#### Copyright Disclaimer:Section 107 of the Copyright Act Fair Use Contents . We are forwarding content link(s) from our website to content website & We are not serving any contents. Main Source:Google.com.All the Content PDF link(s) is/are obtained from GoogleSearch for the purpose of Education & Teaching Intention. Not for commercial purpose. Technicalsymposium.com is not liable/responsible for any copyright issues.

1. What is the angle between the two hands of a clock when the time shown by the

clock is 5.30 p.m. ?

a) 00 b) 50 c)30 d)150

2. By how many degrees does the minute hand move in the same time, in which the

hour hand move by 18 degree ?

a)168 degree b)216 degree c) 196 degree d) 276 degree

3. A watch, which loses time uniformly, was observed to be 5 minutes fast at 8.00 p.m.
on Thursday. It was noticed to be 7 minutes slow at 8.00 a.m. on the subsequent
Monday. When did the watch show the correct time ?

a) 7 a.m. Saturday b) 7 a.m. on Friday

c) 10a.m. on Sunday d) 11 a.m. on Friday

4. In the time in which the second hand covers 3960 degrees, how many degrees does

the hour hand move?

a) 11 b) 5.5 c)3/4 d) 55

5. By how many degrees does the hour hand lag behind the minute hand in a span of

42 minutes, if initially they are at the same position?

a) 2330 b)211 degree c)231 degree d)None of these

6. March 1st is Wednesday. Which month of the same year has the same calendar

a)July b) November c) December d)October

7. If today is Thursday , after 730 days which will be the day of the week ?

a)Thursday b)Friday c)Saturday d)Monday

8. A year starting with Monday and ending with Tuesday. How many days are there
from 16th January to 15th March of that year.

a) 58 b)60 c)59 d)61

9. The last day of a century cannot be :

a) Monday b) Wednesday c) Friday d) Saturday

10. What will be the day of the week on 1st January, 2010 ?

a) Friday b) Saturday c) Sunday d) Monday

1. q = 11/2 m – 30h

= 11/2 *30 - 30 *5

= 165-150 = 150

2. 18*2 *6 = 216 degree

3. The number of hours from 8:00 p.m. on Thursday to 8:00 a.m. on Monday = 84
hours.

In 84 hours, the clock gained 12 minutes.

But to show the correct time, the clock has to gain 5 minutes.

:. 5/12 *84 = 35 hours.

35 hours from 8:00 p.m. on Thursday is 7:00 a.m. on Saturday.

4. 3960/(360*2) = 5.50

5. 42*6-(42/2) = 252-21 = 231 degree

6. Odd days from March to October : 3 + 2 +3 +2 +3 +3 +2 +3 = 21/7 = 0 . If odd day
is 0 then next month has the same calendar.

7. 730/7 = 2 odd days. So Saturday will be the day of the week.

8. It is a leap year. So 16(Jan)+ 29(Feb) + 15(March) = 60 days

9. 100 years contains 5 odd days. So last day of 1st century is ‘Friday’.

200 years contains (5*2) =10 = 3 odd days. So last day of second century is
‘Wednesday’. 300 years contain 15 odd days = 1 odd day.

. : Last day of 3rd century is ‘Monday’. 400 years contains 0 odd day.

: . Last day of 4th century is ‘Sunday’.

Since the order is continually kept in successive cycles, we see that the last day of a
century cannot be Tuesday, Thursday or Saturday

10. 2000 years have 2 odd days.

Number of odd days from 2001 –2009 = 11 odd days = 4 odd days. 1 st January
2010 has 1 odd day. Total number of odd days = (2 + 4 + 1) = 7 = 0 odd day

: . 1st January, 2010 will be a Sunday

Important Formula and Equations

Minute Spaces: The face or dial of watch is a circle whose circumference is divided into 60 equal parts, called minute spaces.

Hour Hand and Minute Hand: A clock has two hands, the smaller one is called the hour hand or short hand while the larger one is called minute hand or long hand .

In 60 minutes, the minute hand gains 55 minutes on the hour on the hour hand.

In every hour, both the hands coincide once.

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.

Angle traced by hour hand in 12 hrs = 360°

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

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 .

Odd Days: We are supposed to find the day of the week on a given date. For this, we use the concept of 'odd days'. In a given period, the number of days more than the complete weeks are called odd days .

Leap Year: (i). Every year divisible by 4 is a leap year, if it is not a century. (ii). Every 4th century is a leap year and no other century is a leap year.

Ordinary Year: The year which is not a leap year is called an ordinary years . An ordinary year has 365 days.

Counting of Odd Days: 1 ordinary year = 365 days = (52 weeks + 1 day.) . 1 ordinary year has 1 odd day.

1 leap year = 366 days = (52 weeks + 2 days) 1 leap year has 2 odd days.
100 years = 76 ordinary years + 24 leap years
= (76 x 1 + 24 x 2) odd days = 124 odd days.
= (17 weeks + days) = 5 odd days.

Number of odd days in 100 years = 5

Number of odd days in 200 years = (5 x 2)= 3 odd days.

Number of odd days in 300 years = (5 x 3) = 1 odd day.

Number of odd days in 400 years = (5 x 4 + 1) = 0 odd day.

Similarly, each one of 800 years, 1200 years, 1600 years, 2000 years etc. has 0 odd days.

Day of the Week Related to Odd Days (Assuming that 1AD January 1st is a Sunday):

 No. of days: 0 1 2 3 4 5 6 Day: Sun. Mon. Tues. Wed. Thurs. Fri. Sat.

Key Notes:

Clocks Concepts :

The dial of the clock is circular in shape and was divided into 60 equal minute spaces

60 minute spaces traces an angle of 3600. Therefore, 1minute space traverses an angle of 60

In 1 hour, Minute hand traverses 60 minute space or 3600 ,Hour hand traverses 5 minute space or 300

The hands of the clock are perpendicular in 15 minute spaces apart

The hands of the clock are in straight line and opposite to each other in 30 minute spaces apart.

The hands of the clock are in straight line when they coincide or opposite to each other.

The hands of the clock are perpendicular to each other for 22 times in 12 hours and for 44 times in a day.

The hands of the clock are opposite to each other for 11 times in 12 hours and 22 times in a day.

The hands of the clock coincides with each other for 11 times in 12 hours and 22 times per day

The hands of the clock are 44 times in a straight line per day

The minute hand gain 55 minutes over hour hand per hour.

Hence x minute space to be gained by minute hand over hour hand can be calculated as x.(60/55) or x.(12/11)

Ex : At what time between 2'O clock and 3'O clock the hands of the clock
are opposite to each other.
1. 34( 6/11 ) past 2'Oclock 2. 43( 7/11 ) past 2'Oclock
3. 56( 8/11 ) past 2'Oclock 4. 64(9/11past 2'Oclock
Sol At 2'O clock the minute hand will be at 12 as shown below

The minutes hand to coincide with the hour hand it should trace at first 10 minute spaces
And then the hands of the clocks to be opposite to each other minute hand should trace 30 minute spaces i.e. totally it should gain 10+30=40 minute spaces to be opposite to that of hour hand
We know that,
Minute hand gains 55 minute spaces over hour hand in 1 hour
Therefore, Minute hand gain 40 minute spaces over hour hand in 40 × (60/55) = 43(7/11)
Hence the hand of the clock will minutes be opposite to each at 43( 7/11 ) past 2'Oclock
Therefore, Correct option is 2'

When clock is too fast, too slow
If a clock or watch indicates 6 hr 10 min when the correct time is 6, it is said that the clock is 10 min too fast
If it indicates 6. 40 when the correct time is 7, it is said to be
20 min too slow.
Now let us have an example based on this concept

Ex. My watch, which gains uniformly, is 2 min, & show at noon on Sunday, and is 4 min 48 seconds fast at 2 p.m on the following Sunday when was it correct ?

Sol: From Sunday noon to the following Sunday at 2 p.m there are 7 days 2 hours or 170 hours.
The watch gains 2+4 4/5 min in 170 hrs.
Therefore, the watch gains 2 min in 2 *170 hrs i.e., 50 hours
6 4/5
Now 50 hours = 2 days 2 hrs.
Therefore, 2 days 2 hours from Sunday noon = 2 p.m on Tuesday.

Calendars Concept :

The time in which the earth travels round the sun is a solar year and is equal to 365 days 5 hrs. 48 minutes and 47 1/2 seconds

Year is 365.2422 days approximately.

The common year consists of 365 days.

The difference between a common year and a solar year is therefore 0.2422 of a day and we consider it by adding a whole day to every fourth year.

Consequently in every 4th year there are 366 days.

The years which have the extra day are called leap years. The day is inserted at the end of February, The difference between 4 common years and 4 solar years is 0.969 of a day.

If therefore, we add a whole day to every 4th year, we add too much by 0.0312 of a day. To take account of this, we omit the extra day three times every 400 years,

The thing is to ensure that each season may fall at the same time of the year in all years. In course of time, without these corrections, we should have winter in July and summer in January also .

With the very small variation, the present divisions of the year are those given in B.C 46 by Julius Caesar . The omission of the extra day three times in 400 years is called the Gregorian Correction. This correction was adopted at once in 1582 in Roman Catholic Countries. but not in England until, 1752.
The Gregorian mode of reckoning is called the New Style, the former, the Old Style.
The New Style has not yet been adopted in Russia , so that they are now 13 days behind us as an Example What we call Oct. 26th they call 13th Oct . They have Christmas day on 7th of January and we have on 25th December every year.

In an ordinary year there are 365 days i.e., 52 weeks + 1 day
Therefore an ordinary years contains 1 odd day.
A leap year contains 2 odd days.

100 year = 76 ordinary years + 24 leap years.
= 76 odd days + 48 odd days
= 124 odd days = 17 weeks + 5 days. (in the consideration of weeks)
Therefore, 100 years contain 5 odd days.
200 years contains 3 odd days.
300 years contain 1 odd days

Since there are 5 odd days in 100 years, there will be 20 days in 400 years. But every 4th century is a leap year.
Therefore, 400 years contain 21 days. Here 400 years contain no odd days.

As First January 1 AD was Monday. we must count days from Sunday
i.e. Sunday for 0 odd days , Monday for 1 odd day , Tuesday for 2 odd days and so on.

Last day of a century cannot be either Tuesday. Thursday or Saturday.

The first day of a century must either be Monday. Tuesday, Thursday or Saturday.

Now let us observe the Examples

Ex How many times does the 29th days of the month occur in 400 consecutive years
1) 97 times 2) 4400 times 3) 4497 times 4) none
Sol: In 400 consecutive years there are 97 leap years. Hence in 400 consecutive years, February has the 29th day 97 times, and the remaining 11 months have the 29th day 400 x 11 or 4400 times.
Therefore, 29th day of the month occurs (4400 + 97) or 4497 times

Ex Given that on 10th November 1981 is Tuesday, what was the day on 10th November 1581
1) Monday 2) Thursday 3) Sunday 4) Tuesday
Sol: After every 400 years, the same day comes.
Thus if 10th November1981 was Tuesday, before 400 years i.e on 10th November 1581, it has to be Tuesday.

Exercise questions

1. What is the angle between the two hands of a clock when the time shown by the clock is 6.30 p.m. ?

a) 00
b) 50
c)30
d)150

Ans: option d
Explanation: q = 11/2 m – 30h
= 11/2 *30 - 30 *6
= /165-180/ = 150

2. At what time between 3 and 4 o’clock will the minute hand and the hour hand are on the same straight line but facing opposite directions.

a) 3:49
b)3:15
c)3:39 1/11
d)3:49 1/11

Ans: option d
Explanation : On straight line means 180 degree angle.
180 = 11/2m- 30h
180 = 11/2 m –30*3
180 = 11/2m-90
(180+90)2 = 11m
m = 540/11 = 49 1/11

3. By how many degrees does the minute hand move in the same time, in which the hour hand move by 280 ?

a)168
b)336
c) 196
d) 376

Ans: option b
Explanation: 28*2 *6 = 3360

4. At what time, between 3 o’clock and 4 o’clock, both the hour hand and minute hand coincide each other?

a) 3:30
b) 3:16 4/11
c) 3:1611/4
d) 3:16 7/11

Ans: Option b
Explanation : Coincide means 00 angle.
0 =11/2m –30*3
11m = 90*2 = 180
m= 180/11 = 16 4/11
So time = 3 : 16 4/11

5. How many degrees will the minute hand move, in the same time in which the second hand move 4800 ?
a ) 60
b) 90
c) 40
d) 80

Ans: Option d
Explanation : Minute hand covers 480/60= 80

6. How many years have 29 days in February from 2001 to 2100.

a)26
b)25
c)23
d)24

Ans: option d
Explanation: 100th year is not a leap year. So 24 February’s has 29 days

7. 2012 January 1st is Sunday, then which day is the Indian Independence day of the same year.

a) Saturday
b) Wednesday
c) Thursday
d) Friday

Ans: Option b
Explanation: 30+ 29+ 31 + 30 + 31 + 30 + 31+15 = 227/7 = reminder = 3
So Independence day is Wednesday

8. Which year has the same calendar as 1700 ?

a) 1705
b)1706
c)1707
d)1708

Ans: Option b

Explanation:
Year : 1700 1701 1702 1703 1704 1705
Odd days : 1 1 1 1 2 1

9. If Arun’s birthday is on May 25 which is Monday and his sister’s birthday is on July 13. Which day of the week is his sister’s birthday?

a) Monday
b) Wednesday
c) Thursday
d) Friday

Ans: option a
Explanation: Reference day : May 25th Monday
Days from May 25th to July 13 = 6 + 30 +13 = 49
No of odd days : 49/7 = 0

10. March 1st is Wednesday. Which month of the same year starts with the same day?

a) October
b) November
c) December
d) None of these

Ans: Option b
Explanation:
Month : Mar April May June July Aug Sept. Oct.
Odd dys : 3 2 3 2 3 3 2 3
Total 21 odd days. 21/7 = 0. So November has start with the same day

Day sequence Test: In these question a date given with the day of the week (Monday, etc.....) using that information you will be asked to find out day of other date.

Approach:

1)Find the No. of years and No. of days between the "given date" and "asked date".

2)Divide the No. of days by 7.

3)Find the reminder of the division.

4)If the reminder is '0' then the required day is as same as "given days".

5)If the reminder is '1' then the required day is the next day to given day in question date?

6)If the reminder is '2' then the required day is two days after the "given date", so on for the 3, 4, 5 and 6 reminders.

Examples:

1) If 15th August is Monday, 17th November comes on which day?

Answer: August month has 31 day, September has 30 days and October has 31 days.
The number of days between 15th August to 17th November= 16 + 30 + 31 + 17= 94.
94/7, it leaves reminder 3. 17th November= 3 days after Monday.
i.e. Thursday.

2) If 7th February of 2006 is Saturday, 24th April 2006 comes on which day?

Answer: 2006 is not a leap year, so February has 28 days, March 2006 has 31 days.
The No. of days between 7th February 2006 to 24th April 2006 = 21 + 31 + 24 = 76.
76/7, it leaves reminder 6, 24 th April 2006 = 6 days after Saturday = Friday.

3) 14th December 2004 is Monday, 25th November 2005 comes on which day?

Answer: The No. of days between 14th December 2004 to 25 th November 2005 = 365 - 19 (days gap between 14th December to 25 th November = 19 and 2005 is a normal year) = 346.
346/7, it leaves reminder 3; 25th November 2005 = 3 days after Monday = Thursday.

4) If 26th March 2006 is Friday, 18th May 2008 is comes on which day?

Answer: Days between Step 1 : from 26th March 2006 to 25th March 2007 - 365 days
Step 2 : from 26 th March 2007 to 25 th March 2008 – 366 days 26 th March 2008 to 18 th May 2008 = 31 + 22 = 53 days Total days = 365 + 366 + 53 = 784.
784/7 = 112, it leaves no reminder.
18th May 2008 is also Friday.

Tips:

A year has 365 days

A leap year has 366 days

Any year is divisible by 4 is a leap year e.g. 2004, 2008,.......

In normal year, February has 28 days but leap year has 29 days. Hence the leap year has one day extra than normal year.

In a normal year, if the year begins with Sunday, it ends with Sunday. The next year starting day will be Monday.

In a Leap year, if the year begins with Sunday, it ends with Monday, the starting day of the next year of the leap year will be Tuesday.

The no. of days in January = 31 days

The no. of days in February = 28 or 29 days

The no. of days in March = 31 days

The no. of days in April = 30 days

The no. of days in May = 31 days

The no. of days in June = 30 days

The no. of days in July = 31 days

The no. of days in August = 31 days

The no. of days in September = 30 days

The no. of days in October = 31 days

The no. of days in November = 30 days

The no. of days in December = 31 days

In a normal year, it has 52 weeks and one day

In a leap year, it has 52 weeks and 2 days.

For every 400 years, the day repeats. e.g. 14th April 1604 is Wednesday, 14th April 2004 is also Wednesday.

1. What is the angle between the two hands of a clock when the time shown by the

clock is 5.30 p.m. ?

a) 00 b) 50 c)30 d)150

2. By how many degrees does the minute hand move in the same time, in which the

hour hand move by 18 degree ?

a)168 degree b)216 degree c) 196 degree d) 276 degree

3. A watch, which loses time uniformly, was observed to be 5 minutes fast at 8.00 p.m.
on Thursday. It was noticed to be 7 minutes slow at 8.00 a.m. on the subsequent
Monday. When did the watch show the correct time ?

a) 7 a.m. Saturday b) 7 a.m. on Friday

c) 10a.m. on Sunday d) 11 a.m. on Friday

4. In the time in which the second hand covers 3960 degrees, how many degrees does

the hour hand move?

a) 11 b) 5.5 c)3/4 d) 55

5. By how many degrees does the hour hand lag behind the minute hand in a span of

42 minutes, if initially they are at the same position?

a) 2330 b)211 degree c)231 degree d)None of these

6. March 1st is Wednesday. Which month of the same year has the same calendar

a)July b) November c) December d)October

7. If today is Thursday , after 730 days which will be the day of the week ?

a)Thursday b)Friday c)Saturday d)Monday

8. A year starting with Monday and ending with Tuesday. How many days are there
from 16th January to 15th March of that year.

a) 58 b)60 c)59 d)61

9. The last day of a century cannot be :

a) Monday b) Wednesday c) Friday d) Saturday

10. What will be the day of the week on 1st January, 2010 ?

a) Friday b) Saturday c) Sunday d) Monday

1. q = 11/2 m – 30h

= 11/2 *30 - 30 *5

= 165-150 = 150

2. 18*2 *6 = 216 degree

3. The number of hours from 8:00 p.m. on Thursday to 8:00 a.m. on Monday = 84
hours.

In 84 hours, the clock gained 12 minutes.

But to show the correct time, the clock has to gain 5 minutes.

:. 5/12 *84 = 35 hours.

35 hours from 8:00 p.m. on Thursday is 7:00 a.m. on Saturday.

4. 3960/(360*2) = 5.50

5. 42*6-(42/2) = 252-21 = 231 degree

6. Odd days from March to October : 3 + 2 +3 +2 +3 +3 +2 +3 = 21/7 = 0 . If odd day
is 0 then next month has the same calendar.

7. 730/7 = 2 odd days. So Saturday will be the day of the week.

8. It is a leap year. So 16(Jan)+ 29(Feb) + 15(March) = 60 days

9. 100 years contains 5 odd days. So last day of 1st century is ‘Friday’.

200 years contains (5*2) =10 = 3 odd days. So last day of second century is
‘Wednesday’. 300 years contain 15 odd days = 1 odd day.

. : Last day of 3rd century is ‘Monday’. 400 years contains 0 odd day.

: . Last day of 4th century is ‘Sunday’.

Since the order is continually kept in successive cycles, we see that the last day of a
century cannot be Tuesday, Thursday or Saturday

10. 2000 years have 2 odd days.

Number of odd days from 2001 –2009 = 11 odd days = 4 odd days. 1 st January
2010 has 1 odd day. Total number of odd days = (2 + 4 + 1) = 7 = 0 odd day

: . 1st January, 2010 will be a Sunday.

Clocks and Calendars Aptitude basics, practice questions, answers and explanations
Prepare for companies tests and interviews

Important Formula and Equations

Minute Spaces: The face or dial of watch is a circle whose circumference is divided into 60 equal parts, called minute spaces.

Hour Hand and Minute Hand: A clock has two hands, the smaller one is called the hour hand or short hand while the larger one is called minute hand or long hand .

In 60 minutes, the minute hand gains 55 minutes on the hour on the hour hand.

In every hour, both the hands coincide once.

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.

Angle traced by hour hand in 12 hrs = 360°

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

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 .

Odd Days: We are supposed to find the day of the week on a given date. For this, we use the concept of 'odd days'. In a given period, the number of days more than the complete weeks are called odd days .

Leap Year: (i). Every year divisible by 4 is a leap year, if it is not a century. (ii). Every 4th century is a leap year and no other century is a leap year.

Ordinary Year: The year which is not a leap year is called an ordinary years . An ordinary year has 365 days.

Counting of Odd Days: 1 ordinary year = 365 days = (52 weeks + 1 day.) . 1 ordinary year has 1 odd day.

1 leap year = 366 days = (52 weeks + 2 days) 1 leap year has 2 odd days.
100 years = 76 ordinary years + 24 leap years
= (76 x 1 + 24 x 2) odd days = 124 odd days.
= (17 weeks + days) = 5 odd days.

Number of odd days in 100 years = 5

Number of odd days in 200 years = (5 x 2)= 3 odd days.

Number of odd days in 300 years = (5 x 3) = 1 odd day.

Number of odd days in 400 years = (5 x 4 + 1) = 0 odd day.

Similarly, each one of 800 years, 1200 years, 1600 years, 2000 years etc. has 0 odd days.

Day of the Week Related to Odd Days (Assuming that 1AD January 1st is a Sunday):

 No. of days: 0 1 2 3 4 5 6 Day: Sun. Mon. Tues. Wed. Thurs. Fri. Sat.

Key Notes:

Clocks Concepts :

The dial of the clock is circular in shape and was divided into 60 equal minute spaces

60 minute spaces traces an angle of 3600. Therefore, 1minute space traverses an angle of 60

In 1 hour, Minute hand traverses 60 minute space or 3600 ,Hour hand traverses 5 minute space or 300

The hands of the clock are perpendicular in 15 minute spaces apart

The hands of the clock are in straight line and opposite to each other in 30 minute spaces apart.

The hands of the clock are in straight line when they coincide or opposite to each other.

The hands of the clock are perpendicular to each other for 22 times in 12 hours and for 44 times in a day.

The hands of the clock are opposite to each other for 11 times in 12 hours and 22 times in a day.

The hands of the clock coincides with each other for 11 times in 12 hours and 22 times per day

The hands of the clock are 44 times in a straight line per day

The minute hand gain 55 minutes over hour hand per hour.

Hence x minute space to be gained by minute hand over hour hand can be calculated as x.(60/55) or x.(12/11)

Ex : At what time between 2'O clock and 3'O clock the hands of the clock
are opposite to each other.
1. 34( 6/11 ) past 2'Oclock 2. 43( 7/11 ) past 2'Oclock
3. 56( 8/11 ) past 2'Oclock 4. 64(9/11past 2'Oclock
Sol At 2'O clock the minute hand will be at 12 as shown below

The minutes hand to coincide with the hour hand it should trace at first 10 minute spaces
And then the hands of the clocks to be opposite to each other minute hand should trace 30 minute spaces i.e. totally it should gain 10+30=40 minute spaces to be opposite to that of hour hand
We know that,
Minute hand gains 55 minute spaces over hour hand in 1 hour
Therefore, Minute hand gain 40 minute spaces over hour hand in 40 × (60/55) = 43(7/11)
Hence the hand of the clock will minutes be opposite to each at 43( 7/11 ) past 2'Oclock
Therefore, Correct option is 2'

When clock is too fast, too slow
If a clock or watch indicates 6 hr 10 min when the correct time is 6, it is said that the clock is 10 min too fast
If it indicates 6. 40 when the correct time is 7, it is said to be
20 min too slow.
Now let us have an example based on this concept

Ex. My watch, which gains uniformly, is 2 min, & show at noon on Sunday, and is 4 min 48 seconds fast at 2 p.m on the following Sunday when was it correct ?

Sol: From Sunday noon to the following Sunday at 2 p.m there are 7 days 2 hours or 170 hours.
The watch gains 2+4 4/5 min in 170 hrs.
Therefore, the watch gains 2 min in 2 *170 hrs i.e., 50 hours
6 4/5
Now 50 hours = 2 days 2 hrs.
Therefore, 2 days 2 hours from Sunday noon = 2 p.m on Tuesday.

Calendars Concept :

The time in which the earth travels round the sun is a solar year and is equal to 365 days 5 hrs. 48 minutes and 47 1/2 seconds

Year is 365.2422 days approximately.

The common year consists of 365 days.

The difference between a common year and a solar year is therefore 0.2422 of a day and we consider it by adding a whole day to every fourth year.

Consequently in every 4th year there are 366 days.

The years which have the extra day are called leap years. The day is inserted at the end of February, The difference between 4 common years and 4 solar years is 0.969 of a day.

If therefore, we add a whole day to every 4th year, we add too much by 0.0312 of a day. To take account of this, we omit the extra day three times every 400 years,

The thing is to ensure that each season may fall at the same time of the year in all years. In course of time, without these corrections, we should have winter in July and summer in January also .

With the very small variation, the present divisions of the year are those given in B.C 46 by Julius Caesar . The omission of the extra day three times in 400 years is called the Gregorian Correction. This correction was adopted at once in 1582 in Roman Catholic Countries. but not in England until, 1752.
The Gregorian mode of reckoning is called the New Style, the former, the Old Style.
The New Style has not yet been adopted in Russia , so that they are now 13 days behind us as an Example What we call Oct. 26th they call 13th Oct . They have Christmas day on 7th of January and we have on 25th December every year.

In an ordinary year there are 365 days i.e., 52 weeks + 1 day
Therefore an ordinary years contains 1 odd day.
A leap year contains 2 odd days.

100 year = 76 ordinary years + 24 leap years.
= 76 odd days + 48 odd days
= 124 odd days = 17 weeks + 5 days. (in the consideration of weeks)
Therefore, 100 years contain 5 odd days.
200 years contains 3 odd days.
300 years contain 1 odd days

Since there are 5 odd days in 100 years, there will be 20 days in 400 years. But every 4th century is a leap year.
Therefore, 400 years contain 21 days. Here 400 years contain no odd days.

As First January 1 AD was Monday. we must count days from Sunday
i.e. Sunday for 0 odd days , Monday for 1 odd day , Tuesday for 2 odd days and so on.

Last day of a century cannot be either Tuesday. Thursday or Saturday.

The first day of a century must either be Monday. Tuesday, Thursday or Saturday.

Now let us observe the Examples

Ex How many times does the 29th days of the month occur in 400 consecutive years
1) 97 times 2) 4400 times 3) 4497 times 4) none
Sol: In 400 consecutive years there are 97 leap years. Hence in 400 consecutive years, February has the 29th day 97 times, and the remaining 11 months have the 29th day 400 x 11 or 4400 times.
Therefore, 29th day of the month occurs (4400 + 97) or 4497 times

Ex Given that on 10th November 1981 is Tuesday, what was the day on 10th November 1581
1) Monday 2) Thursday 3) Sunday 4) Tuesday
Sol: After every 400 years, the same day comes.
Thus if 10th November1981 was Tuesday, before 400 years i.e on 10th November 1581, it has to be Tuesday.

Exercise questions

1. What is the angle between the two hands of a clock when the time shown by the clock is 6.30 p.m. ?

a) 00
b) 50
c)30
d)150

Ans: option d
Explanation: q = 11/2 m – 30h
= 11/2 *30 - 30 *6
= /165-180/ = 150

2. At what time between 3 and 4 o’clock will the minute hand and the hour hand are on the same straight line but facing opposite directions.

a) 3:49
b)3:15
c)3:39 1/11
d)3:49 1/11

Ans: option d
Explanation : On straight line means 180 degree angle.
180 = 11/2m- 30h
180 = 11/2 m –30*3
180 = 11/2m-90
(180+90)2 = 11m
m = 540/11 = 49 1/11

3. By how many degrees does the minute hand move in the same time, in which the hour hand move by 280 ?

a)168
b)336
c) 196
d) 376

Ans: option b
Explanation: 28*2 *6 = 3360

4. At what time, between 3 o’clock and 4 o’clock, both the hour hand and minute hand coincide each other?

a) 3:30
b) 3:16 4/11
c) 3:1611/4
d) 3:16 7/11

Ans: Option b
Explanation : Coincide means 00 angle.
0 =11/2m –30*3
11m = 90*2 = 180
m= 180/11 = 16 4/11
So time = 3 : 16 4/11

5. How many degrees will the minute hand move, in the same time in which the second hand move 4800 ?
a ) 60
b) 90
c) 40
d) 80

Ans: Option d
Explanation : Minute hand covers 480/60= 80

6. How many years have 29 days in February from 2001 to 2100.

a)26
b)25
c)23
d)24

Ans: option d
Explanation: 100th year is not a leap year. So 24 February’s has 29 days

7. 2012 January 1st is Sunday, then which day is the Indian Independence day of the same year.

a) Saturday
b) Wednesday
c) Thursday
d) Friday

Ans: Option b
Explanation: 30+ 29+ 31 + 30 + 31 + 30 + 31+15 = 227/7 = reminder = 3
So Independence day is Wednesday

8. Which year has the same calendar as 1700 ?

a) 1705
b)1706
c)1707
d)1708

Ans: Option b

Explanation:
Year : 1700 1701 1702 1703 1704 1705
Odd days : 1 1 1 1 2 1

9. If Arun’s birthday is on May 25 which is Monday and his sister’s birthday is on July 13. Which day of the week is his sister’s birthday?

a) Monday
b) Wednesday
c) Thursday
d) Friday

Ans: option a
Explanation: Reference day : May 25th Monday
Days from May 25th to July 13 = 6 + 30 +13 = 49
No of odd days : 49/7 = 0

10. March 1st is Wednesday. Which month of the same year starts with the same day?

a) October
b) November
c) December
d) None of these

Ans: Option b
Explanation:
Month : Mar April May June July Aug Sept. Oct.
Odd dys : 3 2 3 2 3 3 2 3
Total 21 odd days. 21/7 = 0. So November has start with the same day

Simple way to solve clock based problems in Aptitude Section

Clock based problems are one of the frequently asked questions in most of the competitive

exam. To solve these problems, it is always better to understand some of the basic principles and the types of problems that get asked. In this post I hereby explained simple tricks and some simple formulas for solving clock based problems.

In every competitive exams clock questions are categorized in to two ways.

• Problems in angles
• Problems on incorrect clocks

Problems in angles

Method :1

Before we actually start solving problems on angles, we need to know couple of basic facts clear:

• Speed of the hour hand = 0.5 degrees per minute (dpm)
• Speed of the minute hand = 6 dpm
• At ‘n’ o’ clock, the angle of the hour hand from the vertical is 30n

The questions based upon these could be of the following types

Example : 1

What is the angle between the hands of the clock at 7:20

At 7 o’ clock, the hour hand is at 210 degrees from the vertical.

In 20 minutes,

Hour hand = 210 + 20*(0.5) = 210 + 10 = 220 {The hour hand moves at 0.5 dpm}

Minute hand = 20*(6) = 120 {The minute hand moves at 6 dpm}

Difference or angle between the hands = 220 – 120 = 100 degrees

Method : 2

Example :2

Find the reflex angle between the hands of a clock at 05.30?

The above problem are solved by the bellow formula

Angle between X and Y =|(X*30)-((Y*11)/2)|

Angle between hands at 5:30

Step 1: X=5 , Y=30

Step 2: 5*30=150

Step 3: (30*11)/2 = 165

Step 4: 165-150=15

Thus, angle between hands at 5:30 is 15 degrees.

Method : 3

Example : 3

At what time 3&4’o clock in the hands of clock together.

Approximately we know at 03:15 hands of the clock together

So 15*60/55=16.36 min

Problems on incorrect clocks

Such sort of problems arise when a clock runs faster or slower than expected pace. When solving these problems it is best to keep track of the correct clock.

Example : 4

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?

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

Important Notes

• Two right angles per hour(Right angle = 90, Straight angle=180)
• Forty four right angles per day
• Between every two hours the hands of the clock coincide with each other for one time except between 11, 12 and 12, 1.In a day they coincide for 22 times.
• Between every two hours they are perpendicular to each other two times except between 2, 3 and 3, 4 and 8, 9 and 9, 10.In a day they will be perpendicular for 44 times.
• Between every two hours they will be opposite to each other one time except between 5, 6 and 6, 7.In a day they will be opposite for 22 times.

Type 1: Finding the time when the angle between the two hands is given.
Type 2: Finding the angle between the 2 hands at a given time.
Type 3: Questions on clocks gaining/losing time.

###### Basic Concept of Clocks:

A clock is a complete circle having 360 degrees. It is divided into 12 equal parts i.e. each part is 360/12 = 30°.
As the minute hand takes a complete round in one hour, it covers 360° in 60 minutes.
In 1 minute it covers 360/60 = 6°/ minute.
Also, as the hour hand covers just one part out of the given 12 parts in one hour. This implies it covers 30° in 60 minutes i.e. ½° per minute.
This implies that the relative speed of the minute hand is 6 - ½ = 5 ½ degrees.
We will use the concept of relative speed and relative distance while solving problems on clocks.

• Every hour, both the hands coincide once. In 12 hours, they will coincide 11 times. It happens due to only one such incident between 12 and 1'o clock.
• The hands are in the same straight line when they are coincident or opposite to each other.
• When the two hands are at a right angle, they are 15-minute spaces apart. In one hour, they will form two right angles and in 12 hours there are only 22 right angles. It happens due to right angles formed by the minute and hour hand at 3’o clock and 9'o clock.
• When the hands are in opposite directions, they are 30-minute spaces apart.
• If both the hour hand and minute hand move at their normal speeds, then both the hands meet after 65 minutes.

Now, let's apply the above concept to some questions.

Important Formulas - Clock

1. Minute Spaces

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

2. 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.

3. In 60 minutes, minute hand gains 55 minute spaces over the hour hand.

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

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

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

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

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

8. Angle traced by hour hand in 12 hrs = 360°

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

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

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

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

13. The hands of a clock coincide 22 times in a day.

14. The hands of a clock are straight 44 times in a day.

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

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?

 144º 150º 168º 180º

The reflex angle between the hands of a clock at 10.25 is:

180º

 192 1 º 2

195º

 197 1 º 2

A clock is started at noon. By 10 minutes past 5, the hour hand has turned through:

 145º 150º 155º 160º

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:

 59 7 min. past 3 12

4 p.m.

 58 7 min. past 3 11
 2 3 min. past 4 11

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

 32 8 min. 11
 36 5 min. 11

90 min.

96 min.

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

5 min. past 7

 5 2 min. past 7 11
 5 3 min. past 7 11
 5 5 min. past 7 11

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

 43 5 min. past 5 11
 43 7 min. past 5 11

40 min. past 5

45 min. past 5

The angle between the minute hand and the hour hand of a clock when the time is 4.20, is:

 0º 10º 5º 20º

At what angle the hands of a clock are inclined at 15 minutes past 5?

 58 1 º 2

64º

 67 1 º 2
 72 1 º 2

At 3:40, the hour hand and the minute hand of a clock form an angle of:

 120° 125° 130° 135°

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

 5 1 " 11
 12 4 " 11
 13 4 " 11
 16 4 " 11

How many times do the hands of a clock coincide in a day?

 20 21 22 24

How many times in a day, the hands of a clock are straight?

 22 24 44 48

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?

 2 p.m. on Tuesday 2 p.m. on Wednesday 3 p.m. on Thursday 1 p.m. on Friday