If 12th January 2015 was Monday, then what day of the week was it on 26th January 2016?
Tuesday
Solving Calendar Problems: Finding the Day of the Week
This question asks us to determine the day of the week for a specific date, 26th January 2016, given the day for another date, 12th January 2015. We can solve this type of calendar problem by calculating the total number of days between the two dates and then finding the number of 'odd days'.
Understanding Odd Days in Calendar Calculations
An 'odd day' is the remainder left after dividing the total number of days by 7. Since the days of the week repeat every 7 days, the position in the cycle (and thus the day of the week) shifts by the number of odd days.
- A normal year has 365 days. \({365 \div 7 = 52}\) weeks and 1 day. So, a normal year has 1 odd day.
- A leap year has 366 days (due to 29th February). \({366 \div 7 = 52}\) weeks and 2 days. So, a leap year has 2 odd days.
The year 2016 is a leap year because it is divisible by 4.
Step-by-Step Calculation
We need to find the total number of days from 12th January 2015 to 26th January 2016. We can break this down into two parts:
- Number of days from 12th January 2015 to 31st December 2015.
- Number of days from 1st January 2016 to 26th January 2016.
Calculating Days Remaining in 2015
The number of days remaining in January 2015 from 12th onwards (i.e., from 13th Jan to 31st Jan) is \({31 - 12 = 19}\) days.
The number of days from 1st February 2015 to 31st December 2015 is the total number of days in 2015 minus the days from 1st Jan to 12th Jan.
Total days in 2015 (normal year) = 365 days.
Days from 1st Jan 2015 to 12th Jan 2015 = 12 days.
Days from 13th Jan 2015 to 31st Dec 2015 = \({365 - 12 = 353}\) days.
Alternatively, using the first method:
- Days in Jan 2015 (from 13th): \({31 - 12 = 19}\) days
- Days in Feb 2015: 28 days
- Days in Mar to Dec 2015: \({31+30+31+30+31+31+30+31+30+31 = 306}\) days
- Total days from 12th Jan 2015 to 31st Dec 2015 = \({19 + 28 + 306 = 353}\) days.
Calculating Days in 2016 up to 26th January
The number of days from 1st January 2016 to 26th January 2016 is simply 26 days.
Calculating Total Days
Total number of days from 12th January 2015 to 26th January 2016 is the sum of days remaining in 2015 and days in 2016:
Total days = (Days from 12th Jan 2015 to 31st Dec 2015) + (Days from 1st Jan 2016 to 26th Jan 2016)
Total days = \({353 + 26 = 379}\) days.
Calculating Odd Days
Now, we find the number of odd days by dividing the total days by 7:
Odd days = \({379 \pmod{7}}\)
\({379 \div 7 = 54}\) with a remainder.
\({7 \times 54 = 378}\)
Remainder = \({379 - 378 = 1}\) day.
So, there is 1 odd day.
Determining the Final Day
The starting day was Monday (on 12th January 2015).
We add the number of odd days to the starting day:
Final day = Starting day + Odd days
Final day = Monday + 1 day
Final day = Tuesday.
Therefore, 26th January 2016 was a Tuesday.
| Date | Day |
|---|---|
| 12th January 2015 | Monday |
| Total Days Elapsed (12/01/2015 to 26/01/2016) | 379 |
| Odd Days (\({379 \pmod{7}}\)) | 1 |
| Day on 26th January 2016 | Monday + 1 day = Tuesday |
Revision Table: Key Concepts
| Concept | Description | Example |
|---|---|---|
| Odd Days | Remainder when total days divided by 7. Determines the shift in the day of the week. | 10 days = \({10 \pmod{7}} = 3\) odd days. If today is Monday, 10 days later is Monday + 3 = Thursday. |
| Normal Year | 365 days. Has 1 odd day. | 2015, 2017, 2018, 2019 are normal years. |
| Leap Year | 366 days (Feb 29th). Has 2 odd days. Occurs every 4 years, except for years divisible by 100 but not by 400. | 2016, 2020, 2024 are leap years. 1900 was not a leap year; 2000 was a leap year. |
Additional Information on Calendar Calculations
Calendar problems often involve calculating the number of days over different months and years, accounting for the varying number of days in each month and the occurrence of leap years.
- Months with 31 days: January, March, May, July, August, October, December.
- Months with 30 days: April, June, September, November.
- February: 28 days in a normal year, 29 days in a leap year.
To calculate the day of the week for a distant date from a known date, calculating the total number of odd days over the period is the most efficient method. This involves summing the odd days for each year or breaking down the period into convenient chunks (like full years and remaining days).
When crossing a leap year, remember to include the extra day if the period includes 29th February. In this problem, the period from 12th Jan 2015 to 26th Jan 2016 includes Feb 2016, which had 29 days, thus contributing to the total number of days correctly calculated as 379.
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