Quantitative Aptitude · Counting, Probability, Calendar & Clocks
Calendar — Questions & Answers
Calendar questions come down to counting 'odd days' — the remainder after dividing a day count by 7. Ordinary years contribute 1 odd day, leap years 2.
- 5 solved questions
- Step-by-step AI walkthroughs
- Key formulas included
Key formulas
Odd days: divide the total number of days by 7 and take the remainder
A leap year is divisible by 4, except century years, which must be divisible by 400
Day codes (0 = Sunday): 100 years give 5 odd days, 200 give 3, 300 give 1, 400 give 0
5 solved Calendar questions
How many odd days are there in an ordinary (non-leap) year?
Solution — answer C
365=52×7+1, so an ordinary year has 1 odd day (a leap year has 2).
4 steps · reveal one at a timeStep 1: Define Odd Days
Odd days are the number of days remaining after dividing the total number of days by 7. This is because a week has 7 days, and we want to find out how many 'extra' days there are beyond full weeks.
Step 2: Days in a Year
First, we need to know the total number of days in an ordinary year. An ordinary year, which is a non-leap year, consists of 365 days. This is the value we will use for our calculation.
Step 3: Calculate Odd Days
To find the number of odd days, we divide the total days in an ordinary year by 7. We are interested in the remainder of this division. So, we calculate 365÷7.
Step 4: The Answer
So the answer is (C) 1. When you divide 365 days by 7, you get 52 weeks with a remainder of 1. This remainder is the number of odd days.
Which of the following is a leap year?
Solution — answer B
A century year is a leap year only if it is divisible by 400. 2000÷400=5, so 2000 is a leap year; 1900 and 2100 are not.
4 steps · reveal one at a timeStep 1: Understand Leap Years
A year is generally a leap year if it is divisible by 4. For example, 1996 and 2004 are leap years. However, there's a special rule for century years that we need to consider.
Step 2: Century Year Rule
A century year (a year ending in '00') is a leap year only if it is divisible by 400. This is an important exception to the general divisibility by 4 rule.
Step 3: Apply the Rule
Let's check the given years. 1990 is not a century year and not divisible by 4. For 1900, 1900÷400 is not a whole number. For 2100, 2100÷400 is also not a whole number.
Step 4: The answer
So the answer is (B) 2000. This is because 2000 is a century year, and it is perfectly divisible by 400 (2000÷400=5), making it a leap year.
If 1 January 2008 was a Tuesday, what day of the week was 1 January 2009? (2008 is a leap year.)
Solution — answer D
2008 is a leap year, contributing 2 odd days, so the day advances from Tuesday by 2 to Thursday.
4 steps · reveal one at a timeStep 1: Understand day shifts
A normal year has 365 days, which is 52 weeks and 1 odd day. This means the day of the week advances by one. A leap year has 366 days, which is 52 weeks and 2 odd days, so the day advances by two.
Step 2: Identify the year type
The question explicitly states that 2008 is a leap year. This is a crucial piece of information, as it tells us how many days are in the year we are considering.
Step 3: Calculate the day shift
Since 2008 is a leap year, it has 366 days. This means that from 1 January 2008 to 1 January 2009, there will be a shift of 2 days in the week, due to the 2 odd days.
Step 4: The answer
So the answer is (D) Thursday. If 1 January 2008 was a Tuesday, advancing by 2 days (Tuesday + 1 day = Wednesday, Wednesday + 1 day = Thursday) brings us to Thursday for 1 January 2009.
1 March 2010 was a Monday. What day of the week was 1 March 2011?
Solution — answer C
The 12 months from 1 March 2010 to 1 March 2011 contain 365 days (no 29 February), i.e. 1 odd day, so the day moves from Monday to Tuesday.
4 steps · reveal one at a timeStep 1: Identify the period
We need to find the day of the week exactly one year after 1 March 2010. This period is from 1 March 2010 to 1 March 2011.
Step 2: Count total days
This period includes the entire year 2010. Since 2010 is not a leap year, February 2011 does not have 29 days. Therefore, the total number of days in this period is 365.
Step 3: Calculate day shift
To find how the day of the week changes, we divide the total number of days by 7 (days in a week). The remainder tells us the shift. 365÷7=52 with a remainder of 1.
Step 4: The answer
So the answer is (C) Tuesday. This remainder of 1 means the day of the week will shift forward by one day from Monday, making it Tuesday.
In an ordinary (non-leap) year, how many months contain a 29th day?
Solution — answer A
Every month has a 29th day except February in an ordinary year (28 days), so the 29th occurs in 12−1=11 months.
5 steps · reveal one at a timeStep 1: Understand the Question
We need to find out how many months in an ordinary (non-leap) year have at least 29 days. This means we are looking for months that contain the 29th day of that month.
Step 2: Recall Month Lengths
Think about the number of days in each month. Most months have 30 or 31 days. February is the special case, as its length varies depending on whether it's a leap year or an ordinary year.
Step 3: Consider Ordinary Year
In an ordinary (non-leap) year, February has only 28 days. All other months have either 30 or 31 days. This means that every month except February will definitely contain a 29th day.
Step 4: Calculate the Total
There are 12 months in a year. Since February is the only month that does not have a 29th day in an ordinary year, we subtract 1 from the total number of months: 12−1=11.
Step 5: The answer
So the answer is (A) 11. Every month has a 29th day except February in an ordinary year (28 days), making it 11 months.
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