If it is Friday on January 1, 1998 then January 1, 2000 would have been:
(d) Sunday
Understanding the Problem: Calculating Days
This question asks us to determine the day of the week for a specific date (January 1, 2000) based on the day of the week for another date (January 1, 1998). To solve this type of problem, we need to understand the concept of "odd days" in a calendar.
What are Odd Days?
Odd days are the number of days left over after dividing the total number of days by 7 (the number of days in a week). For example, if there are 10 days, $10 = 1 \times 7 + 3$, so there are 3 odd days. Adding a certain number of odd days to a starting day shifts the day of the week forward by that number of days.
- An ordinary year has 365 days. $365 = 52 \times 7 + 1$. So, an ordinary year has 1 odd day.
- A leap year has 366 days (due to February 29). $366 = 52 \times 7 + 2$. So, a leap year has 2 odd days.
Identifying Leap Years
A year is a leap year if it is divisible by 4, unless it is a century year (like 1900, 2000) which must be divisible by 400 to be a leap year.
- 1998: Not divisible by 4. Ordinary year.
- 1999: Not divisible by 4. Ordinary year.
- 2000: Divisible by 400. Leap year.
When calculating the day for January 1, 2000, starting from January 1, 1998, we need to consider the full years in between, which are 1998 and 1999. The leap day of 2000 (Feb 29) does not fall within the period from Jan 1, 1998, up to and including Jan 1, 2000. We are calculating the day *on* Jan 1, 2000, not after Feb 29, 2000.
Calculating Odd Days Between January 1, 1998 and January 1, 2000
We need to find the total number of odd days for the period from January 1, 1998, up to December 31, 1999 (to find the day on Jan 1, 2000).
- Year 1998 (Jan 1, 1998 to Dec 31, 1998): This is an ordinary year. It contributes 1 odd day.
- Year 1999 (Jan 1, 1999 to Dec 31, 1999): This is an ordinary year. It contributes 1 odd day.
Total odd days = (Odd days in 1998) + (Odd days in 1999)
Total odd days = $1 + 1 = 2$ odd days.
Alternatively, consider the total number of days from Jan 1, 1998 to Dec 31, 1999:
- Days in 1998: 365 (ordinary year)
- Days in 1999: 365 (ordinary year)
Total days = $365 + 365 = 730$ days.
To find the odd days, we calculate $730 \pmod{7}$:
$\text{Odd days} = 730 \div 7 \text{ remainder}$
$730 = 104 \times 7 + 2$
So, there are 2 odd days.
Determining the Day on January 1, 2000
The starting day is Friday, January 1, 1998.
We have calculated a total of 2 odd days between January 1, 1998, and January 1, 2000.
Starting day + Total odd days = Day on January 1, 2000
Friday + 2 days
- Friday + 1 day = Saturday
- Saturday + 1 day = Sunday
Therefore, January 1, 2000, was a Sunday.
| Period | Year Type | Odd Days |
|---|---|---|
| Jan 1, 1998 - Dec 31, 1998 | Ordinary (1998) | 1 |
| Jan 1, 1999 - Dec 31, 1999 | Ordinary (1999) | 1 |
| Total Odd Days | $1 + 1 = 2$ |
Starting Day: Friday
Shift by: +2 days
Resulting Day: Sunday
Revision Table: Calendar Concepts
| Concept | Description | Calculation Impact |
|---|---|---|
| Ordinary Year | 365 days | 1 odd day ($365 \div 7$, remainder 1) |
| Leap Year | 366 days | 2 odd days ($366 \div 7$, remainder 2) |
| Odd Days | Extra days beyond a full week (remainder when divided by 7) | Determines the shift in the day of the week |
| Leap Year Rule | Divisible by 4 (unless divisible by 100 but not 400) | Adds an extra day (Feb 29), increasing odd days for the year |
Additional Information: Calendar Day Calculations
Calendar day calculations involve understanding the pattern of days and the cycle of weeks. Every 7 days, the day of the week repeats. The difference in the day of the week between two dates depends on the total number of odd days in the period between them.
- Moving forward in time: Add the total odd days to the starting day.
- Moving backward in time: Subtract the total odd days from the starting day.
- A shift of 0 or 7 odd days means the day of the week is the same.
- A shift of 1 odd day means the day shifts forward by one day (e.g., Friday becomes Saturday).
- A shift of 2 odd days means the day shifts forward by two days (e.g., Friday becomes Sunday).
- And so on, up to a shift of 6 odd days. A shift of 7 is the same as 0.
When calculating odd days over several years, it's crucial to correctly identify which years in the period are leap years and add 2 odd days for each leap year and 1 odd day for each ordinary year. Pay attention to the start and end dates to ensure the leap day (Feb 29) is included in the count if the period spans across it in a leap year.
Important Questions from Sitting Arrangement
Four students P, Q, R, and S are playing Carrom. In this game, P makes a pair with R, and Q makes a pair with S. S is the right of R whose face is towards west. Which direction is S facing?
Read the following information carefully to answer the given question. Five friends P, Q, R, S, and T are sitting on a bench and wearing different coloured jackets, i.e., brown, red, yellow, white, and grey (not necessarily in the order). (A) P is sitting next to Q and wearing a red jacket. (B) R is sitting next to S and not wearing either a brown or grey jacket. (C) S is not sitting with T and T is wearing a brown jacket. (D) T is on the left end of the bench. (E) R is on the second position from the right. (F) P is on the right of Q and T, and Q is wearing a grey jacket. (G) P and R are sitting together. (H) S is not wearing a white jacket. What is the color of R’s jacket?
Six persons A, B, C, D, E, and F are sitting in two parallel rows of three persons in each row, all facing north. C is sitting to the right of E. B is sitting to the right of D, who is sitting opposite to A. F is sitting diagonally opposite to A, who is at the left of E. Who is sitting opposite to B?
Which of the following is not a member of ‘SAARC’?
Five students are standing in a circle and facing the center. Ajay is between Amar and Aman. Alok is on the left of Amit. Amar is on the immediate left of Alok. Who is sitting immediate right of Ajay?
