Understanding Day of the Week Calculations

To find the day of the week on a future date based on a known date, we need to calculate the total number of days between the two dates and then find the number of "odd days". Odd days are the remainder when the total number of days is divided by 7. Each odd day shifts the day of the week forward by one day.

The given information is that 31 January 2009 was a Saturday. We need to find the day of the week on 30 January 2013.

Step-by-Step Day Calculation

Let's calculate the number of days between 31 January 2009 and 30 January 2013. We can break this down year by year, considering leap years.

• From 31 January 2009 to 31 January 2010: This period covers the year 2009 (from February onwards). 2009 is a normal year. So, 365 days.
• From 31 January 2010 to 31 January 2011: This period covers the year 2010. 2010 is a normal year. So, 365 days.
• From 31 January 2011 to 31 January 2012: This period covers the year 2011. 2011 is a normal year. So, 365 days.
• From 31 January 2012 to 31 January 2013: This period covers the year 2012. 2012 is a leap year (divisible by 4). So, 366 days because February 2012 has 29 days.

The total number of days from 31 January 2009 to 31 January 2013 is the sum of days in these four periods:

Total days until 31 Jan 2013 = $365 + 365 + 365 + 366 = 1461$ days.

We need to find the day on 30 January 2013, which is one day before 31 January 2013.

Total days from 31 January 2009 to 30 January 2013 = $1461 - 1 = 1460$ days.

Calculating Odd Days for Day Shift

Now, we need to find the number of odd days in 1460 days. We do this by dividing 1460 by 7 (the number of days in a week) and finding the remainder.

Number of odd days = $1460 \div 7$ remainder.

$1460 = 7 \times 208 + 4$

The remainder is 4. So, there are 4 odd days.

Determining the Final Day of the Week

The day on 31 January 2009 was Saturday. We need to move forward by the number of odd days, which is 4.

• Saturday + 1 day = Sunday
• Saturday + 2 days = Monday
• Saturday + 3 days = Tuesday
• Saturday + 4 days = Wednesday

Therefore, the day of the week on 30 January 2013 was Wednesday.

Revision Table: Key Concepts

Additional Information on Calendar Calculations

Calendar problems often involve calculating the number of days between two dates and using the concept of odd days to find the day of the week. A year has either 365 days (normal year) or 366 days (leap year). Leap years occur every 4 years, except for years divisible by 100 but not by 400.

The day of the week repeats every 7 days. So, if a certain date falls on a Monday, the same date next week will also fall on a Monday. This property is fundamental to using odd days for calendar calculations.

To find the number of odd days in a period:

• Number of odd days in a normal year (365 days): $365 = 52 \times 7 + 1$. So, 1 odd day.
• Number of odd days in a leap year (366 days): $366 = 52 \times 7 + 2$. So, 2 odd days.

In our calculation, instead of finding odd days per year, we calculated the total number of days and then found the odd days for the total period, which is also a valid and often easier approach for periods spanning multiple years. The final day is found by moving forward from the start day by the number of odd days.