Clock Basics and Hand Speeds
- Minute Hand: Covers 360° in 60 minutes, moving at 6° per minute.
- Hour Hand: Covers 30° in 60 minutes, moving at 0.5° per minute.
- Relative Speed: The minute hand travels faster, gaining 5.5° per minute on the hour hand.
Essential Clock Formulas
- Angle Formula: Use θ = |30H − 5.5M| where H is the hour and M is the minutes. At 3:00 (H = 3, M = 0), the angle is 30 × 3 = 90°.
- Reflex Angle: If the calculated angle is greater than 180°, subtract it from 360°.
- Coincidence Frequency: Hands overlap 11 times in 12 hours and 22 times in a full day (24 hours).
- Right Angles (90°): Formed 22 times in 12 hours and 44 times in 24 hours (twice per hour except around 3:00 and 9:00).
- Faulty Clocks: Calculate correct time using: True Time = Clock Time × (Total Actual Hours / Total Faulty Hours).
Calendar Basics and Odd Days
- An odd day represents the remainder left after dividing total days by 7, which dictates how weekdays shift.
- Ordinary Year: 365 days = 52 weeks + 1 odd day. January 1 shifts forward by 1 weekday each year.
- Leap Year: 366 days = 52 weeks + 2 odd days, featuring an extra day in February (29 days).
Leap Year Identification Rules
- Regular Years: A year is a leap year if it is completely divisible by 4 (e.g., 2024).
- Century Years: Must be divisible by 400. While 2000 is a leap year, 1900 is not because it fails the 400-rule, creating an 8-year gap between leap years across century transitions (1896 to 1904).
Calendar Shortcut Rules
- Century Odd Days: 400 years = 0 odd days; 300 years = 1; 200 years = 3; 100 years = 5.
- Calendar Repetition: Non-leap year calendars repeat after 11, 6, or 11 years. Leap year calendars repeat every 28 years.
- Date-to-Day Calculation: Sum up odd days from centuries, completed years, and months up to the given date, then find the final remainder modulo 7 (0 = Sunday, 1 = Monday, etc.).
SECTION: EASY LEVEL MCQs
Q1. What is the angle between the hour hand and the minute hand of a clock at 3:00?
Correct Answer: C) 90°
Explanation: At 3:00, the hour hand is at 3 and the minute hand is at 12. Angle = 3 × 30° = 90°.
Q2. How many degrees does the minute hand of a clock turn in 20 minutes?
Correct Answer: C) 120°
Explanation: The minute hand moves at 6° per minute. In 20 minutes, angle moved = 20 × 6° = 120°.
Q3. How many degrees does the hour hand move in 4 hours?
Correct Answer: C) 120°
Explanation: The hour hand moves 30° per hour. In 4 hours, angle moved = 4 × 30° = 120°.
Q4. How many times do the hands of a clock coincide in a single day (24 hours)?
Correct Answer: B) 22
Explanation: The hands coincide 11 times in 12 hours (between 11 and 1 they coincide only once at 12:00). In 24 hours, they coincide 11 × 2 = 22 times.
Q5. How many times are the hands of a clock at right angles to each other in 12 hours?
Correct Answer: B) 22
Explanation: In 12 hours, hands form a 90° angle 22 times (twice every hour, except around 3:00 and 9:00).
Q6. How many odd days are there in a standard (non-leap) year of 365 days?
Correct Answer: B) 1
Explanation: 365 ÷ 7 = 52 weeks and 1 remainder day. So, a non-leap year has 1 odd day.
Q7. Which of the following is NOT a leap year?
Correct Answer: C) 1900
Explanation: A century year is a leap year only if it is divisible by 400. 1900 is divisible by 100 but not by 400, so it is an ordinary year.
Q8. If January 1, 2023 was a Sunday, what day of the week was January 1, 2024?
Correct Answer: B) Monday
Explanation: 2023 is a non-leap year, which has 1 odd day. Sunday + 1 day = Monday.
Q9. How many odd days are there in a leap year?
Correct Answer: B) 2
Explanation: 366 ÷ 7 = 52 weeks and 2 remainder days. So, a leap year has 2 odd days.
Q10. If today is Friday, what day of the week will it be after 63 days?
Correct Answer: B) Friday
Explanation: 63 ÷ 7 = 9 exact weeks (remainder = 0 odd days). Therefore, the day will remain Friday.
SECTION: MEDIUM LEVEL MCQs
Q11. Find the angle between the hour hand and minute hand at 4:20 PM.
Correct Answer: B) 10°
Explanation: Formula: θ = |30H - 5.5M|. For 4:20, H = 4, M = 20. θ = |30(4) - 5.5(20)| = |120 - 110| = 10°.
Q12. At what time between 7 and 8 o'clock will the hands of a clock be together?
Correct Answer: D) 7:38 2/11
Explanation: At 7:00, minute hand is 35 minutes spaces behind. Time taken to gain 35 min spaces = 35 × (12/11) = 420/11 = 38 2/11 minutes past 7.
Q13. At what time between 2 and 3 o'clock will the hands of a clock be at right angles?
Correct Answer: A) 2:27 3/11
Explanation: At 2:00, hands are 10 min spaces apart. For 90° (15 min spaces apart), minute hand must gain (10 + 15) = 25 min spaces. Time = 25 × (12/11) = 300/11 = 27 3/11 min past 2.
Q14. Find the reflex angle between the hands of a clock at 10:25.
Correct Answer: B) 197.5°
Explanation: Angle θ = |30(10) - 5.5(25)| = |300 - 137.5| = 162.5°. Reflex angle = 360° - 162.5° = 197.5°.
Q15. A clock is set right at 5 AM. It loses 16 minutes in 24 hours. What will be the true time when the clock indicates 10 PM on the 4th day?
Correct Answer: B) 11 PM
Explanation: Time from 5 AM 1st day to 10 PM 4th day = 89 hours. 23 hrs 44 min (1424/60 hrs) of this clock = 24 hrs of true time. True time = 89 × 24 / (1424/60) = 90 hours. 90 hours after 5 AM is 11 PM.
Q16. What day of the week was March 15, 2020?
Correct Answer: B) Sunday
Explanation: 2000 years = 0 odd days. Years 2001-2019: 19 yrs (4 leap, 15 ordinary) = 4(2)+15(1) = 23 = 2 odd days. Jan(3) + Feb(1) + Mar(15) = 19 days = 5 odd days. Total odd days = 2 + 5 = 7 = 0 odd days = Sunday.
Q17. The calendar for the year 2007 will be identical to the calendar for which year?
Correct Answer: C) 2018
Explanation: Count odd days starting from 2007: 2007(1), 2008(2), 2009(1), 2010(1), 2011(1), 2012(2), 2013(1), 2014(1), 2015(1), 2016(2), 2017(1). Total odd days = 14 = 0 remainder. Thus 2018 will have the same calendar as 2007.
Q18. If 15th August 2011 was a Monday, what day was 15th August 2016?
Correct Answer: D) Saturday
Explanation: Number of years = 5. Leap years included = 2 (2012 and 2016). Odd days = 3 ordinary years (3) + 2 leap years (4) = 7 odd days - wait: from 15 Aug 2011 to 15 Aug 2016 has 6 total odd days (2012=2, 2013=1, 2014=1, 2015=1, 2016=1 till Aug). Monday + 6 days = Saturday.
Q19. What is the maximum gap (in years) between two successive leap years?
Correct Answer: B) 8 years
Explanation: When a century year is not divisible by 400 (e.g., 1896 to 1904), 1900 is not a leap year. The gap becomes 1904 - 1896 = 8 years.
Q20. How many times does the 29th day of a month occur in 400 consecutive years?
Correct Answer: C) 4497
Explanation: In 400 years, 11 months have a 29th day every year = 400 × 11 = 4400. February has 29 days in 97 leap years. Total = 4400 + 97 = 4497 times.
SECTION: HARD LEVEL MCQs
Q21. A clock gains 5 minutes every hour. If it is set correctly at 12:00 PM on Monday, what exact time will it show when the actual time is 12:00 PM on Wednesday?
Correct Answer: B) 4:00 PM Wednesday
Explanation: Mon 12 PM to Wed 12 PM = 48 actual hours. Clock gains 5 min/hr. Total gain = 48 × 5 = 240 minutes = 4 hours. Clock shows 12:00 PM + 4 hrs = 4:00 PM Wednesday.
Q22. The minute hand of a clock overtakes the hour hand at intervals of 65 minutes of correct time. How much time does the clock gain or lose in a day?
Correct Answer: A) Gains 10 10/143 min
Explanation: Standard overtake interval = 65 5/11 min. Given interval = 65 min. Gain in 65 min = 5/11 min. Gain in 24 hrs = (5/11) × (24 × 60 / 65) = 1440 / 143 = 10 10/143 minutes.
Q23. At what time between 5:30 and 6:00 will the hands of a clock be at right angles?
Correct Answer: B) 5:43 7/11
Explanation: At 5:00, minute hand is 25 min spaces behind. For 90° (15 min spaces), minute hand must gain 25 + 15 = 40 min spaces. Time = 40 × (12/11) = 480/11 = 43 7/11 min past 5 (5:43 7/11).
Q24. Two clocks are set correctly at 9:00 AM. Clock A gains 2 minutes per hour, while Clock B loses 3 minutes per hour. What will be the time difference between the two clocks at 9:00 PM on the same day?
Correct Answer: C) 60 minutes
Explanation: Time elapsed from 9:00 AM to 9:00 PM = 12 hours. Combined rate of separation = 2 + 3 = 5 minutes per hour. Total difference = 12 × 5 = 60 minutes (1 hour).
Q25. A watch which gains uniformly is 5 minutes slow at 8 AM on Sunday and is 5 minutes 48 seconds fast at 8 PM on following Sunday. When was it correct?
Correct Answer: A) 7:20 PM Wednesday
Explanation: Total time = 180 hrs. Total gain = 10.8 min. Rate = 10.8 min in 180 hrs. Time to gain 5 min = (180 / 10.8) × 5 = 83 hrs 20 min = 3 days 11 hrs 20 min. 8 AM Sunday + 3 days 11 hrs 20 min = 7:20 PM Wednesday.
Q26. What was the day of the week on 15th August 1947?
Correct Answer: B) Friday
Explanation: 1600 yrs = 0 odd days. 300 yrs = 1 odd day. 46 yrs = 1 odd day. Total till 1946 = 2 odd days. Jan(3)+Feb(0)+Mar(3)+Apr(2)+May(3)+Jun(2)+Jul(3)+Aug(15) = 31 days = 3 odd days. Total = 2 + 3 = 5 odd days = Friday.
Q27. Which year in the 21st century will have the exact same calendar as the year 2024?
Correct Answer: C) 2052
Explanation: A leap year calendar repeats every 28 years (provided no non-leap century is crossed). 2024 + 28 = 2052.
Q28. If the first day of a non-leap year is Friday, what day of the week will be the 300th day of that year?
Correct Answer: B) Wednesday
Explanation: 300th day = 1st day + 299 days. 299 ÷ 7 = 42 weeks + 5 remainder odd days. Friday + 5 days = Wednesday.
Q29. In a given century, what is the maximum number of times that a specific date (e.g., May 12) can fall on a Sunday?
Correct Answer: B) 15
Explanation: In 100 years, there are 24 or 25 leap years, meaning a day repeats roughly every 5, 6, or 11 years. Across 100 years, any given date falls on a specific weekday either 14 or 15 times.
Q30. If February 1, 2004 was a Sunday, what day of the week was March 1, 2008?
Correct Answer: A) Saturday
Explanation: Feb 1, 2004 to Feb 1, 2008 = 5 odd days (2004 leap, 2005, 2006, 2007). Feb 1, 2008 = Friday. Feb 1 to Mar 1, 2008 (29 days in leap year) = 1 odd day. Friday + 1 = Saturday.