clock and calendar

1. QA09 CLOCK

2. • A clock has 2 hands, the smaller one is called
the hour hand or short hand while the larger
one is called the minute hand or long hand.
• The face of a clock is a circle which subtends
an angle of 360° at the centre

3. • In every hour
(a) Both the hands coincide once. At this point
the angle between them is 0°.
(b)The hands are straight (point in opposite
directions) once. At this point the angles
between them are 180°.
(c) The hands are twice perpendicular to each
other. At this point the angle between them
is 90°.

5. • When the two hands are at right angles, they
are15min spaces apart.
• When the two hands are in opposite
directions, they are 30 min spaces apart.
• In 60 min the minute hand gains 55 min on
the hour hand.

6. • The minute hand moves 12 times as fast as
the hour hand.
• Too Fast: If a clock indicates 4 : 30, when the
correct time is 4:20, it is said to be 10 min too
fast.
• Too Slow: If a clock indicates 7 : 30, when the
correct time is 7: 45 , it is said to be 15 min
too slow.

8. • Q2: A watch which gains uniformly is 4 min slow at 7 pm on
Monday and is 4 min 24 s fast at 7 pm on following
Monday.
• When was the watch correct ?
• Solution.
• Total time in hours from Monday at 7 pm to the following
Monday at 7 pm = (7 × 24 ) = 168 hours.
• In 168 hours the watch gains : (4+(22/5)) = 42/5mins
• Then 4 mins are gained in = ((168× (5/42) × 4 )= 80 hours =
3days and 8 hours.
• Watch is correct 3 days and 8 h after 7 pm Monday
• i.e, it will be correct at 3 am on Friday.

9. • Q3 :How many times in a day, are the hands of a
clock in straight line but opposite in direction?
• Solution:
• The hands of a clock point in opposite directions
(in the same straight line) 11 times in every 12
hours.
• (Because between 5 and 7 they point in opposite
directions at 6 o clock only).
• So, in a day, the hands point in the opposite
directions 22 times.

10. • If you look at a clock and the time is 3:15.
• What is the angle between the hour and the minute
hands?
• At 3:15 minute hand will be perfectly horizontal
pointing towards 3. Whereas hour hand will be
towards 4.
• Also, hour hand must have covered 1/4 of angle
between 3 and 4.
• The angle between two adjacent digits is 360/12 = 30
degrees.
• Hence 1/4 of it is 7.5 degrees.

11. Adding Time
• Let's add 1 hr and 35 minutes and 3 hr 55 minutes together.
• What you do is this:
• make the 1 hr 35 minutes into one number, which will give us 135 and do
the same for the other number, 3 hours 55 minutes, giving us 355
• Now you want to add these two numbers together:
• 135
• 355
• ____
• 490
• What you need to do to this and all sub totals is add the time constant of
40.
• No matter what the hours and minutes are, just add the 40 time constant
to the sub total.
• 490 + 40 = 530
• so we can now see our answer is 5 hrs and 30 minutes!

12. Calendar

13. 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.

14. 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.
• Note: A leap year has 366 days.
• Examples:
• Each of the years 1948, 2004, 1676 etc. is a leap year.
• Each of the years 400, 800, 1200, 1600, 2000 etc. is a leap year.
• None of the years 2001, 2002, 2003, 2005, 1800, 2100 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.

15. 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.

16. Day of the Week Related to Odd Days
No. of
days:
0 1 2 3 4 5 6
Day: Sun. Mon. Tues. Wed. Thurs. Fri. Sat.

17. • If 6th March 2005 is Monday, what was the day of the
week on 6th March, 2004?
• The year 2004 is a leap year. So, it has 2 odd days.
• But Feb 2004 not included because we are calculating
from March 2004 to March 2005.
• So it has 1 odd day only.
• The day on 6th March, 2005 will be 1 day beyond the
day on 6th March, 2004.
• Given that, 6th March, 2005 is Monday, 6th March, 2004
is Sunday
• (1 day before to 6th March, 2005.)

18. • On what dates of April, 2001 did Wednesday fall?
• We shall find the day on 1st April,2001.
• 1st April 2001 = (2000 years + Period from 1.1.2001 to
1.4.2001)
• Odd days in 2000 years = 0
• Jan,Feb,Mar,Apr
• (31+28+31+1) = 91 days = o odd days.
• Total number of odd days = 0.
• On 1st April 2001 it was Sunday.
• In April 2001, Wednesday falls on 4th,11th,18th and 25th.

19. • The last day of a century cannot be?
• 100 years contain 5 odd days.
• Last day of 1st century is Friday.
• 200 years contain (5x2) 3 odd days.
• Last day of 2nd century is Wednesday.
• 300 years contain (5x3) 1 odd day.
• Last day of 3rd century is Monday.
• 400 years contain 0 odd day
• Last day of 4th century is Sunday.
• This cycle is repeated.
• Hence Last day of a century cannot be Tuesday or Thursday
or Saturday.

20. • On 8th Feb, 2005 it was Tuesday.
• What was the day of the week on 8th Feb
2004?
• The year 2004 is a leap year. It has 2 odd days.
• The day on 8th Feb 2004 is 2 days before the
day on 8th Feb.2005.
• Hence this day is Sunday.

21. JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
0 3 3 6 1 4 6 2 5 0 3 5
Step1: Ask for the Date. Ex: 23rd June 1986
Step2: Number of the month on the list, June is 4.
Step3: Take the date of the month, that is 23
Step4: Take the last 2 digits of the year, that is 86.
Step5: Find out the number of leap years. Divide the last 2 digits of the year by 4, 86
divide by 4 is 21.
Step6: Now add all the 4 numbers: 4 + 23 + 86 + 21 = 134.
Step7: Divide 134 by 7 = 19 remainder 1.
The reminder tells you the day.
No. of
days:
0 1 2 3 4 5 6
Day: Sun. Mon. Tues. Wed. Thurs. Fri. Sat.
1600/2000 1700/2100 1800/2200 1900/2300
0 6 4 2

22. • 4/4, 6/6, 8/8, 10/10, 12/12, and the last day of February all occur on the
same day of the week in any given year. Applying the Doomsday algorithm
involves three steps:
• Determine the "anchor day" for the century.
• Use the anchor day for the century to calculate the doomsday for the year.
• Choose the closest date out of the ones that always fall on the doomsday
(e.g. 4/4, 6/6, 8/8), and count the number of days (modulo 7) between
that date and the date in question to arrive at the day of the week.
• This technique applies to both the Gregorian calendar A.D. and the Julian
calendar, although their doomsdays will usually be different days of the
week.
• Since this algorithm involves treating days of the week like numbers
modulo 7, John Conway suggests thinking of the days of the week as
"Noneday" or "Sansday" (for Sunday), "Oneday", "Twosday", "Treblesday",
"Foursday", "Fiveday", and "Six-a-day".

23. • As mentioned above, the last day of February always falls on the
doomsday, as do the double dates 4/4, 6/6, 8/8, 10/10, and 12/12.
• Four of the odd month dates (May 9, September 5, July 11, and November
7) can be remembered with the mnemonic "I work from 9 to 5 at the 7-
11."
• For March, one can remember the pseudo-date "March 0", which refers to
the day before March 1, i.e. the last day of February; one can alternately
remember the date a week later, March 7, or March 21 which is
traditionally regarded as the first day of spring in the northern hemisphere
and autumn in the southern hemisphere (although after 2007 the
Northward equinox will not fall on that date in Europe again until 2102).[6]
• For January, January 11 is a Doomsday during leap years, while January 10
is a Doomsday during common years; January 3 is a doomsday during
common years and January 4 a Doomsday during leap years, which can be
remembered as "the 3rd during 3 years in 4, and the 4th in the 4th".

24. Month Memorable date Month/Day Mnemonic
January
January 3 (common years),
January 4 (leap years)
January 11 (leap years)
1/3 or 1/4
1/11
the 3rd 3 years in 4 and the
4th in the 4th
all ones
February
February 14 (common years)
February 22 (leap years)
February 28 (common years),
February 29 (leap years)
2/14
2/22
2/28 or 2/29
Valentine's Day
all twos
last day of February
March
"March 0"
March 14
March 21
3/0
3/14
3/21
last day of February
Pi day
traditionally first day of
spring/autumn
April April 4 4/4 even month
May May 9 5/9 9-to-5 at 7-11
June
June 6
June 20
6/6
6/20
even month
traditionally first day of
summer/winter

25. July
July 4
July 11
7/4
7/11
Independence Day
(United States)
9-to-5 at 7-11
August August 8 8/8 even month
September September 5 9/5 9-to-5 at 7-11
October
October 10
October 31
10/10
10/31
even month
Halloween
November November 7 11/7 9-to-5 at 7-11
December
December 12
December 26
12/12
12/26
even month
Boxing Day

26. Century Anchor day Mnemonic Index (day of week)
1800–1899 Friday - 5 (Fiveday)
1900–1999 Wednesday
We-in-dis-day
(most living people
were born in that
century)
3 (Treblesday)
2000–2099 Tuesday
Y-Tue-K or Twos-
day
(Y2K was at the
head of this
century)
2 (Twosday)
2100–2199 Sunday
Twenty-one-day is
Sunday
(2100 is the start of
the next century)
0 (Noneday)
We first take the anchor day for the century. For the purposes of the
Doomsday rule, a century starts with '00 and ends with '99. The following
table shows the anchor day of centuries 1800–1899, 1900–1999, 2000–
2099 and 2100–2199.

27. Month Dates Week numbers *
January (common years) 3, 10, 17, 24, 31 1–5
January (leap years) 4, 11, 18, 25 1–4
February (common years) 7, 14, 21, 28 6–9
February (leap years) 1, 8, 15, 22, 29 5–9
March 7, 14, 21, 28 10–13
April 4, 11, 18, 25 14–17
May 2, 9, 16, 23, 30 18–22
June 6, 13, 20, 27 23–26
July 4, 11, 18, 25 27–30
August 1, 8, 15, 22, 29 31–35
September 5, 12, 19, 26 36–39
October 3, 10, 17, 24, 31 40–44
November 7, 14, 21, 28 45–48
December 5, 12, 19, 26 49–52