14th Feb 2022 was Monday. What day of the week on was 14th Feb 2020?
Friday
Finding the Day of the Week: 14th Feb 2020
The problem asks us to determine the day of the week for 14th Feb 2020, given that 14th Feb 2022 was a Monday. This involves working backward in time across a couple of years, which includes a leap year. To solve this type of calendar problem, we need to calculate the number of "odd days" between the two dates.
Understanding Odd Days and Leap Years
- A normal year has 365 days. $365 \div 7 = 52$ weeks and $1$ day. This extra day is called an odd day.
- A leap year has 366 days (due to 29th February). $366 \div 7 = 52$ weeks and $2$ days. A leap year has $2$ odd days.
- To find the day of the week after a certain number of days, we calculate the total odd days and move forward or backward from the starting day.
Step-by-Step Calculation for Day of Week
We are going from 14th Feb 2022 back to 14th Feb 2020.
Let's consider the period between these two dates year by year, moving backward:
- From 14th Feb 2022 to 14th Feb 2021: This is a period of exactly one year. The year 2021 is a normal year (not divisible by 4). So, this period has $365$ days, which means $1$ odd day.
- From 14th Feb 2021 to 14th Feb 2020: This is also a period of exactly one year. The year 2020 is a leap year because it is divisible by 4 ($2020 \div 4 = 505$). When going from 14th Feb 2021 backward to 14th Feb 2020, we cross the leap day (29th Feb 2020). Therefore, this period has $366$ days, which means $2$ odd days.
Alternatively, we can calculate the total number of days between 14th Feb 2020 and 14th Feb 2022.
- Number of days from 14th Feb 2020 to 14th Feb 2021 = 366 days (since 2020 is a leap year and Feb 29th is included in this interval).
- Number of days from 14th Feb 2021 to 14th Feb 2022 = 365 days (since 2021 is a normal year).
- Total number of days from 14th Feb 2020 to 14th Feb 2022 = $366 + 365 = 731$ days.
Now, let's find the total number of odd days in $731$ days.
$731 \div 7$
We perform the division:
$731 = 104 \times 7 + 3$
The remainder is $3$. So, there are $3$ odd days between 14th Feb 2020 and 14th Feb 2022.
Determining the Day
We know that 14th Feb 2022 was a Monday. We are calculating the day for 14th Feb 2020, which is before 14th Feb 2022. Therefore, we need to subtract the total number of odd days from Monday.
Day on 14th Feb 2020 = Day on 14th Feb 2022 - Total Odd Days
Day on 14th Feb 2020 = Monday - 3 days
- Monday - 1 day = Sunday
- Sunday - 1 day = Saturday
- Saturday - 1 day = Friday
So, the day of the week on 14th Feb 2020 was Friday.
| Date | Day of Week |
|---|---|
| 14th Feb 2022 | Monday (Given) |
| 14th Feb 2021 | Monday - 1 odd day (Normal Year 2021) = Sunday |
| 14th Feb 2020 | Sunday - 2 odd days (Leap Year 2020 interval including Feb 29) = Friday |
Final Answer Derivation
Based on our calculation of 3 odd days between 14th Feb 2020 and 14th Feb 2022, and working backward from Monday, we arrive at Friday.
Revision Table: Key Concepts for Calendar Problems
| Concept | Description | Odd Days |
|---|---|---|
| Normal Year | 365 days | 1 |
| Leap Year | 366 days (includes Feb 29th) | 2 |
| Calculating Odd Days | Divide total days by 7, the remainder is the number of odd days. | Remainder of (Total Days ÷ 7) |
| Moving Forward | Add odd days to the current day. | Current Day + Odd Days |
| Moving Backward | Subtract odd days from the current day. | Current Day - Odd Days |
Additional Information: Identifying Leap Years
A year is a leap year if it satisfies one of the following conditions:
- The year is divisible by 400. For example, 2000, 2400 are leap years.
- The year is divisible by 4 but NOT by 100. For example, 2020, 2024, 2028 are leap years.
Years divisible by 100 but not by 400 are NOT leap years. For example, 1900, 2100, 2200 are not leap years.
In this problem, 2020 is divisible by 4 but not by 100, so it is a leap year. This is crucial for correctly calculating the odd days between the dates.
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