CLOCKSThe face or dial of a watch is a circle whose circumference is divided

into 60 equal parts, called minute spaces. A clock has two hands, the smaller one is called the hour hand or short hand while the larger one is called the minute hand or long hand.

Minute Hand Hour Hand Space b/w hour & minute hand in 1

minute= 6 °−12

°

=112

°360 ° 60 minute 360 ° 12 hour

1 minute 6 ° 1 minute 12

°

1 minute=112

° 1 °= 211min Two hands are overlap at 12 o’clock. Next time

when both hands are overlap after travel360 °.

360°=360×211

=72011

=65 511minute

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

When the two hands are at right angles, they are 15 minute spaces apart.

When the hands are in opposite directions, they are 30 minute spaces apart.

The two hands overlap each other every 65 5/11 minutes. In a 12-hour gap the two hands would overlap /coincide 11times and

hence in a 24-hour gap 22 overlap would take place. In a 12-hour gap the two hands would be perpendicular to each other

22 times. In a 12-hour gap the two hands would be in a straight line 22 times. In a 12-hour gap the two hands would be in a straight line but opposite

in direction is 11 times. In 60 minutes, the minute hand gains 55 minutes on the hour hand. Too fast and too slow: If a watch or a clock indicates 8.15, when the

correct time is 8, it is said to be 15 minutes too fast.On the other hand, if it indicates 7.45, when the correct time is 8, it is said to be 15 minutes too slow.

Finding the angle between the hands of a clock is,

Angle (θ )=30H−112M

Angle (θ )=112M−30H

The minute hand of a clock overtakes the hour hand at intervals of M minutes of correct time. The clock gains or losses in a day by

(65 511−M )(60× 24M ) Minutes

SOLVED PROBLEMS

1. An accurate clock shows 8 o'clock in the morning. Through how may degrees will the hour hand rotate when the clock shows 2 o'clock in the afternoon?A. 154° B. 180° C. 170° D. 160°

Explanation : We know that Angle traced by hour hand in 12 hrs = 360° , From 8 to 2, there are 6 hours, The angle traced by the hour hand in 6 hours = 6×360/12=180°

2. Find at what time between 8 and 9 o’clock will the hands of a clock be in the same straight line but not together.  Same straight line but not together means angle is 180°.

Angle (θ )=30H−112M, 180°= 30(8)-

112M ,

112M=240-180=60°.

M=60× 211

=12011

=10 1011minute past 8hour .

3. What is the angle between the two hands of a clock at 3:27 p.m.?

Angle (θ )=30H−112M , θ = 30×3 –

112

(27) = 90o – 148.5o = 58.5o (here no need

to represent the negative in that)

4. At what time between 2 a.m. and 3 a.m. is the angle between the two hands of a clock 28°?There are two possibilities:1. Minute hand is ahead of the hour hand.2. Hour hand is ahead of the minute hand.

CASE 1 :Angle (θ )=30H−112M , 28=30(2)-11/2 M, M=

211×88=16min

CASE 2 : Angle (θ )=112M−30H, 28=11/2 M - 30H, M=

211×32=5 9

11min

Thus, there are two points of time at which the angle between the two hands

is 28° and these two points of time are 16 minutes past 2 a.m. and 5911

minutes past 2 a.m.

5. A watch which gains uniformly is 5 minute slow at 8 o’clock in the morning on Sunday and it is 5 minute 48 sec fast at 8 p.m. on following Sunday. When was it correct? Time from 8 a.m. on Sunday to 8 p.m. on following Sunday.7 days 12 hours = 180 hours.

The watch gains [5+545

] min. or 545

min. in 180 hrs.

Now 54/5 min. is gained in 180 hrs.

5 min. are gained in (180× 554 ×5)hrs. = 83 hrs 20 min. = 3 days 11 hrs 20

min.It will be correct at 20 min. past 7 p.m. on Wednesday.

6. The minute hand of a clock overtakes the hour hand at intervals of 64 minutes of the correct time. How much a day does the clock gain or lose?

55min spaces are covered in 60min, spaces are covered in (6055×60) =

65511minute .

Loss in 64min = 65511

−64=1611

min.

Loss in 24 hrs = {1611 × 2464 ×60}=32 811minute .7. 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?

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.23 hrs 44 min = 356/15 hrs of this clock = 24 hours of correct clock.

89 hrs of this clock = (24×15356

×89) hrs of correct clock = 90 hrs of correct

clock.So, the correct time is 11 p.m.

8. 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?

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.1456

Hrs of this clock = 24 hrs of the correct clock.

29 hrs of this clock = (24×6145

×29) hrs of the correct clock = 28 hrs 48 min.

The correct time is 28 hrs 48 min. after 8 a.m. This is 48 min. past 12.

9. 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. 3 pm B. 3.45 pm C. 3.30 pm D. 4 pm

Explanation:

Time from 7 am to 4.15 pm = 9 hours 15 minutes = 914

 hours= 374

 hours

=>374×60=555minute.

3 minute 5 seconds of the given clock = 3 minutes of a normal clock.

3560

 Min=>3712

 Minutes of the given clock = 3 minutes of a normal clock.

1 min of the given clock= 3×1237of normalclock .

555 min of the given clock= 555×3×1237

=540minute normal time.

540 min= 9 hrs, 7am to 9 hr is 4pm.

10. The minute hand of a clock overtakes the hour hand at intervals of 65 minutes. How much a day does the clock gain or loss?

A. 109143

minutes B. 119143

minutes C.1110143

minutes D. 1010143

minutes

Exp: (65 511−M )(60× 24M )  = (65 511−65)( 60×2465) =

511 (12×2413 )=1440143

=10 10143

minutes

11. If a clock takes 7sec to strike 7, how long will the same clock take to strike 10?

Exp: The clock strikes for the first time at the start and takes 7 seconds for 6 intervals-thus for one interval time taken=7/6.

Therefore, for 10 seconds there are 9 intervals and time taken is 9*7/6=10 and 1/2 seconds.

12. At what time the clock hands’ position is a mirror image of the hands’ position at 4:08 along the line crossing the 12 and 6 hour marks?

Exp: The clock shown in Fig. 1 is at 4:08. The hands’ position of the clock shown in Fig. 2 is a mirror image of that in Fig. 1.

The minute hand

The difference between the 12 o’clock position and the minute hand’s position at 4:08 is 8 minutes. Therefore, the minute hand’s mirror image must be 8 minutes to the left of the 12 o’clock position or 8 minutes before the hour. See Fig. 3. This is the same as saying that the time in Fig. 2 is 52 minutes past the hour (since 60 min − 8 min = 52 min).

The hour hand

The hour hand’s position at 4:08 is between the 4 and 5 hour marks. These marks are horizontally across the 8 and 7 hour marks. Thus, the hour hand at 4:08 must have a mirror image (along the vertical line crossing the 12 and 6 o'clock position) that is between the 7 and 8 hour marks.

Therefore, the mirror image's time is 7:52.