Chapter 5 · Topic 5.1 · Prelims + Mains1 of 3
General Mental Ability: Clocks and calendars
Key points
- Clock angles and overlapping hands.
- Calendars and repeating years.
Prelims box: facts and traps
| Item | Value |
|---|---|
| Clock questions, CSAT 2014 to 2026 | About 15 |
| Calendar and day questions | About 9 |
| Cube and dice questions | About 11. |
- Move one day per year, two after a leap year: 2026 Thu, 2027 Fri, 2028 Sat (leap), 2029 Mon, 2030 Tue, 2031 Wed. 2031 is ordinary too.
- CSAT 2024, Q38: The calendar for the year 2025 is same for Prelims 2024 2025 starts on a Wednesday and is ordinary.
- Step 2: Add odd days year by year from 2026: 2026 (1), 2027 (1), 2028 (2), 2029 (1), 2030 (1), 2031 (1): running total 7, so 2032 starts on the same weekday, but 2032 is a leap year, so it fails.
- Trap (UPSC Prelims 2014): “the hour and minute hands of a clock move without jerking” is incorrect. “the clock shows a time between 8 o'clock and 9 o'clock” is incorrect. “the two hands of the clock are one above the other. After how many minutes (nearest integer) will the two hands be again lying one above the other ?” is incorrect.
- Trap: (The leap day of 2028 comes in February, after 1 January, so it does not count here.)
Clocks: angles and overlapping hands
A clock face is a circle of 360° with 60 minute marks. Two plain facts solve almost every clock question: the minute hand moves 6° a minute (360 ÷ 60), and the hour hand moves 30° an hour (360 ÷ 12), that is half a degree a minute.
| Item | Value | Type |
|---|---|---|
| Clock questions, CSAT 2014 to 2026 | About 15 | Battle IAS keyword search of the CSAT PYQ book (approximate) |
| Calendar and day questions | About 9 | Same search (approximate) |
| Cube and dice questions | About 11 | Same search (approximate) |
Battle IAS keyword search of the CSAT PYQ book question text; counts are approximate.
CSAT 2024, Q40: What is the angle between the minute hand and hour hand when the clock shows 4:25 hours? Prelims 2024 Minute hand: 25 × 6 = 150° from 12. Hour hand: 4 × 30 = 120°, plus 25 × ½ = 12.5°, so 132.5°. Gap 150 − 132.5 = 17.5°. Answer (c).
What is the angle between the minute hand and hour hand when the clock shows 4:25 hours?
- 12·5°
- 15°
- 17·5°
- 20°
Show answer
Answer: (c) · UPSC official key
CSAT 2015, Q68: Between 6 PM and 7 PM the minute hand of a clock will be ahead of the hour hand by 3 minutes at Prelims 2015 Check the options. At 6:36 the hour hand has moved from the 30-minute mark by 36 ÷ 12 = 3 marks, so it is at 33. The minute hand is at 36: 3 minutes ahead. Answer (c).
Between 6 PM and 7 PM the minute hand of a clock will be ahead of the hour hand by 3 minutes at
- 6:15 PM
- 6:18 PM
- 6:36 PM
- 6:48 PM
Show answer
Answer: (c) · UPSC official key
CSAT 2014, Q12: Assume that Each minute the minute hand gains 6 − ½ = 5½° on the hour hand. To be on top again it must gain a full 360°: 360 ÷ 5½ = 65 and a bit. Answer (c), 65.
Assume that 1. the hour and minute hands of a clock move without jerking. 2. the clock shows a time between 8 o’clock and 9 o’clock. 3. the two hands of the clock are one above the other. After how many minutes (nearest integer) will the two hands be again lying one above the other ?
- 60
- 62
- 65
- 67
Show answer
Answer: (c) · UPSC official key
CSAT 2024, Q37: How many times the hour hand and the minute hand coincide in a clock between 10:00 a.m. and 2:00 p.m. (same day)? Prelims 2024 The hands are together about every 65 minutes: near 10:54, at 12:00 and near 1:05. The next is near 2:11, after 2:00. That is 3 times. Answer (a).
How many times the hour hand and the minute hand coincide in a clock between 10:00 a.m. and 2:00 p.m. (same day)?
- 3 times
- 4 times
- 5 times
- 6 times
Show answer
Answer: (a) · UPSC official key
CSAT 2019, Q49: A wall clock moves 10 minutes fast in every 24 hours. The clock was set right to show the correct time at 8:00 a.m. on Monday. When the clock shows the time 6:00 p.m. on Wednesday, what is the correct time? Prelims 2019 In 24 real hours the clock shows 24 hours 10 minutes: 145 clock minutes for every 144 real minutes. From Monday 8 a.m. to Wednesday 6 p.m. the clock shows 58 hours = 3,480 minutes. Real time = 3,480 ÷ 145 × 144 = 24 × 144 = 3,456 minutes = 57 hours 36 minutes, so the clock is 24 minutes ahead. Answer (a), 5:36 p.m.
A wall clock moves 10 minutes fast in every 24 hours. The clock was set right to show the correct time at 8:00 a.m. on Monday. When the clock shows the time 6:00 p.m. on Wednesday, what is the correct time?
- 5:36 p.m.
- 5:30 p.m.
- 5:24 p.m.
- 5:18 p.m.
Show answer
Answer: (a) · UPSC official key
Calendars: odd days and repeating years
An ordinary year is 52 weeks and 1 day, so the next year starts one weekday later. A leap year (366 days) pushes it two weekdays later. Two years have the same calendar when they start on the same weekday and are both ordinary or both leap.
CSAT 2024, Q38: The calendar for the year 2025 is same for Prelims 2024 2025 starts on a Wednesday and is ordinary. Move one day per year, two after a leap year: 2026 Thu, 2027 Fri, 2028 Sat (leap), 2029 Mon, 2030 Tue, 2031 Wed. 2031 is ordinary too. Answer (c).
The calendar for the year 2025 is same for
- 2029
- 2030
- 2031
- 2033
Show answer
Answer: (c) · UPSC official key
CSAT 2019, Q59: Which year has the same calendar as that of 2009? Prelims 2019 2009 starts on a Thursday. 2010 Fri, 2011 Sat, 2012 Sun (leap), 2013 Tue, 2014 Wed, 2015 Thu, and 2015 is ordinary. Answer (d), 2015.
Which year has the same calendar as that of 2009?
- 2018
- 2017
- 2016
- 2015
Show answer
Answer: (d) · UPSC official key
CSAT 2017, Q11: If second and fourth Saturdays and all the Sundays are taken as only holidays for an office, what would be the minimum number of possible working days of any month of any year? Prelims 2017 Every month has exactly two 2nd/4th Saturdays. The fewest days with the most Sundays: February with 28 days has 4 Sundays: 28 − 4 − 2 = 22; a leap February with 5 Sundays also gives 29 − 5 − 2 = 22. Longer months give more. Answer (b), 22.
If second and fourth Saturdays and all the Sundays are taken as only holidays for an office, what would be the minimum number of possible working days of any month of any year?
- 23
- 22
- 21
- 20
Show answer
Answer: (b) · UPSC official key
Mains box: analysis and answer points
- A clock face is a circle of 360° with 60 minute marks.
- Hour hand: 4 × 30 = 120°, plus 25 × ½ = 12.5°, so 132.5°.
- CSAT 2015, Q68: Between 6 PM and 7 PM the minute hand of a clock will be ahead of the hour hand by 3 minutes at Prelims 2015 Check the options.
- Likely next angle: Check options
Practice
Practice MCQs (8)
The mirror image of a wall clock shows the time as 3:40. What is the actual time on the clock?
Answer (a) & explanation
Step 1: For a mirror image, actual time = 11:60 − mirror time (that is, 12:00 minus the time shown). Step 2: 11:60 − 3:40 = 8:20. Step 3: Check: at 8:20 the minute hand points to 4; a mirror sends 4 to 8, i.e. 40 minutes, and the hour hand just past 8 appears just before 4, i.e. 3:40.
1 January 2026 is a Thursday. On which day of the week will 1 January 2028 fall?
Answer (d) & explanation
Step 1: Moving from 1 January 2026 to 1 January 2028 crosses the whole of 2026 and 2027. Step 2: Neither 2026 nor 2027 is a leap year, so each adds 1 odd day. Step 3: Total 2 odd days: Thursday + 2 = Saturday. (The leap day of 2028 comes in February, after 1 January, so it does not count here.)
Consider the following years:
1. 1900
2. 2000
3. 2100
4. 2024
How many of the years given above are leap years?
Answer (b) & explanation
Rule: a year is a leap year if it is divisible by 4, except century years, which must be divisible by 400. Step 1: 1900 is a century year not divisible by 400: not leap. Step 2: 2000 is divisible by 400: leap. Step 3: 2100 is a century year not divisible by 400: not leap. Step 4: 2024 is divisible by 4 and not a century year: leap. Two leap years.
Using the method of odd days, find the day of the week on which 26 January 1950 fell.
Answer (c) & explanation
Step 1: Count odd days in the 1949 complete years before 1950. 1600 years give 0 odd days; 300 years give 15 odd days, i.e. 1 (15 mod 7). Step 2: The remaining 49 years (1901- 1949) have 12 leap years and 37 ordinary years: 12 × 2 + 37 × 1 = 61 odd days, i.e. 5. Step 3: In 1950, the days up to 26 January give 26 days, i.e. 5 odd days. Step 4: Total = 1 + 5 + 5 = 11, i.e. 4 odd days. Taking 0 = Sunday, 4 = Thursday.
What is the smaller angle between the hour hand and the minute hand of a clock at 10:15?
Answer (a) & explanation
Step 1: Angle = |30H − 5.5M| = |300 − 82.5| = 217.5°. Step 2: This is more than 180°, so the smaller angle is 360° − 217.5° = 142.5°. Check: the minute hand is at 3 (90°); the hour hand is a quarter of the way from 10 to 11, at 300° + 7.5° = 307.5°; going the short way round, 90° + (360° − 307.5°) = 142.5°.
How many times in a day (24 hours) are the hour hand and the minute hand of a clock at right angles to each other?
Answer (c) & explanation
Step 1: In 12 hours the minute hand laps the hour hand 11 times (not 12), because the hour hand also moves forward. Step 2: In each lap the hands are at right angles twice (once at 90° ahead, once at 90° behind), giving 22 times in 12 hours. Step 3: In 24 hours: 2 × 22 = 44.
Which is the next year after 2026 whose calendar will be exactly the same as that of 2026?
Answer (c) & explanation
Step 1: Two years share a calendar only if they start on the same weekday and are both leap or both ordinary. 2026 is an ordinary year. Step 2: Add odd days year by year from 2026: 2026 (1), 2027 (1), 2028 (2), 2029 (1), 2030 (1), 2031 (1): running total 7, so 2032 starts on the same weekday, but 2032 is a leap year, so it fails. Step 3: Continue: 2032 (2), 2033 (1), 2034 (1), 2035 (1), 2036 (2): the total from 2026 reaches 14, a multiple of 7, at the start of 2037. Step 4: 2037 is ordinary, so 2037 is the answer.
Consider the following statements:
1. In any month that has a 29th day, the 1st and the 29th of that month fall on the same day of the week.
2. In a non-leap year, 1 January and 31 December fall on the same day of the week.
3. A non-leap year that begins on a Sunday has 53 Sundays and 53 Mondays.
Which of the statements given above is/are correct?
Answer (b) & explanation
Step 1 (Statement 1): The 29th is 28 days after the 1st, and 28 is a multiple of 7, so the weekday is the same. Correct. Step 2 (Statement 2): 31 December is 364 days after 1 January in a non-leap year, and 364 = 52 × 7, so the weekday is the same. Correct. Step 3 (Statement 3): A non-leap year has 52 weeks + 1 extra day. That extra day is the starting day, Sunday, so there are 53 Sundays but only 52 Mondays. Incorrect.
0 / 0 correct
Write an answer
In 150 words, explain the significance of General Mental Ability with one recent example.
Flashcards (11)
Minute hand speed?
6° a minute.
Hour hand speed?
30° an hour, ½° a minute.
How much does the minute hand gain each
minute? 5½°.
Angle at 4:25?
17.5°.
Hands coincide about every?
65 5/11 minutes (about 65).
Times hands coincide from 10 a.m. to 2
p.m.? 3.
Start day shift after an ordinary year?
One weekday.
After a leap year?
Two weekdays.
Year with the same calendar as 2025?
2031.
Year with the same calendar as 2009?
2015.
Fewest working days in a month (2nd/4th
Saturdays and Sundays off)? 22.
Next topic · 5.2 →General Mental Ability: Cubes: cuts, painted faces and weighing