Reasoning & Quantitative Aptitude Calendar & Clocks
MODULE OBJECTIVE:
The module calendar is used to find many problems related to odd days,leap year, and counting of odd days and many.
The module clock is used to find many problems related to find angle between hour and minute hand of a clock, at what time the hands of clock will be together and many.
Prerequisites:
CALENDAR
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.
Note: A leap year has 366 days.
Examples:
i. Each of the years 1948, 2004, 1676 etc. is a leap year.
ii. Each of the years 400, 800, 1200, 1600, 2000 etc. is a leap year.
iii. 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.
Counting of Odd Days:
1. 1 ordinary year = 365 days = (52 weeks + 1 day.)
1 ordinary year has 1 odd day.
2. 1 leap year = 366 days = (52 weeks + 2 days)
1 leap year has 2 odd days.
3. 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:
No. of days: 0 1 2 3 4 5 6
Day: Sun. Mon. Tues. Wed. Thurs. Fri. Sat.
Some codes o remember the months and weeks:
a) Week
Sunday – 1
Monday – 2Tuesday – 3
Wednesday – 4
Thursday – 5
Friday – 6
Saturday – 0
b) Month
jan – 1 july – 0
feb – 4 Aug – 3
Mar – 4 Sep – 6
Apr – 0 Oct – 1
May – 2 Nov – 4
june – 5 Dec – 6
CLOCK:
The 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..
i) In 60 minutes, the minute hand gains 55 minutes on the hour hand.
ii) In every hour, both the hands coincide once.
iii) The hands are in the same straight line when they are coincident or opposite to each
other.
iv) When the two hands are at right angles, they are 15 minute spaces apart.
v)When the hand's are in opposite directions, they are 30 minute spaces apart.
vi)Angle traced by hour hand in 12 hrs = 360°.
vii)Angle traced by minute hand in 60 min. = 360°.
viii)Too Fast and Too Slow: If a watch or a clock indicates 8.15, when the correct time ,
8 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.
Solved Examples:
CALENDAR:
1. It was Sunday on Jan 1, 2006. What was the day of the week Jan 1, 2010?
A.Sunday B.Saturday
C.Friday D.Wednesday
Answer: Option C [ RRB 2001]
Solution:
On 31st December, 2005 it was Saturday.
Number of odd days from the year 2006 to the year 2009 = (1 + 1 + 2 + 1) = 5 days.
On 31st December 2009, it was Thursday.
Thus, on 1st Jan, 2010 it is Friday.
2. What was the day of the week on 28th May, 2006?
A.Thursday B.Friday
C.Saturday D.Sunday
Answer: Option D
Solution:
28 May, 2006 = (2005 years + Period from 1.1.2006 to 28.5.2006)
Odd days in 1600 years = 0
Odd days in 400 years = 0
5 years = (4 ordinary years + 1 leap year) = (4 x 1 + 1 x 2) 6 odd days
Jan. Feb. March April May
(31 + 28 + 31 + 30 + 28 ) = 148 days
148 days = (21 weeks + 1 day) 1 odd day.
Total number of odd days = (0 + 0 + 6 + 1) = 7 0 odd day.
Given day is Sunday.
3.What was the day of the week on 17th June, 1998?
A.Monday B.Tuesday
C.Wednesday D.Thursday [Sathyam 2000]
Answer: Option C
Solution:
17th June, 1998 = (1997 years + Period from 1.1.1998 to 17.6.1998)
Odd days in 1600 years = 0
Odd days in 300 years = (5 x 3) 1
97 years has 24 leap years + 73 ordinary years.
Number of odd days in 97 years ( 24 x 2 + 73) = 121 = 2 odd days.
Jan. Feb. March April May June
(31 + 28 + 31 + 30 + 31 + 17) = 168 days
168 days = 24 weeks = 0 odd day.
Total number of odd days = (0 + 1 + 2 + 0) = 3.
Given day is Wednesday.
4. What will be the day of the week 15th August, 2010?
A.Sunday B.Monday
C.Tuesday D.Friday [ TCS 2006]
Answer: Option A
Solution:
15th August, 2010 = (2009 years + Period 1.1.2010 to 15.8.2010)
Odd days in 1600 years = 0
Odd days in 400 years = 0
9 years = (2 leap years + 7 ordinary years) = (2 x 2 + 7 x 1) = 11 odd days 4 odd days.
Jan. Feb. March April May June July Aug.
(31 + 28 + 31 + 30 + 31 + 30 + 31 + 15) = 227 days
227 days = (32 weeks + 3 days) 3 odd days.
Total number of odd days = (0 + 0 + 4 + 3) = 7 0 odd days.
Given day is Sunday.
5. Today is Monday. After 61 days, it will be:
A.Wednesday B.Saturday
C.Tuesday D.Thursday
Answer: Option B
Solution:
Each day of the week is repeated after 7 days.
So, after 63 days, it will be Monday.
After 61 days, it will be Saturday
6. If 6th March, 2005 is Monday, what was the day of the week on 6th March, 2004?
A.Sunday B.Saturday
C.Tuesday D.Wednesday
Answer: Option A [ I-CET-2007]
Solution:
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).
7. On what dates of April, 2001 did Wednesday fall?
A.1st, 8th, 15th, 22nd, 29th B.2nd, 9th, 16th, 23rd, 30th
C.3rd, 10th, 17th, 24th D.4th, 11th, 18th, 25th
Answer: Option D [INFOSYS 2003]
Solution:
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 1600 years = 0
Odd days in 400 years = 0
Jan. Feb. March April
(31 + 28 + 31 + 1) = 91 days 0 odd days.
Total number of odd days = (0 + 0 + 0) = 0
On 1st April, 2001 it was Sunday.
In April, 2001 Wednesday falls on 4th, 11th, 18th and 25th.
8. How many days are there in x weeks x days?
A.7x2 B.8x
C.14x D.7
Answer: Option B
Solution:
x weeks x days = (7x + x) days = 8x days.
CLOCK:
9. 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.144º B.150º [RRB 2009]
C.168º D.180º
Answer: Option D
Explanation:
Angle traced by the hour hand in 6 hours =
360/12 x 6 =180
10.The reflex angle between the hands of a clock at 10.25 is:
A.180º B.192 1/2
C.195º D.197 1/2
Answer: Option D
Explanation:
Angle traced by hour hand in 6 hours 125/12 hrs =
(360/12 x125/12)=312 1/2
Angle traced by minute hand in 25 min =
(360/60 x 25) º= 150º.
Reflex angle = 360º -( 312 1/2- 150) º= 360º - 162 1/2º
= 197 1/2
11. A clock is started at noon. By 10 minutes past 5, the hour hand has turned through:
A.145º B.150º
C.155º D.160º
Answer: Option C
Explanation:
Angle traced by hour hand in 12 hrs = 360º.
Angle traced by hour hand in 5 hrs 10 min. i.e.,
31/6hrs =(360/12 x 31/6)º= 155º.
12. 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.59 7/12 min. past 3
B.4 p.m. [AAO EXAM 2003]
C.58 7/11 min. past 3
D.2 3/11 min. past 4
Answer: Option B
Explanation:
Time from 7 a.m. to 4.15 p.m. = 9 hrs 15 min. =37/4hrs.
3 min. 5 sec. of this clock = 3 min. of the correct clock.
37/720 hrs of this clock =1/20 hrs of the correct clock.
37/4 hrs of this clock =1/20 hrs of the correct clock.
4 20 37 4
= 9 hrs of the correct clock.
The correct time is 9 hrs after 7 a.m. i.e., 4 p.m.
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