This is the aptitude questions and answers section on ‘Clocks and Calendars’ with solutions and detailed explanation. Questions on clocks focus on important pointers of an analog clock and questions on calendars focus on the calendar math involved in ascertaining the day of the week of a given date.

**Clocks – Important Terms:**

**Hour hand and minute hand**

The smaller hand of a clock, which rotates around the clock once in every 12 hours is called the hour hand or shorthand and the larger hand ( instead of one ), which rotates around the clock once in every hour is called minute hand or long hand.

**Minute Space**

The complete circumference of the face or dial of the clock is divided into 60 equal parts called minute space.

**Minute Gain**

In every 60 minutes, the minute hand gains 55 minutes on the hour on the hour hand called the minute gain

**Overlap**

In each hour, both the hands coincide only once. That coincidence is called an overlap.

**Straight Line**

The minute and hour hands are in the same straight line when they are coincident or opposite to each other forming a straight line.

**Clock Too Fast**

When a watch or a clock indicates 8.15, but the correct time is 8, the clock is said to be 15 minutes too fast.

**Clock Too Slow**

When a clock indicates 8:45, but the correct time is 8, the clock is said to be 15 minutes too slow.

**Things to keep in mind**

- 60 minute space = 360° = 1 hour ( minute hand)
- 1 minute space = 6° = 1 minute ( minute hand)
- 5 minute space = 6° x 5 = 30° = 5 minutes (minute hand)
- Right Angle or Perpendicular formed between the hour and minute hand = 15 minute spaces apart
- Right Angle or Perpendicular formed between the hour and minute hand = 22 times in 12 hours or 44 times in 1 day.
- Straight Angle or Straight Line or 180°formed between the hour and minute hand = 30-minute space apart
- Straight Angle formed between the hour and minute hand = 11 times in 12 hours or 22 times in 1 day
- Angle traced by the hour hand in 12 hrs = 360°
- Angle traced by the minute hand in 60 min. = 360°
- Speed of hour hand = ½ or 0.5 degree per minute
- Speed of the minute hand = 6 degree per minute
- Angle of the hour hand from vertical at N o’clock = 30 degrees.

**Concepts & Types:**

There are various types of questions under the clock.

Type 1: Find the time when the angle between the two hands is known.

Type 2: Find the angle between the 2 hands at known time.

Type 3: Questions on clocks when the clock is gaining or losing time.

**Concept**

- The hand clock is a complete circle with a rotation of 360 degrees. The clock is divided into 12 equal parts and each part is 360/12 = 30°.
- When the minute hand takes a complete round in one hour, it covers 360° in 60

minutes.

- In 1 minute, the minute hand covers 360/60 = 6°/minute.
- 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 means that the relative speed of the minute hand is 6 – ½ = 5 ½ degrees.
- Also, we will be using the concept of relative speed and relative distance for solving problems related to clocks.

**Tips:**

- Tip 1: It would be easy to calculate the angle between the minute and the hour hand by using the formula given below,

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

- Tip 2: You can use this short formula for calculating the time when the angle is known

Angle = (minutes) -30 (hours)

**Examples on**

**Example 1:**

What would be the mirror image of the clock when the time is 01:40

Solution→

We need to subtract the time ahead of 12:00 from 12:00 to get mirror image time Mirror image of 01:40

Hence,

11:60 – 01:40 = 10:20

The mirror image of 1:40 would be 10:20.

**Example 2:**

When Rena looked at a clock,it was showing 6:00 in the morning. By how much angle will the hour’s hand rotate when she again looks at the clock at 12:00 in the noon?

Solution→

In 12 hours, the hour’s hand turns 360°

Hence, the difference between time = 6 hours

Therefore, the required angle = 360/12x 6 = 180°

### Concept of an Odd Day

Number of odd days in a month

- January has 31 days, irrespective of whether it’s an ordinary year or leap year. The division of the number 31 by 7 provides the remainder 3 hence January has 3 odd days. On generalising, any month which has 31 days has 3 odd days and any month which has 30 days has 2 odd days.
- The only exception happens in the case of February. The February month of an ordinary year has 28 days, division of 28 by 7 provides zero as remainder. Hence, the number of odd days in February of an ordinary year will have 0 odd days and that of leap years will have 1 odd day as February in a leap year has 29 days.
- Refer to the following three tables for odd days:

### Table 1

Month | Number of Odd Days |

January | 3 |

February | 0/1 |

March | 3 |

April | 2 |

May | 3 |

June | 2 |

July | 3 |

August | 3 |

September | 2 |

October | 3 |

November | 2 |

December | 3 |

### Table 2

Day | Code of the day |

Sunday | 0 |

Monday | 1 |

Tuesday | 2 |

Wednesday | 3 |

Thursday | 4 |

Friday | 5 |

Saturday | 6 |

### Table 3

Century | Number of odd days |

100 | 5 |

200 | 3 |

300 | 1 |

400 | 0 |

**Calendars – Important Terms:**

**Ordinary year**

The year which is not a special one that means the leap year is called an ordinary year. An ordinary year consists of 365 days.

**Leap Year**

A leap year has 366 days. It has an extra day i.e.29th day in the month of February.

Every year which is divisible by 4 is a leap year, if it is not a century. Every 4th year is a leap year and no other year could be a leap year.

**Odd days**

When we are supposed to find the day of the week on a given date, we use the concept of ‘odd days’. In a given time, the number of days more than the complete weeks are called odd days.

**Things to keep in mind.**

- A leap year consists of 366 days.
- Every year which is divisible by 4 is a leap year, if it is not a century.
- Every 4th century is a leap year and no other century other than the 4th one is a leap year.

*Counting odd days*

- 1 ordinary year = 365 days = (52 weeks + 1 day)
- Number of odd days in an ordinary year= 1.
- 1 leap year = 366 days = (52 weeks + 2 days)
- Number of odd days in the leap year= 2.
- 100 years = (76 ordinary years + 24 leap years)

⇒ (76 x 1 + 24 x 2) odd days

⇒ 124 odd days.

⇒ (17 weeks + 5 days)

⇒ 5 odd days.

Therefore the number of odd days in 100 years = 5.

- Number of odd days in 200 years = (5 x 2) = 10 = 3 odd days

**Tricks**:

- 100 years has 5 odd days.
- 200 years has 5 x 2 = 10 – 7 (one week) = 3 odd days.
- 300 years has 5 x 3 = 15 – 14 (two weeks) = 1 odd day.
- 400 years has {5 x 4 + 1 (leap century)} – 21} (three weeks) = 0 odd days.
- Month of January has 31 – 28 = 3 odd days. Same for the months of March, May, July, August, October and December.
- Month of February has 28 – 28 = 0 odd days in an ordinary year.
- Month of February has 1 odd day in a leap year.
- Month of March has 30-28 = 2 odd days. Same for April, June, September and November.

**Important Note:**

When you start counting from the beginning, that means for 1st January, 0001,

- 1 odd day mean Monday is the odd day
- 2 odd days mean Tuesday is the odd day
- 3 odd days mean Wednesday and so on till 6 odd days which means Saturday.

**Examples**

**Example 1:**

If today is Monday. After 57 days, it would be:

Solution→

It is well known that each day of the week is repeated after 7 days. So, after 57 days, it will be a Monday again.

**Example 2 **

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

Solution→

17th June, 1998 = (1997 years + Period from 1/1/1998 to 17/6/1998)

Odd days in 1600 years = 0 odd day

Odd days in 300 years = 1 odd day

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, and 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 odd days.

It was Wednesday on 17th June, 1998.

## Problems on Clocks and Calendars

**QUESTION 1: It was Sunday on Jan 1, 2006. What was the day of the week Jan 1, 2010?**

- Sunday
- Monday
- Thursday
- Friday

**QUESTION 2: What was the day of the week on 28th May, 2006?**

- Thursday
- Friday
- Saturday
- Suday

**QUESTION 3: What was the day of the week on 17th June, 1998?**

- Monday
- Thursday
- Wednesday
- Tuesday

**QUESTION 4: What will be the day of the week 15th July, 2001?**

- Sunday
- Monday
- Friday
- Tuesday

**QUESTION 5: Today is Saturday. After 59 days, it will be:**

- Sunday
- Monday
- Tuesday
- Saturday

**QUESTION 6: If 8th April, 2005 is Monday, what was the day of the week on 8th April, 2004?**

- Sunday
- Monday
- Thrusday
- Saturday

**QUESTION 7: On what dates of May, 2001 did Thursday fall?**

- 1st, 8th, 15th, 22nd, 29th
- 2nd, 9th, 16th, 23rd, 30th
- 3rd, 10th, 17th, 24th, 31st
- 4th, 11th, 18th, 25th

**QUESTION 8: How many days are there in y weeks y days?**

- 7y^2
- 8y
- 14y
- 7

**QUESTION 9: An accurate clock shows 2 o’clock in the morning. Through how may degrees will the hour hand rotate when the clock shows 9 o’clock in the evening?**

- 144º
- 210º
- 168º
- 570º

**QUESTION 10: A boy saw the clock when it is 5 a.m. The clock loses 8 minutes in half a day. What will be the true time when he sees the clock at 10 p.m. on 4th day?**

- 9 pm
- 10 pm
- 11 pm
- 12 pm

**QUESTION 11: A clock is started at 12 noon. By 600 seconds past 5 p.m, the hour hand has turned through how many degrees?**

- 145º
- 150º
- 155º
- 160º

##### Read Also: Number System

**QUESTION 12: The reflex angle between the hands of a clock at 10.25 is ?**

- 180
- 192½°
- 195
- 197½°

**QUESTION 13: How many times in 24 hours, are the hands of a clock in a straight line but opposite in direction?**

- 20
- 22
- 24
- 48

**QUESTION 14: How many times are the hands of a clock at 90 degrees in 12 hours?**

- 44
- 24
- 28
- 22

**QUESTION 15: 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 pm on Tuesday
- 2 pm on Wednesday
- 3 pm on Thrusday
- 1 pm on Friday

**QUESTION 16: How many times in a day, the hands of a clock are straight?**

- 22
- 24
- 44
- 48

**QUESTION 17: At what angle the hands of a clock are inclined at 15 minutes past 5?**

- 58½°
- 64½°
- 67½°
- 72½°

**QUESTION 18: The angle between the hour hand and the Minute hand of a clock when the time is 8.30 is?**

- 80
- 75
- 60
- 105

**QUESTION 19: At 3:20, the hour hand and the minute hand of a clock form an angle of?**

- 10
- 20
- 15
- 30

**QUESTION 20: The angle between the minute hand and the hour hand of a clock when the time is 4.20, is?**

- 0
- 5
- 10
- 20

**QUESTION 21: Find at what time between 8 and 9 o’clock will the hands of a clock be in 180°.**

- 110/11 min
- 100/11 min
- 200/11 min
- 120/11 min

**QUESTION 22: At what time between 2 and 3 o’clock will the hands of a clock be at 0°?**

- 100/11
- 200/11
- 120/11
- 110/11

**QUESTION 23: How much does a clock lose per day, if its hands coincide ever 64 minutes?**

- 256/11
- 288/11
- 92
- 90

**QUESTION 24: The last day of a century cannot be?**

- Monday
- Wednesday
- Tuesday
- Friday

**QUESTION 25: The calendar for the year 2007 will be the same for the year?**

- 2014
- 2016
- 2018
- 2017

**QUESTION 26: Which of the following is not a leap year?**

- 800
- 2000
- 700
- 1200

**QUESTION 27: On 22nd Nov, 2007 Thursday falls. What day of the week was it on 8th Nov, 2006?**

- Sunday
- Thursday
- Tuesday
- Wednesday

**QUESTION 28: January 1, 2007 was Monday. What day of the week lies on Jan. 1, 2008?**

- Monday
- Thrusday
- Tuesday
- Wednesday

**QUESTION 29: Calculate the angle between the two hands of clock when the clock shows 5 : 25 p.m. ?**

- 12 degree
- 12.5 degree
- 15 degree
- None of these

**QUESTION 30: How many non-leap years are there from 1801 to 1900?**

- 74
- 75
- 76
- 77

Read Also: Syllogism

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