Calender and Clock

Calender and Clock

Calender

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.

4^{th}

(ii). Every  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 :

i. 1 ordinary year = 365 days = (52 weeks + 1 day.)

∴   1 ordinary year has 1 odd day.

ii. 1 leap year = 366 days = (52 weeks + 2 days)

∴    1 leap year has 2 odd days.

iii. 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:

Question 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

Explanation:

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.

Question 2:

What was the day of the week on 28th May, 2006?

A. Thursday

B. Friday

C. Saturday

D. Sunday

Answer: Option D

Explanation:

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) = 70 odd day.

Given day is Sunday.

Question 3 :

What was the day of the week on 17th June, 1998?

A. Monday

B. Tuesday

C. Wednesday

D. Thursday

Answer: Option C

Explanation:

17^{th}

 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.

Question 4 :

15^{th}

What will be the day of the week August, 2010?

A. Sunday

B. Monday

C. Tuesday

D. Friday

Answer: Option A

Explanation:

15^{th}

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.

Question 5 :

Today is Monday. After 61 days, it will be:

 A. Wednesday

B. Saturday

C. Tuesday

D. Thursday

Answer: Option B

Explanation:

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.

Question 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

Explanation:

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.

6^{th}

∴The day on  March, 2005 will be 1 day beyond the day on 6th

March, 2004.

6^{th}

Given that, March, 2005 is Monday.

6^{th}
6^{th}

∴ March, 2004 is Sunday (1 day before to March, 2005).

Question 7 :

On what dates of April, 2001 did Wednesday fall?

1^{st}
8^{ht}
15^{th}
22^{nd}
29^{th}

A. , , , ,

2^{nd}
9^{th}
16^{th}
23^{rd}
30^{th}

B. , , , ,

3^{rd}
10^{th}
17^{th}
24^{th}

C. , , ,

4^{th}
11^{th}
18^{th}
25^{th}

D. , , ,

Answer: Option D

Explanation:

1^{st}

We shall find the day on April, 2001.

1^{st}

 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

1^{st}

On April, 2001 it was Sunday.

8^{th}
11^{th}
18^{th}
25^{th}

In April, 2001 Wednesday falls on ,, and .

Question 8 :

How many days are there in x weeks x days?

x^{2}

A. 7

B. 8x

C. 14x

D. 7

Answer: Option B

Explanation:

x weeks x days = (7x + x) days = 8x days.

Question 9 :

The last day of a century cannot be

A. Monday

B. Wednesday

C. Tuesday

D. Friday

Answer: Option C

Explanation:

100 years contain 5 odd days.

∴Last day of 1st century is Friday.

200 years contain (5 x 2) 3 odd days.

∴Last day of 2nd century is Wednesday.

300 years contain (5 x 3) = 15 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.

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

Question 10 :

8^{th}
8^{th}

On Feb, 2005 it was Tuesday. What was the day of the week on Feb, 2004?

 A. Tuesday

B. Monday

C. Sunday

D. Wednesday

Answer: Option C

Explanation:

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.

Question 11 :

The calendar for the year 2007 will be the same for the year:

A. 2014

B. 2016

C. 2017

D. 2018

Answer: Option D

Explanation:

Count the number of odd days from the year 2007 onward to get the sum equal to 0 odd day.

Sum = 14 odd days 0 odd days.

∴ Calendar for the year 2018 will be the same as for the year 2007.

Question 12:

Which of the following is not a leap year?

A. 700

B. 800

C. 1200

D. 2000

Answer : Option A

Explanation:

The century divisible by 400 is a leap year.

The year 700 is not a leap year.

Question 13 :

8^{th}
8^{th}

On Dec, 2007 Saturday falls. What day of the week was it on Dec, 2006?

A. Sunday

B. Thursday

C. Tuesday

D. Friday

Answer : Option D

Explanation:

The year 2006 is an ordinary year. So, it has 1 odd day.

8^{th}
8^{th}
8^{th}

So, the day on Dec, 2007 will be 1 day beyond the day on Dec, 2006.But, Dec, 2007 is Saturday.

8^{th}

 Dec, 2006 is Friday.

Question 14 :

January 1, 2008 is Tuesday. What day of the week lies on Jan 1, 2009?

A. Monday

B. Wednesday

C. Thursday

D. Sunday

Answer : Option C

Explanation:

The year 2008 is a leap year. So, it has 2 odd days.

1^{st}

 day of the year 2008 is Tuesday (Given)

1^{st}

So, day of the year 2009 is 2 days beyond Tuesday.

Hence, it will be Thursday.

Question 15 :

January 1, 2007 was Monday. What day of the week lies on Jan. 1, 2008?

A. Monday

B. Tuesday

C. Wednesday

D. Sunday

Answer : Option B

Explanation:

The year 2007 is an ordinary year. So, it has 1 odd day.

1^{st}

day of the year 2007 was Monday.

1^{st}

 day of the year 2008 will be 1 day beyond Monday.

Hence, it will be Tuesday.

Clock

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.

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

Question 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. 144º

B. 150º

C. 168º

D. 180º

Answer: Option D

Explanation:

(frac{360}{12}times6)

Angle traced by the hour hand in 6 hours = º= 180º.

Question 2 :

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

A. 180º

192frac{1}{2}

B. º

C. 195º

197frac{1}{2}

D. º

Answer: Option D

Explanation:

frac{125}{12}
(frac{360}{12}timesfrac{125}{12})
frac{1}{2}

Angle traced by hour hand in  hrs =  º= 312º

(frac{360}{60}times25)

Angle traced by minute hand in 25 min = º= 150º.

(312frac{1}{2}times-150)
162frac{1}{2}
frac{1}{2}

Reflex angle = 360º – º = 360º- º= 197

Question 3 :

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.

frac{31}{6}
(frac{360}{12}timesfrac{31}{6})

Angle traced by hour hand in 5 hrs 10 min. i.e., hrs = º = 155º.

Question 4 :

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:

59frac{7}{12}min. past3

A.

B. 4 p.m.

58frac{7}{11}min. past3

C.

2frac{3}{11}min. past4

D.

Answer : Option B

Explanation:

frac{37}{4}

Time from 7 a.m. to 4.15 p.m. = 9 hrs 15 min. =hrs.

3 min. 5 sec. of this clock = 3 min. of the correct clock.

Rightarrow frac{37}{720}
frac{1}{20}

 hrs of this clock = hrs of the correct clock.

Rightarrow frac{37}{4}
(frac{1}{20}timesfrac{720}{37}times frac{37}{4})

 hrs of this clock = hrs of the correct clock.

= 9 hrs of the correct clock.

∴ The correct time is 9 hrs after 7 a.m. i.e., 4 p.m.

Question 5 :

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

32frac{8}{11}

A.  min.

36frac{5}{11}

B.  min.

C. 90 min.

D. 96 min.

Answer: Option A

Explanation:

55 min. spaces are covered in 60 min.

(frac{60}{55}times60)_{{min}}
(65frac{5}{11})_{{min}}

60 min. spaces are covered in  . = 

(65frac{5}{11}-64)_{{min}}
frac{16}{11}_{{min}}

Loss in 64 min. = =

(frac{16}{{11}}times frac{1}{64}times24times60)_{{min}}
32frac{8}{{11}}_{{min}}

Loss in 24 hrs = = 

Question 6 :

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

A. 5 min. past 7

5frac{2}{{11}}_{{min}}

B.  . past 7

5frac{3}{{11}}_{{min}}

C.  . past 7

5frac{5}{{11}}_{{min}}

D. min. past 7

Answer: Option D

Explanation:

When the hands of the clock are in the same straight line but not together, they are 30 minute spaces apart.

At 7 o’clock, they are 25 min. spaces apart.

∴ Minute hand will have to gain only 5 min. spaces.

55 min. spaces are gained in 60 min.

(frac{60}{55}times5)_{{min}}
(5frac{5}{11})_{{min}}

5 min. spaces are gained in  = 

(5frac{5}{11})_{{min}}

∴ Required time =  . past 7.

Question 7 :

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

43frac{5}{11}

A.  min. past 5

43frac{7}{11}

B. min. past 5

C. 40 min. past 5

D. 45 min. past 5

Answer: Option B

Explanation:

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

minute hand has to gain (25 + 15) = 40 min. spaces. 55 min. spaces are gained in 60 min.

(frac{60}{55}times40)_{{min}}
43frac{7}{11}

40 min. spaces are gained in  =  min.

43frac{7}{11}

∴Required time =  min. past 5.

Question 8 :

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

A. 0º

B. 10º

C. 5º

D. 20º

Answer: Option B

Explanation:

frac{13}{3}
(frac{360}{12}times frac{13}{3})

Angle traced by hour hand in hrs = º = 130º.

(frac{360}{60}times 20)

Angle traced by min. hand in 20 min. =º= 120º.

∴ Required angle = (130 – 120)º = 10º.

Question 9 :

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

58frac{1}{2}

A. º

B. 64º

67frac{1}{2}

C.  º

72frac{1}{2}

D.  º

Answer: Option C

Explanation:

frac{21}{4}
(frac{360}{{12}}timesfrac{21}{4})
frac{1}{2}

Angle traced by hour hand in  hrs = º= 157º

Angle traced by min. hand in 15 min. =360x 15º= 90º.60

frac{1}{2}
frac{1}{2}

Required angle = 157º- 90º = 67º

Question 10 :

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

A. 120°

B. 125°

C. 130°

D. 135°

Answer: Option C

Explanation:

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

frac{11}{3}
(frac{360}{12}times frac{11}{3})

Angle traced by it in  hrs = °= 110°.

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

(frac{360}{60}times40)

Angle traced by it in 40 min. =°= 240°

∴ Required angle (240 – 110)° = 130°.

Question 11 :

How many times are the hands of a clock at right angle in a day?

A. 22

B. 24

C. 44

D. 48

Answer: Option C

Explanation:

In 12 hours, they are at right angles 22 times.

∴ In 24 hours, they are at right angles 44 times.

Question 12 :

The angle between the minute hand and the hour hand of a clock when the time is 8.30, is:

A. 80º

B. 75º

C. 60º

D. 105º

Answer: Option B

Explanation:

frac{17}{2}
(frac{360}{12}times frac{17}{2})

Angle traced by hour hand in hrs = º = 255.

(frac{360}{60}times 30)

Angle traced by min. hand in 30 min. =º= 180.60

∴ Required angle = (255 – 180)º = 75º.

Question 13 :  

How many times in a day, are the hands of a clock in straight line but opposite in direction?

A. 20

B. 22

C. 24

D. 48

Answer: Option B

Explanation:

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’clcok only). So, in a day, the hands point in the opposite directions 22 times.

Question 14 :

At what time between 4 and 5 o’clock will the hands of a watch point in opposite directions?

A. 45 min. past 4

B. 40 min. past 4

50frac{4}{11}

C.  min. past 4

54frac{6}{11}

D.  min. past 4

Answer: Option D

Explanation:

At 4 o’clock, the hands of the watch are 20 min. spaces apart.

To be in opposite directions, they must be 30 min. spaces apart.

∴Minute hand will have to gain 50 min. spaces.

55 min. spaces are gained in 60 min.

(frac{60}{55}times50)_{min}
54frac{6}{11}

50 min. spaces are gained in. or  min

54frac{6}{11}

∴ Required time =  min. past 4.

Question 15 :

At what time between 9 and 10 o’clock will the hands of a watch be together?

A. 45 min. past 9

B. 50 min. past 9

49frac{1}{11}

C.  min. past 9

48frac{2}{11}

D.  min. past 9

Answer: Option C

Explanation:

To be together between 9 and 10 o’clock, the minute hand has to gain 45 min. spaces.

55 min. spaces gained in 60 min.

(frac{60}{55}times 45)_{{min}}
49frac{1}{11}

45 min. spaces are gained in   or  min.

49frac{1}{11}

∴ The hands are together at  min. past 9.