Calendar Practice Test Questions - Quantitative Aptitude

1996 was a leap year, so the period from 26 Jan 1996 to 26 Jan 1997 (which includes 29 Feb 1996) adds 2 odd days. Friday + 2 odd days = Sunday.

1st January, 1993 was a Friday. The day count from 1 Jan to 31 July is 212 days (211 days after 1 Jan). 211 mod 7 = 1, so 31 July, 1993 = Friday + 1 day = Saturday.

A year must be divisible by 4 to be a leap year, but a century year must additionally be divisible by 400. 2100 is divisible by 4 but not by 400 (2100400 = 5.25), so it is not a leap year.

For the 13th of consecutive months to fall on the same weekday, the first month must have a number of days that is a multiple of 7. Only February in a non-leap year has 28 days (a multiple of 7), so the months are February and March.

54 days = 7 weeks and 5 days, so the day after 54 days is 5 days after Sunday.
Sunday + 5 days = Friday.

61 days = 8 weeks + 5 days, so the day after 61 days is 5 days after Monday.
Monday + 5 days = Saturday.

Count odd days. 2000 years give 0 odd days. The years 2001 to 2009 have 7 ordinary years and 2 leap years (2004, 2008): 7 + 2 × 2 = 11 odd days = 4 odd days.
From 1 January to 15 August 2010: 31 + 28 + 31 + 30 + 31 + 30 + 31 + 15 = 227 days = 32 weeks + 3 days, so 3 odd days.
Total = 4 + 3 = 7 = 0 odd days, so 15 August 2010 was a Sunday.

3p weeks = 3p × 7 = 21p days.
Adding 2p days: 21p + 2p = 23p days.

3p weeks = 21p days
∴ 21p + 2p = 23p days ✓

(x + 2) weeks = 7(x + 2) = 7x + 14 days.
Adding x days: 7x + 14 + x = 8x + 14 days.

(x + 2) weeks = 7x + 14 days
∴ 7x + 14 + x = 8x + 14 days ✓

Number of days from 1 January 2005 to 1 January 2006 = 365.
365 = 7 × 52 + 1, so the number of odd days is 1.
Given day = Saturday. The required date is 1 day beyond it.
So the day is Sunday.

365 days = 7 × 52 + 1
Saturday + 1
∴ Sunday ✓

Number of days from 1 January 2019 to 1 January 2021 = 731.
731 = 7 × 104 + 3, so the number of odd days is 3.
The period includes one 29 February (in 2020), so it has 365 + 366 = 731 days.
Given day = Tuesday. The required date is 3 days beyond it.
So the day is Friday.

731 days = 7 × 104 + 3
Tuesday + 3
∴ Friday ✓

Number of days from 22 November 2010 to 22 November 2011 = 365.
365 = 7 × 52 + 1, so the number of odd days is 1.
Given day = Tuesday. The required date is 1 day before it.
So the day was Monday.

365 days = 7 × 52 + 1
Tuesday − 1
∴ Monday ✓

Number of days from 9 October 2014 to 9 October 2016 = 731.
731 = 7 × 104 + 3, so the number of odd days is 3.
The period includes one 29 February (in 2016), so it has 365 + 366 = 731 days.
Given day = Sunday. The required date is 3 days before it.
So the day was Thursday.

731 days = 7 × 104 + 3
Sunday − 3
∴ Thursday ✓

Number of days from 12 May 2008 to 12 May 2009 = 365.
365 = 7 × 52 + 1, so the number of odd days is 1.
Given day = Tuesday. The required date is 1 day before it.
So the day was Monday.

365 days = 7 × 52 + 1
Tuesday − 1
∴ Monday ✓

Number of days from 15 March 2012 to 15 March 2013 = 365.
365 = 7 × 52 + 1, so the number of odd days is 1.
Given day = Friday. The required date is 1 day before it.
So the day was Thursday.

365 days = 7 × 52 + 1
Friday − 1
∴ Thursday ✓

Number of days from 5 February 2008 to 5 February 2009 = 366.
366 = 7 × 52 + 2, so the number of odd days is 2.
The period includes 29 February (a leap day), so the period contains 366 days.
Given day = Thursday. The required date is 2 days before it.
So the day was Tuesday.

366 days = 7 × 52 + 2
Thursday − 2
∴ Tuesday ✓

Number of days from 1 February 2022 to 30 November 2022 = 302.
302 = 7 × 43 + 1, so the number of odd days is 1.
Given day = Tuesday. The required date is 1 day beyond it.
So the day is Wednesday.

302 days = 7 × 43 + 1
Tuesday + 1
∴ Wednesday ✓

Years completed = 1999 = 1600 + 300 + 99.
Odd days in 1600 years = 0 (every 400 years has 0 odd days).
Odd days in 300 years = 1.
The remaining 99 years have 24 leap years and 75 ordinary years, so odd days = 24 × 2 + 75 = 123, which leaves 4 odd days after dividing by 7.
Days from 1 January to 1 February 2000 = 31 + 1 = 32, so 4 odd days.
Total odd days = 1 + 4 + 4 = 9, which leaves 2 odd days after dividing by 7 (0 = Sunday, 1 = Monday, and so on).
So 1 February 2000 is a Tuesday.
1 February 2000 is itself a Tuesday, so the Tuesdays are 1, 8, 15, 22 and, because 2000 is a leap year with 29 days in February, also 29.

1 Feb 2000 = Tuesday
every 7 days; 2000 is a leap year (29 days)
∴ 1, 8, 15, 22, 29 ✓

100 years contain 5 odd days, 200 years contain 3, 300 years contain 1 and 400 years contain 0, and this cycle repeats.
So the last day of a century can only be Friday, Wednesday, Monday or Sunday.
The first day of the next century is the day after, so it can only be Saturday, Thursday, Tuesday or Monday.
Wednesday is not in this list, so the first day of a century cannot be Wednesday.

Last days: Fri, Wed, Mon, Sun
First days = next day: Sat, Thu, Tue, Mon
∴ Cannot be Wednesday ✓

Number of days from 1 January 2000 to 1 January 2020 = 7305.
7305 = 7 × 1043 + 4, so the number of odd days is 4.
The period includes five leap days (2000, 2004, 2008, 2012 and 2016), so it has 20 × 365 + 5 = 7305 days.
Given day = Saturday. The required date is 4 days beyond it.
So the day is Wednesday.

7305 days = 7 × 1043 + 4
Saturday + 4
∴ Wednesday ✓

Years completed = 1988 = 1600 + 300 + 88.
Odd days in 1600 years = 0 (every 400 years has 0 odd days).
Odd days in 300 years = 1.
The remaining 88 years have 22 leap years and 66 ordinary years, so odd days = 22 × 2 + 66 = 110, which leaves 5 odd days after dividing by 7.
Days from 1 January to 9 November 1989 = 31 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 9 = 313, so 5 odd days.
Total odd days = 1 + 5 + 5 = 11, which leaves 4 odd days after dividing by 7 (0 = Sunday, 1 = Monday, and so on).
So 9 November 1989 is a Thursday.

Odd days: years before + days into the year
total = 4 (0 = Sunday)
∴ Thursday ✓

Years completed = 1947 = 1600 + 300 + 47.
Odd days in 1600 years = 0 (every 400 years has 0 odd days).
Odd days in 300 years = 1.
The remaining 47 years have 11 leap years and 36 ordinary years, so odd days = 11 × 2 + 36 = 58, which leaves 2 odd days after dividing by 7.
Days from 1 January to 30 January 1948 = 30, so 2 odd days.
Total odd days = 1 + 2 + 2 = 5, which leaves 5 odd days after dividing by 7 (0 = Sunday, 1 = Monday, and so on).
So 30 January 1948 is a Friday.

Odd days: years before + days into the year
total = 5 (0 = Sunday)
∴ Friday ✓

Years completed = 2011 = 2000 + 11.
Odd days in 2000 years = 0 (every 400 years has 0 odd days).
The remaining 11 years have 2 leap years and 9 ordinary years, so odd days = 2 × 2 + 9 = 13, which leaves 6 odd days after dividing by 7.
Days from 1 January to 29 February 2012 = 31 + 29 = 60, so 4 odd days.
Total odd days = 0 + 6 + 4 = 10, which leaves 3 odd days after dividing by 7 (0 = Sunday, 1 = Monday, and so on).
So 29 February 2012 is a Wednesday.

Odd days: years before + days into the year
total = 3 (0 = Sunday)
∴ Wednesday ✓

Years completed = 2019 = 2000 + 19.
Odd days in 2000 years = 0 (every 400 years has 0 odd days).
The remaining 19 years have 4 leap years and 15 ordinary years, so odd days = 4 × 2 + 15 = 23, which leaves 2 odd days after dividing by 7.
Days from 1 January to 29 February 2020 = 31 + 29 = 60, so 4 odd days.
Total odd days = 0 + 2 + 4 = 6, which leaves 6 odd days after dividing by 7 (0 = Sunday, 1 = Monday, and so on).
So 29 February 2020 is a Saturday.

Odd days: years before + days into the year
total = 6 (0 = Sunday)
∴ Saturday ✓

Years completed = 1911 = 1600 + 300 + 11.
Odd days in 1600 years = 0 (every 400 years has 0 odd days).
Odd days in 300 years = 1.
The remaining 11 years have 2 leap years and 9 ordinary years, so odd days = 2 × 2 + 9 = 13, which leaves 6 odd days after dividing by 7.
Days from 1 January to 15 April 1912 = 31 + 29 + 31 + 15 = 106, so 1 odd day.
Total odd days = 1 + 6 + 1 = 8, which leaves 1 odd day after dividing by 7 (0 = Sunday, 1 = Monday, and so on).
So 15 April 1912 is a Monday.

Odd days: years before + days into the year
total = 1 (0 = Sunday)
∴ Monday ✓