If the 20th day of the month falls on the 4th day before Monday, then what will be the day of the week on the 1st of that month?
Saturday
Understanding the Calendar Day Problem
This question asks us to determine the day of the week for the 1st day of a month, given information about the 20th day of that same month. We are told that the 20th day falls on the 4th day before Monday.
Finding the Day of the Week for the 20th
Let's first figure out which day is the 4th day before Monday.
- 1st day before Monday is Sunday.
- 2nd day before Monday is Saturday.
- 3rd day before Monday is Friday.
- 4th day before Monday is Thursday.
So, the 20th day of the month is a Thursday.
Calculating the Difference in Days
Now we need to find the day of the week for the 1st day. To do this, we need to know the number of days between the 1st and the 20th.
Number of days between the 1st and the 20th (inclusive of the 20th, counting backwards from 20 to 1):
Difference = 20th day - 1st day = $20 - 1 = 19$ days.
We need to go back 19 days from Thursday.
Using Modulo 7 for Day Calculation
The days of the week repeat every 7 days. To find the day 19 days before Thursday, we can find the remainder when 19 is divided by 7.
$19 \div 7$
$19 = 2 \times 7 + 5$
The remainder is 5. This means that 19 days before Thursday is the same day of the week as 5 days before Thursday.
Determining the Day for the 1st
Now, let's count back 5 days from Thursday:
- Thursday - 1 day = Wednesday
- Thursday - 2 days = Tuesday
- Thursday - 3 days = Monday
- Thursday - 4 days = Sunday
- Thursday - 5 days = Saturday
Therefore, the 1st day of the month is a Saturday.
Summary of Steps
| Step | Description | Result |
|---|---|---|
| 1 | Find the day for the 20th. | Thursday (4 days before Monday) |
| 2 | Calculate the difference in days from 20th to 1st. | $20 - 1 = 19$ days |
| 3 | Find the remainder when 19 is divided by 7. | $19 \pmod{7} = 5$ |
| 4 | Count back the remainder days from the 20th's day. | Thursday - 5 days = Saturday |
Following these steps, we conclude that the 1st day of the month falls on a Saturday.
Revision Table: Calendar Day Calculations
| Concept | Explanation | Application |
|---|---|---|
| Days before/after | Counting days forward or backward from a given day. | Finding the day 4 days before Monday. |
| Difference in days | Calculating the total number of days between two dates. | Finding the 19 days between the 1st and 20th. |
| Modulo 7 | Used because the week cycle is 7 days. The remainder tells us how many days offset the result is within the week. | $19 \pmod{7} = 5$, meaning we shift 5 days back. |
Additional Information: Working with Weeks and Days
Calendar problems often involve working with cycles of 7 days. The key idea is that adding or subtracting multiples of 7 days from a date results in the same day of the week. For example, 7 days after a Monday is a Monday, 14 days after a Monday is a Monday, and so on. Similarly, 7 days before a Monday is a Monday.
When you need to find the day of the week for a date that is $N$ days away from a known date, you calculate $N \pmod{7}$.
- If the remainder is 0, it's the same day.
- If the remainder is 1, it's 1 day later (if counting forward) or 1 day earlier (if counting backward).
- If the remainder is 2, it's 2 days later/earlier, and so on.
In our problem, we needed to find the day 19 days *before* the 20th. Since $19 \pmod{7} = 5$, we look for the day 5 days *before* the day of the 20th.
Important Questions from Clock and Calendar
If 19 July 2000 was a Wednesday, then what would be the day of the week on 15 June 2012?
What day of the week was 31 st January 2007?
What was the day of the week on 10 June 2011?
What day of the week was 5 February 2008?
What day of the week was 29 June 2010?
