**Introduction**

We use a calendar in our everyday life to check the day and the month of the year. Do you know that there were there only 19 days in September 1752 in Great Britain but 30 in Spain? Also, 1800 and 1900 were not considered as leap years. In earlier times, calendars were lunisolar in nature, which meant that they were based on both solar as well as lunar cycles. It is was in 1858 that Pope Gregory XIII as a reform of the Julian calendar proclaimed the Gregorian calendar, also known as the new calendar. This calendar is used the world over. This calendar tells about the months and days of the year.

**The Calendar**

A calendar is a chart or series of pages that shows the days, weeks, and months of a particular year. Observe the calendar carefully. You will see that the calendar has 12 months. Let us learn more about these months.

Let us start from a day. We have learnt about time and know that there are 24 hours in a day. So, we have,

1 day = 24 hours

1 week = 7 days

1 month = 30 / 31 days

1 years = 365 days

1 leap year = 366 days.

Let us learn more about them

**Days, Weeks, Months and Years**

A day comprises from 12:00 midnight of one day to 12:00 midnight of the next day which is equal to 24 hours. There are different days that fall in the order –

**Monday**

**Tuesday**

**Wednesday**

**Thursday**

**Friday**

**Saturday**

**Sunday**

These 7 days together comprise one week. Therefore,

**1 week = 7 days**

**The following is the representation of a week in a month.**

**March 2020**

S | M | T | W | T | F | S |

1 | 2 | 3 | 4 | 5 | 6 | 7 |

8 | 9 | 10 | 11 | 12 | 13 | 14 |

15 | 16 | 17 | 18 | 19 | 20 | 21 |

22 | 23 | 24 | 25 | 26 | 27 | 28 |

29 | 30 | 31 |

There are 12 months in a year. But every month does not have the same number of days. The following are the months and the number of days each month has –

Name of the Month | Number of days |

January | 31 |

February | 28 or 29 |

March | 31 |

April | 30 |

May | 31 |

June | 30 |

July | 31 |

August | 31 |

September | 30 |

October | 31 |

November | 30 |

December | 31 |

The famous rhyme to remember number of days in the respective months is –

30 days has September,

April, June and November.

All the rest have 31

Except February alone,

Which has 28 days clear

And 29 in each leap year.

Another popular method to remember the number of days in each month is the knuckle method.

**The Knuckle method to remember number of days in a month**

The knuckles of the hand can be conveniently used to remember which has 30 days and which ones have 31 days. The following steps should be followed for this purpose –

- Make a fist.
- If the month is on a knuckle, it has 31 days. Otherwise is has 30 or less days.
- Starting with the 1st knuckle as January which has 31 days.
- The space between the first knuckle and the second knuckle is February, which has 28 / 29 days, depending upon the year.
- 2nd Knuckle is March which has 31 days.
- The space been March and the next knuckle is April, which has 30 days.
- The 3rd knuckle is May which has 31 days.
- The space between May and the next knuckle is June, which has 30 days.
- The fourth knuckle is July which has 31 days.
- Since you are now at the end of the fist, start again from the first knuckle. The first knuckle now represents August which has 31 days.
- The space between the first knuckle and the second knuckle is September which has 30 days.
- The second knuckle is October, which has 31 days.
- The space between October and the next knuckle is November, which has 30 days.
- The third knuckle is December, which has 31 days.

We can see above the month of February has either 28 days or 29 days. How do we decide in which year February has 28 days and in which year it has 29 days? This is where the concept of leap year comes in. So, what is leap year?

**Leap Year **

We know that a year has 364 days. However, in actual it has 365 $\frac{1}{4}$ days. This $\frac{1}{4}$ days or the quarter of a day is added every four years to make one full day. Hence, every fourth year we have 366 days in a year instead of 365 days. This additional one day is added to the month of February which otherwise has 28 days. So, in this year the month of February has 29 days and it is called a leap year. How do we know which is this fourth year that forms the leap year? We can simply check by dividing the year by 4. If it is divisible by 4, then the year is a leap year.

Therefore, we can say that –

**Year**** ****Number of days**

2020 366 – Leap Year

2021 365 days

2022 365 days

2023 365 days

2024 366 days – Leap Year

Hence, we can see that after every 4 years we have a leap year in which the month of February has 29 days.

**Changing weeks to days and days to weeks **

We will now learn how to change weeks to days and days to weeks. To do this, we will follow the steps given below –

We know that 1 week = 7 days

Therefore, if we want to change 6 weeks into days, we simply need to multiply 6 by 7. We will get,

6 weeks = ( 6 x 7 ) days = 42 days

Similarly, suppose we wish to change 4 weeks 3 days into days. How will we do so?

4 weeks 3 days = ( 4 x 7 ) days + 3 days = 28 days + 3 days = 31 days

Now, let us change days to weeks. In order to change days to weeks we will divide the given number of days by 7. Let us understand it using an example.

Suppose we wish to change 56 days into weeks. We will simply divide 56 by 7. We know that 56 is completely divisible by 7 and we will get the quotient as 8. This means that

56 days = ( 56 ÷ 7 ) weeks = 8 weeks

Now, what if the number of days given is not completely divisible by 7? Let us take an example. Suppose we wish to change 25 days into weeks. We will first divide 25 by 7. We will get 3 as the quotient and 4 as the remainder. So, we have

25 days = 3 weeks + 4 days

**Changing years to months and months to years**

We will now learn how to change years to months and months to years. To do this, we will follow the steps given below –

We know that 1 year = 12 months

Therefore, if we want to change 5 years into months, we simply need to multiply 5 by 12. We will get,

5 years = ( 5 x 12 ) months = 60 months

Similarly, suppose we wish to change 2 years 8 months into months. How will we do so?

2 years 8 months = ( 2 x 12 ) months + 8 months = 24 months + 8 months = 32 months

Now, let us change months to years. In order to change months to years we will divide the given number of months by 12. Let us understand it using an example.

Suppose we wish to change 48 months into years. We will simply divide 48 by 12. We know that 48 is completely divisible by 12 and we will get the quotient as 4. This means that

48 months = ( 48 ÷ 12 ) years = 4 years

Now, what if the number of months given is not completely divisible by 12? Let us take an example. Suppose we wish to change 21 months into years. We will first divide 21 by 12. We will get 1 as the quotient and 9 as the remainder. So, we have

21 = 1 years + 9 months

**Solved Examples**

**Example 1** Alice’s school closed for the summer vacations on 4 May and reopened on 16 June. For how many days the school close for the summer vacations?

**Solution** We have been given that Alice’s school closed for the summer vacations on 4 May and reopened on 16 June. Now, since Alice’s summer vacations began on 4 May, so she would not go to school on 4 May. Therefore, we will include 4 May as vacation. Also, we know that the month of May as 31 days. Therefore, we have,

Number of vacation days in the month of May = 31 – 3 = 28 days ……. ( 1 )

Now, we have been given that the school reopened on 16 June. This means that Alice went to school on 16 June. Therefore, 16 June will not be included in the vacation days. Hence, we have,

Number of vacation days in the month of June = 25 days ………………. ( 2 )

Now, to obtain the total number of vacation days we need to find the sum of ( 1 ) and ( 2 ). Therefore, we have,

Total number of vacation days = 28 + 15 = 43 days.

**Hence, the school was closed for 43 days for the summer vacations. **

**Example 2**** **What was the date on 27 days after 13 February 2014 ?

**Solution** We know that 2014 was not a leap year. This means that in 2014 the month of February had 28 days.

Hence the number of days to be considered in the month of February = 28 – 13 = 15 days.

We are required to find the date 27 days after 13 February. Now, out of these 27 days, 15 days are consumed in February. Therefore we are left with 27 – 15 = 12 days which need to be forwarded to the next month which is March. Therefore, we have

The date which was 27 days after 13 February 2014 = 12 Mach 2014.

**Example 3** The construction of a new basketball court in Allen’s school was to be completed in 10 weeks 2 days. 3 weeks 4 days have passed since the start of the construction work. How much more time is left?

**Solution** We have been given that the construction of a new basketball court in Allen’s school was to be completed in 10 weeks 2 days. 3 weeks 4 days have passed since the start of the construction work. We need to find out the lift left for construction of the new basketball court. We will have

Total time required for construction = 10 weeks 2 days

Number of days passed since the start of the construction work = 3 weeks 4 days

Number of days left for the completion of the construction = ( 10 weeks 2 days ) – (3 weeks 4 days ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 )

Now, 2 days < 4 days. So we will borrow 1 week from 10 weeks. We will now have 9 weeks 9 days. We can therefore write equation ( 1 ) as

Number of days left for the completion of the construction = ( 9 weeks 9 days ) – (3 weeks 4 days ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 )

This means that Number of days left for the completion of the construction = ( 9 weeks – 3 weeks ) and ( 9 days – 4 days)

⇒ Number of days left for the completion of the construction = 6 weeks 5 days

Hence, 6 weeks 5 days are left for the construction of the new basketball court to be completed.

**Example 4** A mathematics tutor started teaching at the age of 21 years and 5 months. He is now 43 years 2 months old. How long has he been teaching?

**Solution** We have been given that a mathematics tutor started teaching at the age of 21 years and 5 months. He is now 43 years 2 months old. We need to find out how long he has been teaching?

Age at which the mathematics tutor started teaching = 21 years and 5 months . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 )

Present age of the mathematics tutor = 43 years and 2 months . . . . . . . . . . . . . . . . . . . . . . . ( 2 )

Number of years for which the mathematics tutor has been teaching = ( 43 years 2 months ) – ( 21 years 5 months ) . . . . . . . . . . . . . . . . . . . . . . ( 3 )

From ( 1 ) and ( 2 ) we can see that 2 months < 5 months Therefore, we will have to borrow 1 year from 43 years. So, we will have

43 years 2 months = 42 years + 1 year + 2 months . . . . . . . . . . . . . ( 4 )

Now, 1 year = 12 months

Therefore, equation ( 4 ) becomes,

43 years 2 months = 42 years + 12 months + 2 months = 42 years + 14 months . . . . . . . . . . . . . ( 4 )

Substituting the value of 43 years 2 months obtained in the above equation to equation ( 3 ), we get

Number of years for which the mathematics tutor has been teaching = ( 42 years 14 months ) – ( 21 years 5 months ) . . . . . . . . . . . . . . . . . . . . . . ( 5 )

⇒ Number of years for which the mathematics tutor has been teaching = ( 42 years – 21 years ) and ( 14 months – 5 months )

⇒ Number of years for which the mathematics tutor has been teaching = 21 years 9 months

**Hence, the number of years for which the mathematics tutor has been teaching = 21 years 9 months**

**Key Facts and Summary**

- It is was in 1858 that Pope Gregory XIII as a reform of the Julian calendar, proclaimed the Gregorian calendar, also known as the new calendar.
- A calendar is a chart or series of pages that shows the days, weeks, and months of a particular year.
- The calendar has 12 months – January, February, March, April, May, June, July, August, September, October, November, December.
- A year has 364 days.

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