9. For how many times, the two hands of a clock overlap in 12 hours?
(a) 12
(b) 11
(c) 10
(d) 14
(e) 22
10. For how many times, the two hands of a clock overlap in 24 hours
(a) 22
(b) 20
(c) 24
(d) 12
(e) 11
11. For how many times, the two hands of a clock perpendicular to each other in a day?
(a) 40
(b) 48
(c) 44
(d) 22
(e) 11
12. For how many times, the two hands of a clock make an angle of 600 in a 12 hours
(a) 11
(b) 22
(c) 44
(d) 48
(e) 40
13. What is the time shown by a clock if the hour hand is in between the hour division 7 and 8, and the angle between the two hand is 50o
(a) 7 Hr 29 1/11min
(b) 7 hr 47 3/11 min
(c) 7 Hr 43 7/11 min
(d) 7 Hr 32 9/11 min
(e) Both A & B
14. What is the time shown by a clock of the hour hand is in between 5 0’ clock & 6 0’ clock and the angle between the two hands is 100o
(a) 5 hr 45 5/11 min
(b) 5 hr 43 7/11 min
(c) 5 Hr 9 1/11min
(d) 5 hr 35 6/11 min
(e) Both a & c
15. Two clocks are having the correct time at 8 am one clock loses one minute per hour while the other gains two min’s per hour. At 5 Pm on the same day, by how much time will the two clocks differ?
(a) 30 min
(b) 27 min
(c) 24 min
(d) 32 min
(e) 26 min
16. A watch which loses uniformly was observed to be 4 minutes fast at 8 am on a Sunday and two minutes slow at 2 pm on the next days, Monday. When did the watch show the correct time?
(a) 2 am on Monday
(b) 4 am on Monday
(c) 3 am on Monday
(d) 8 am on Monday
(e) 12 am on Monday
17. A watch which gains uniformly was observed to be five minutes slow at 10 am on a Friday & 7 minutes fast at 10 pm on the subsequent Sunday. When did the watch show the correct time?
(a) 11 am on Saturday
(b) 1 pm on Saturday
(c) 10 am on Saturday
(d) 1am on Saturday
(e) None of these
18. A clock strikes once at 1 0’ clock, twice at 2 0’clcok, three times at 3 0’clock and so on. If it takes 4 seconds to strike at 4 0’ clock. Find the time taken by it to strike at 10 0’ clock
(a) 16 sec
(b) 18 sec
(c) 24 sec
(d) 12 sec
(e) 15 sec
19. What is the mirror image of a clock when it is shows 5 hr 20 min?
(a) 6 Hr 20 min
(b) 7 hr 40 min
(c) 6 hr 40 min
(d) 7 hr 20 min
(e) None of these
20. The reflection of a wall clock in mirror is showing the time as 4 hr 44 min. what is the actual time?
(a) 6 hr 16 min
(b) 8 hr 16 min
(c) 7 hr 16 min
(d) 8 hr 14 min
(d) None of these
21. The min’s hand of a clock over takes the hour hand at intervals of 65 min’s of correct time. How much does the clock lose (or) gain in a day?
(a) 11 11/143min gain
(b) 10 10/143 min gain
(c) 9 9/143 min gain
(d) both a & b
(e) None of these
22. The min’s hand of a clock overtakes the hour hand at intervals of 64 min’s of correct time. How much does the clock lose (or) gain in a day?
(a) 23 7/11min fast / gain
(b) 30 6/11 min fast/gain
(c) 32 8/11 min fast/gain
(d) both a & c
(e) None of these
23. The min’s hand of a clock over takes the hour hand at intervals of 66 min’s of correct time. How much does the clock lose (or) gain in a day?
(a) 11 109/121 min gain
(b) 11 109/121 min lose
(c) 10 109/121min gain
(d) 10 109/121 min lose
(e) None of these
24. What is the mirror image of the clock, if the time is 4:30?
(a) 7:30
(b) 6:30
(c) 5:30
(d) 8:20
(e) None of these
25. A clock which loses time uniformly was observed to be 3 min’s fast at 3:00 pm on a Wednesday. On the sub-sequent Friday at 10 pm the clock was 8 min’s slow. When did the clock shows the correct time?
(a) Thursday 8am
(b) Thursday 6 pm
(c) Thursday 6 am
(d) Thursday 8 pm
(e) None of these

 ANSWER KEY 1 c 11 c 21 b 2 a 12 b 22 c 3 b 13 d 23 b 4 d 14 d 24 a 5 b 15 b 25 c 6 a 16 b 7 c 17 a 8 a 18 d 9 b 19 c 10 a 20 c
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