Clocks

Introduction The dial of a clock is a circle whose circumference is

divided into 12 parts, called hour spaces. Each hour space is further divided into 5 parts, called minute spaces. This way, the whole circumference is divided into 12 x 5 = 60 minute spaces.

The time taken by the hour hand (smaller hand) to cover a distance of an hour space is equal to the time taken by the minute hand (longer hand) to cover a distance of the whole circumference. Thus, we may conclude that in 60 min-utes, the minute hand gains 55 minutes on the hour hand.

Note: The above statement (given in bold) is very much useful in solving the problems in this chapter, so it should be remembered. The above statement wants to say that:

"In an hour, the hour-hand moves a distance of 5 minute spaces whereas the minute-hand moves a distance of 60 minute spaces. Thus the minute-hand remains 60 - 5 = 55 minute spaces ahead of the hour-hand."

Some other facts: 1. In every hour, both the hands coincide once. 2. When the two hands are at right angle, they are 15 minute

spaces apart. This happens twice in every hour. 3. When the hands are in opposite directions, they are 30

minute spaces apart. This happens once in every hour. 4. The hands are in the same straight line when they are

coincident or opposite to each other. 5. The hour hand moves around the whole circumference

of clock once in 12 hours. So the minute hand is twelve times faster than hour hand.

6. The clock is divided into 60 equal minute divisions. 360

7. 1 minute division = - apart. 60

8. The clock has 12 hours numbered from 1 to 12 serially arranged.

9. Each hour number evenly and equally separated by five minute divisions (= 5 * 6) = 30 apart.

10. In one minute, the minute hand moves one minute divi-sion or 6.

11. In one minute, the hour hand moves 2

12. In one minute the minute hand gains 5 ^ more than

hour hand. 13. When the hands are together, they are 0 apart.

Hence, Both hands Required Angle to be coincident 0

to be at right angle 90

to be in opposite direction 180

to be in straight line 0 or 180

As per the required angle difference between minute-hand and hour-hand and the initial (or starting) position of the hour-hand, different formulae are used to find out the re-quired time. Now consider the Rules (Quicker Methods) given in the following pages.

Too Fast And Too Slow: I f a watch indicates 9.20, when the correct time is 9.10,

it is said to be 10 minutes too fast. And i f it indicates 9.00, when the correct time is 9.10, it is said to be 10 minutes too slow.

Rule 1 Theorem: Between x and (x + l)o 'clock, the two hands will

f\2\

be together at 5x^~^-J minutes past x.

Illustrative Example Ex: At what time between 4 and 5 o'clock are the hands

of the clock together? Soln: Detail Method: At 4 o'clock, the hour hand is at 4 and

the minute hand is at 12. It means that they are 20 min spaces apart. To be together, the minute hand must gain 20 minutes over the hour hand. Now, we know that 55 min. are gained in 60 min.

682 PRACTICE BOOK ON QUICKER MATHS

^ . 60 240 9 . .-. 20 min are gained in x ^ = - j r - = * 1 yy min.

Therefore, the hands will be together at 2 1 min

past 4. Quicker Method: Applying the above theorem, we have required answer

5x12 240 9 . = "~Tl~ 11 minpast4.

Exercise 1. At what time between 3 and 4 o'clock are the hands of a

clock together?

a) 16 min past 3

c) 16 min past 3

b) 16 YY min past 3

d) None of these

2. At what time between 5 and 6 are the hands of a clock coincident? a) 22 minutes past 5 b) 30 minutes past 5

8 3 c) 22 minutes past 5 d) 2 7 y y minutes past 5

3. At what time between 9 and 10 will the hands of a watch be together? a) 45 minutes past 9 b) 50 minutes past 9

1 2 c) 49 minutes past 9 d) 48 minutes past 9

4. At what time, are the hands of a clock together between 2 and 3?

9 10 a) 10 min past2 b) 10 min past 2

c) 10 min past 2 d) None of these

Answers l .c 2.d 3.c 4.b

Rule 2 Theorem: Between x and (x +1) o 'clock the two hands are

12

at right angle at (5x15)x minutes past x.

Illustrative Example Ex: At what time between 4 and 5 o'clock will the hand of

clock be at right angle? Soln: Detail Method: At 4 o'clock, there are 20 min. spaces

between hour and minute hands.To be at right angle, they should be 15 min spaces apart. So, there are two

cases: Case I : When the minute hand is 15 min spaces be-hind the hour hand. To be in this position, the min hand should have to gain 2 0 - 15 = 5 min spaces.

Now, we know that 55 min spaces are gained in 60 min

.-, 5 min spaces are gained in 55 JT m m

,*, they are at right angle at 5 min past 4.

Cae I I : When the minute hand is 15 min spaces ahead of the hour hand. To be in this position, the min hand should have to gain 20+ 15 = 35 min spaces. Now, we know that 55 min spaces are gained in 60 min

60 2

,', 35 min spaces will be gained in T ^ * - " ~ \

min

.. they are at right angle at 38 min past 4.

Quicker Method: Applying the above rule, we can say that,

12 they wil l be at right angle at (5x4-15)x and

12 5 2 (5x4 + 15)x min past 4 or, 5 min and 38

min past 4. Exercise 1. At what time between 5 and 6 o'clock will the hands of a

clock be at right angle?

10 9 a) min past 4 b) ' ^ J ^ min past 4

v , , 1 0 -c) 11 mm past 4 d) None of these

2. At what times are the hands of a clock at right angles between 7 am and 8 am?

. . . 6 , 9 a) 54 min past 7, 2 1 y y min past 7

5 8 b) 52 min past 7, 2 ' y y min past 7

c) 5 " Y Y m i n past 7, 2 1 min past 7

d) None of these 3. At what time between 5.30 and 6 will the hands of a clock

be at right angles?

Clocks 683

a) 43 minutes past 5 b) 43 minutes past 5

c) 40 minutes past 5 d) 45 minutes past 5 At what time between 10 and 11 O' c lock wi 11 the hand of clock be at right angle?

a) 38 min past 10

c) 38 min past 10

b) 6 YY min past 10

d) 8 min past 10

Answers l.a 2. a 3.b; Hint: In this case direct formula is not applicable be-

cause, given data is not in the form of x and x + 1. At 5 O'clock, the hands are 25 min spaces apart. To be at right angles and that too between 5.30 and 6, the min hand has to gain (25 + 15) or 40 min spaces. Now applying the formula, we have required answer

12 11

x40 43 YY min past 5.

4. a

Rule 3 Theorem: Between x and (x+ l)o 'clock, the two hands are in the same straight line Case I (a) when they are in opposite directions ie, 30 min-

utes spaces apart, at

(5x - 3 0J minutes past x. [where x > 6]

(b) when x < 6, the above formula will become as

12

(5X + 30)JY minutes pastx. [See Ex. 2] Note: At 6 O'clock two hands will be in opposite direction. Case I I : When they coincide (or come together), ie 0 minute

spaces apart at 5 x '12^

minutes pastx (see Rule 1)

Illustrative Examples Ex. 1: Find at what time between 7 and 8 o'clock will the

hands of a clock be in the same straight line. Soln: Detail Method: There are two cases: Case I : When they are in opposite directions, ie, 30 min

spaces apart. At 7 o'clock they are 25 min spaces apart. Therefore, to be in opposite directions the minute hand will have to gain 30 - 25 = 5 min spaces.

I . 60 . . 5 Now, 5 min spaces will be gained in x 5 - 5-

. 5 .-. the hand will in opposite directions at 5 min

past 7. Case I I : When they coincide (or come together), ie, 0 min

spaces apart. (See rule 1). Quicker Method:

Case I : Between x and (x + 1) O'clock the two hands are in opposite directions at

12 12 (5X-30)YJ- min past x = (35-30)x min past 7

= 5 min past 7.

Case I I : Same as Ex 1. Ex.2: At what time between 4 and 5 will the hands of a

watch point in opposite direction? Soln: Applying the above rule, we have,

the required answer = (5x4 + 30) min past 4

= 54 min past 4.

Exercise 1. 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.

, A 1 0 a) 10 min past 8

Find at what time between 2 and 3 O'clock will the hands of a clock be in the same straight line but not together.

b) 10 YY m>n past 8

d) None of these

a) 43 min past 2

c) 43 min past 2

b) 43 min past 2

d) None of these

Find at what time between 9 and 10 O'clock will the hands of a clock be in the same straight line but not together.

a) 16 YY min past 9 3 .

c) 16 min past 9

b) 16-j- min past 9

d) None of these

55 11 min

Answers , o , n \ 1 2 120

1. a; Hint: Required answer = (5x8--^O/yy = "JT m m

,-10 . = 10 min past 8

2. b 3. a

684 PRACTICE BOOK ON QUICKER MATHS

Rule 4

Theorem: Between x and (x+1)0 'clock the two hands will

he't' minutes apart at [px + tj minutes past x.

Illustrative Example Ex: At what time between 4 and 5 are the hands 2 minute

spaces apart? Soln: Detail Method: At 4 o'clock, the two hands are 20 min

spaces apart. Case I: When the minute hand is 2 minute spaces behind the

hour hand. In this case, the minute hand will have to gain (20 -2), ie, 18 minute spaces. Now, we know that 18 min spaces will be gained in

x l8 = = 1 9 m l n . 55 11 I I

7 .-. they will be 2 min apart at ^ min past 4.

Case II: When the minute hand is 2 min spaces ahead of the hour hand. In this case, the min-hand will have to gain (20 + 2), ie, 22 minute spaces. Now, we know that 22 minute spaces will be gained in

x22 = 24 min 55 .-. the hands will be 2 min apart at 24 min past 4. Quicker Method: Applying the above rule, we have the required answer

U , 12 18x12 22x12 = ( 5 x 4 2 ) = or

11 11 11 7

= 1 9 o r 24 min

7

Therefore, they will be 2 min spaces apart at 19yy

min past 4 and 24 min past 4.

Exercise 1. At what time between 5 and 6 are the hands of a clock 3

minutes apart? a) 24 min past 6 b) 26 min past 5 c) 30 min past 5 d) Can't be determined

2. At what time between 4 and 5 are the hands of a clock 4 minutes apart?

a) 2 6 y y min past 4

c) W>- min past 4

b) 2 ( > ~ min past 4

d) None of these

3. At what time between 3 and 4 is the minute-hand 7 min-utes ahead of the hour-hand? a) 24 min past 5 b) 24 min past 7 c) 24 min past 3 d) 24 min past 8

4. At what time between 3 and 4 is the minute-hand 4 min-utes behind the hour-hand? a) 12 minutes past 3 b) 14 minutes past 3 c) 16 minutes past 3 d) None of these

Answers l .a 2. a

3. b; Hint: Required answer = (l5 + 7) =24 min past 7.

4. a; Hint: Required answer = ( 5 x 3 - 4 ) = 12 min past 3.

Rule 5 Theorem: The minute hand of a clock overtakes the hour hand at certain intervals (given in minutes) of correct time. The clock lose or gain in a day is given by

720 11

given interval in minute

60x24 given interval in minutes

according as the sign is +ve or -ve.

Illustrative Example Ex: The minute hand of a clock overtakes the hour hand

at intervals of 63 minutes of correct time. How much a day does the clock lose or gain?

Soln: Detail Method: 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 min over the hour hand.

We know that 60 min are gained in

x60 = 6 5 m n 55 11

But they are together after 63 minutes.

.-. gain in 63 minutes = 65 63 6 11

27 11

min.

27x60x24 8 .-. gain in 24 hrs = l l x 6 3 = 5 6 ^ y min. Quicker Method: Applying the above rule, we have the required answer

720 -63

60x24 63 min

27 60 x 24. Y p x - ^ T - min.

Clocks 685

Since sign is +ve, there is a gain of 56 min. I 77 Exercise

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

a) 1 0 mm

i i 1 0 c) 1 1 m i n

b) 10 min

d) None of these

How much does a watch gain or lose per day, i f its hands coincide every 64 minutes?

a) 32 min

c) 32 min

nswers 2. a

b) 31 ~ min

d) None of these

Rule 6 find the angle between hands of clock, gle between two hands =

30x Difference of hours and

Angle

= 30 Difference of hours and

[See Ex. 3]

Illustrative Examples E\ 1: At what angle are the hands of a clock inclined at 25

minutes past 5? In: Detail Method:

At 25 minutes past 5, the minute hand is at 5 and hour hand sightly ahead of 5. The hour-hand moves by an angle of 30 in 60 min. .-. in 25 minutes, the hour hand moves by an angle of

30 M , 1 x 25 = 1 2 - 60 2 .-. the angle between two hands = 12.5 Quicker Method: Applying the above rule, we have

the required answer = : i d I + 2 5

= 30x0-12 .5=12 .5 Ex. 2: At what angle the hands of a clock are inclined at 15

minutes past 5? Soln: Quicker Method:

Angle = 30 Diff of 5 and + 5 J

5 = 30x2 + = 67.5

2 Ex.3: At what angle are the hands of a clock inclined at 55

minutes past 8? Soln: By the above formula:

A n g l e = 3 0 l D i f f o f 8 a n d ^ l + 5 | 117.5

But it is not correct. I f we think carefully wc find that the angle should be less than 90. In this case, the formula differs and is given below as

Angle = 30^Diff of 8 a n d y 55 i

Minutes Minutes 5 2

If the angle is greater formula, (2)

Minutes Minutes 5 2

= 30(3)-27.5 = 62.5 Note: The two types of formulae work in two different cases.

(1) When hour hand is ahead of the minute hand (like, when the minute hand is at 4, the hour hand should be after 4, ie, between 4-5,5-6,6-7....) we use the for-mula:

30 Diff of hrs and Minutest minutes

use the formula:

30^Diff of hrs and Minutes ] minutes 5 J 2 ~

Exercise 1. Find the time between 3 and 4 O'clock when the angle

between the hands of a watch is one-third of a right angle.

a) 'Oyy m m P a s t 3

c) 1 min past 3

b) 10 min past 3

d) None of these

Find the angle between the two hands of a clock of 15 minutes past 4 O'clock. a) 38.5 b)36.5 c)37.5 d) None of these Find the angle between the two hands of a clock at 4.30 pm. a) 45 b)30 c)60 d) None of these At what angle are the two hands of a clock inclined at 20

686 PRACTICE BOOK ON QUICKER MATHS

minutes past 5? a) 30 b)45 c

Answers c)50 d)40

1. a; Hint: 3 0 3 -M

+ ^ = 30 2

or,(15-M)6-M 2 30 or, 180-12M + M = 60

120 10 M = = 10minpast3.

11 l 1 F 2. c; Hint: At 15 minutes past 4, the minute-hand is behind

the hour-hand. Hence, using the given formula (i), we have the required answer

= 30 x4-15

+ = 30 + 7.5: 2

57.5

a; Hint: At 4.30 pm the minute-hand is ahead of the hour-hand. Therefore, we apply the given formula (ii) .-. required answer

30 H _ 4 5

30 = 30x2-15 = 45

Note: Sometimes, time is given in the form of 24 hours in-stead of 12 hours in order to avoid the confusion of am and pm. For example, 14.20 or 1420 hours. In all such cases when the hour part exceeds 12, we subtract 1200 from it and then solve it. So, 14.20 reduces to 2.20 or 20 minutes past 2.

4.d

Miscellaneous 1. How many times do the hands of a clock point opposite

to each other in 12 hours? a) 6 times b) 10 times c) 11 times d) 12 times

2. How many times are the hands of a clock at right angles in a day? a) 24 times b) 48 times c) 22 times d) 44 times

3. How many times in a day are the hands of a clock straight? a) 48 times b) 24 times c) 44 times d) None of these

4. A watch which gains uniformly, is 5 min slow at 8 O'clock in the morning on Sunday, and is 5 min 48 sec fast at 8 pm on following Sunday. When was it correct? a) 20 min past 7 pm on Tuesday b) 20 min past 7 pm on Wednesday c) 10 min past 7 pm on Tuesday d) 10 min past 7 pm on Wednesday

5. A clock is set right at 8 am. The clock gains 10 minutes in 24 hours. What will be the true time when the clock indi-cates 1 pm on the following day? a) 28 hrs b) 28 hrs 48 min c) 28 hrs 42 min d) None of these

6. A clock is set right at 4 a.m. The clock loses 20 min in 24 hours. What will be the true time when the clock indi-cates 3 a.m. on 4th day? a) 4 am b)5am c)3am d)4pm

7. A watch, which gains uniformly is 2 min slow at noon or Monday, and is 4 min 48 seconds fast at 2 pm on the following Monday. When was it correct? a) 2 pm on Tuesday b) 2 pm on Wednesday c) 3 pm on Thursday d) 1 pm on Friday

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

a) 59 minutes past 3 12

11

b)4pm

7 3 c) 58 minutes past 3 d) 2 minutes past 4

9. How many times do the hands of a clock coincide in a day? a) 24 b)20 c)21 d)22

10. How many times do the hands of a clock point towards each other in a day? a) 24 b)20 c)12 d)22 At what time between 4 and 5 will the hands of a watc" be equidistant from the figure 5.

a) 27 yy min past 4

c) 27 min past 4

8 b) 27 m i n past 4

d) None of these

12. I f the hands of a clock coincide every 65 minutes (tr_: time) how much does the clock gain or lose in 24 houn'

, i 1 0 . a ) l l m i n

c) 1 0 min

, 1 b) 1 0 min

d) None of these

Answers 1. c; Hint: The hands of a clock point opposite to each otbe-

11 times in every 12 hours (because between 5 and 7. a 6 O'clock only they point opposite to each other).

2. d; Hint: In 12 hours, they are at right angles 22 times cause two positions of 3 O'clock and 9 O'clock are c mon). Therefore, in a day they are at right angles for times.

3. c; Hint: The hands coincide or are in opposite direction + 22) ie 44 times in a day.

4. b; Hint: Time between the given interval = 180 hrs

The watch gains = 5 _ "5" m ' n ' n ' ^0 hrs

locks 687

180x5 . .-. 5 min is gained in x -> =83 hrs 20 min

54 = 3 days 11 hrs 20 min .-. it was correct at 20 min past 7 pm on Wednesday,

b; Hint: Time from 8 am on a day to 1 pm on the following day is 29 hrs. Now, 24 hrs 10 min of this clock are the same as 24 hours of the correct clock.

145 ie, hrs of this clock = 24 hrs of correct clock.

6

( 2 4 x 6 ) 29 hrs of this clock = I x 2 " j hrs of correct clock.

= 28 hrs 48 min of correct clock. So, the correct time is 28 hrs 48 min after 8 am or 48 min past 12.

a; Hint: Time from 4 a.m. to 3 a.m. on 3rd day = 24 x 3 - 1 = 71 hrs.

Now, 23 hrs 40 min of this clock = 24 hrs of correct clock.

, , 2 71

or, 23 = - hrs of this clock = 24 hrs of correct clock.

24x3x71 71 hrs of this clock = 72 hrs of correct

71 clock. Therefore, the correct time = 3 a.m. +(72-71) = 4 a.m.

b; Hint: Time from Monday noon to 2 pm on following Monday = 7 days 2 hours = 170 hours

; ( 4^ 34 The watch gains | 2 + 4 ~ | or, min in 170 hours

it will gain 2 min in

5.

170x5 34

x2 hrs

= 50 hrs = 2 days 2 hrs. So, the watch is correct 2 days 2 hours after Monday noon ie, at 2 pm on Wednesday,

b; Hint: Time from 7 am to quarter past 4 = 9 hours 15 min = 555 min.

37 Now, min of this watch = 3 min of the correct watch.

12

555 min of this watch = 3x12

37 x555 min

3x12 555 -x hrs = 9 hrs of the correct watch. V 37 60 J

Correct time is 9 hours after 7 am ie, 4 pm

9. d; Hint: The hands of a clock coincide 11 times in every 12 hours (because between 11 and 1, they coincide only once, at 12 O'clock). So, the hands coincide 22 times in a day.

10. d; Hint: The hands of a clock point towards each other 11 times in every 12 hours (because between 5 and 7, at 6 O'clock only they point towards each other). So, in a day the hands point towards each other 22 times.

11. c; Hint: The hands will be equidistant from the figure 5, (i) when they are coincident between 4 and 5 and (ii) when they are in the position shown in the diagram.

For the 1st case see Rule 1 In the second case suppose that the hour-hand is at A, and the minute-hand at B, so that A5 = 5B. Since the space between 4 and 5 is equal to the space between 5 and 6, .-. 4A = 6B Hence, 12B+4A= 12B + 6B = 30min That is, the two hands, between them have moved through a space of 30 minutes since 4 O'clock. But the minute hand moves 12 times as fast as the hour hand.

Hence 12B= 30 x 12 13

27 m i n .

.-. the required time is 27 min past 4.

12. b; Hint: The minute-hand gains 60 minutes over the hour-

60x60 5 hand in or " y y minutes. Therefore, the hands

of a correct clock coincide every 65 minutes.

But the hands of the clock mentioned in the question coincide every 65 minutes. Hence in 65 minutes, the clock

gains min.

.-. in 60 x 24 min or 24 hours it gains

A x J _ x 6 0 x 2 4 = 1 0 min 11 65 143