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Clocks & Calendars

Angles between hands, gains and losses, days of the week.

46 solved questions · 16 easy · 21 medium · 9 hard

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Shortcuts & formulas

  • Minute hand moves 6 degrees per minute; hour hand moves 0.5 degree per minute (30 degrees per hour).
  • Angle between hands at h:m = |30h - 5.5m|. If it exceeds 180, subtract it from 360.
  • Hands coincide 11 times in 12 hours (22 in a day) and are opposite 11 times in 12 hours. They are at right angles 22 times in 12 hours (44 in a day).
  • Hands coincide every 65 5/11 minutes in a correct clock. Between h and h+1 o'clock they coincide at 60h/11 minutes past h.
  • Mirror image of a clock showing time T is 11:60 minus T (e.g. 3:40 appears as 8:20).
  • Odd days: ordinary year = 1, leap year = 2, 100 years = 5, 200 years = 3, 300 years = 1, 400 years = 0. Week days: 0 Sunday, 1 Monday ... 6 Saturday.
  • Leap year: divisible by 4; a century year must be divisible by 400 (1900 is not leap, 2000 is).
  • Months with 28, 29, 30, 31 days have 0, 1, 2, 3 odd days. Moving N days ahead changes the weekday by N mod 7.

Solved examples

Try each one first, then open the answer and the step-by-step solution.

Easy questions

Q1.

What is the smaller angle between the hour hand and the minute hand of a clock at 3:00?

  1. A90°
  2. B270°
  3. C105°
  4. D60°
Show answer & solution

Answer: A. 90°

Solution

Angle = |30h - 5.5m| = |30 x 3 - 5.5 x 0| = |90 - 0| = 90° Smaller angle = 90° Answer: 90°

Shortcut: Hour hand position = 30h + m/2 degrees; minute hand = 6m degrees; take the difference.

Q2.

What is the smaller angle between the hour hand and the minute hand of a clock at 9:20?

  1. A150°
  2. B160°
  3. C200°
  4. D175°
Show answer & solution

Answer: B. 160°

Solution

Angle = |30h - 5.5m| = |30 x 9 - 5.5 x 20| = |270 - 110| = 160° Smaller angle = 160° Answer: 160°

Shortcut: Hour hand position = 30h + m/2 degrees; minute hand = 6m degrees; take the difference.

Q3.

What is the smaller angle between the hour hand and the minute hand of a clock at 4:30?

  1. A60°
  2. B315°
  3. C45°
  4. D15°
Show answer & solution

Answer: C. 45°

Solution

Angle = |30h - 5.5m| = |30 x 4 - 5.5 x 30| = |120 - 165| = 45° Smaller angle = 45° Answer: 45°

Shortcut: Hour hand position = 30h + m/2 degrees; minute hand = 6m degrees; take the difference.

Q4.

What is the smaller angle between the hour hand and the minute hand of a clock at 7:20?

  1. A90°
  2. B260°
  3. C115°
  4. D100°
Show answer & solution

Answer: D. 100°

Solution

Angle = |30h - 5.5m| = |30 x 7 - 5.5 x 20| = |210 - 110| = 100° Smaller angle = 100° Answer: 100°

Shortcut: Hour hand position = 30h + m/2 degrees; minute hand = 6m degrees; take the difference.

Q5.

What is the smaller angle between the hour hand and the minute hand of a clock at 1:15?

  1. A52.5°
  2. B60°
  3. C307.5°
  4. D67.5°
Show answer & solution

Answer: A. 52.5°

Solution

Angle = |30h - 5.5m| = |30 x 1 - 5.5 x 15| = |30 - 165/2| = 52.5° Smaller angle = 52.5° Answer: 52.5°

Shortcut: Hour hand position = 30h + m/2 degrees; minute hand = 6m degrees; take the difference.

Q6.

How many degrees does the minute hand move in 25 minutes?

  1. A120°
  2. B150°
  3. C125°
  4. D75°
Show answer & solution

Answer: B. 150°

Solution

Minute hand moves 6 degrees per minute. 25 x 6 = 150 degrees Answer: 150°

Q7.

How many degrees does the hour hand move in 3 hours 20 minutes?

  1. A110°
  2. B90°
  3. C100°
  4. D120°
Show answer & solution

Answer: C. 100°

Solution

Hour hand moves 30 degrees per hour = 0.5 degree per minute. Time = 200 minutes 200 x 0.5 = 100 degrees Answer: 100°

Q8.

How many times do the hour hand and the minute hand coincide in 12 hours?

  1. A12
  2. B10
  3. C13
  4. D11
Show answer & solution

Answer: D. 11

Solution

In 12 hours the minute hand gains 11 laps on the hour hand, so the hands coincide 11 times (12:00 to 12:00 counted once). Answer: 11

Q9.

A clock shows 4:25. What time will its mirror image show?

  1. A7:35
  2. B8:35
  3. C7:25
  4. D8:25
Show answer & solution

Answer: A. 7:35

Solution

Mirror image time = 11:60 - 4:25 = 7:35 Answer: 7:35

Q10.

Which of the following is a leap year?

  1. A1900
  2. B2000
  3. C2100
  4. D2022
Show answer & solution

Answer: B. 2000

Solution

A century year is a leap year only if divisible by 400. 1900, 1800 and 2100 are not divisible by 400; 2022 is not divisible by 4. 2000 is divisible by 400, so it is a leap year. Answer: 2000

Q11.

If the 3rd of a month is a Friday, what day is the 25th of that month?

  1. AFriday
  2. BSunday
  3. CSaturday
  4. DMonday
Show answer & solution

Answer: C. Saturday

Solution

25 - 3 = 22 days later; 22 mod 7 = 1 Friday + 1 day = Saturday Answer: Saturday

Q12.

How many odd days are there in an ordinary (non-leap) year?

  1. A0
  2. B2
  3. C3
  4. D1
Show answer & solution

Answer: D. 1

Solution

365 = 52 weeks + 1 day, so there is 1 odd day. Answer: 1

Q13.

26 January 2025 was a Sunday. What day of the week is 26 January 2026?

  1. AMonday
  2. BSunday
  3. CTuesday
  4. DWednesday
Show answer & solution

Answer: A. Monday

Solution

Between 26 Jan 2025 and 26 Jan 2026 there are 365 days (no 29 February). 365 mod 7 = 1 odd day Sunday + 1 = Monday Answer: Monday

Q14.

In a non-leap year, which month always starts on the same day of the week as February?

  1. AJanuary
  2. BMarch
  3. CApril
  4. DMay
Show answer & solution

Answer: B. March

Solution

February has 28 days = exactly 4 weeks (0 odd days), so March 1 falls on the same weekday as February 1. Answer: March

Q15.

At 8:00 exactly, what is the smaller angle between the hour hand and the minute hand?

  1. A240°
  2. B150°
  3. C120°
  4. D100°
Show answer & solution

Answer: C. 120°

Solution

At 8:00 the hands are 8 hours apart one way and 4 hours apart the other way. 4 x 30 = 120 degrees (smaller angle) Answer: 120°

Q16.

15 August 2023 was a Tuesday. What day of the week was 15 August 2024?

  1. AWednesday
  2. BFriday
  3. CTuesday
  4. DThursday
Show answer & solution

Answer: D. Thursday

Solution

From 15 Aug 2023 to 15 Aug 2024 includes 29 February 2024, so 366 days. 366 mod 7 = 2 odd days Tuesday + 2 = Thursday Answer: Thursday

Medium questions

Q17.

Find the smaller angle between the hands of a clock at 8:42.

  1. A9°
  2. B351°
  3. C12°
  4. D15°
Show answer & solution

Answer: A. 9°

Solution

Angle = |30h - 5.5m| = |240 - 231| = 9° Answer: 9°

Q18.

Find the smaller angle between the hands of a clock at 11:24.

  1. A198°
  2. B162°
  3. C174°
  4. D168°
Show answer & solution

Answer: B. 162°

Solution

Angle = |30h - 5.5m| = |330 - 132| = 198° This is more than 180, so smaller angle = 360 - 198° = 162° Answer: 162°

Q19.

Find the smaller angle between the hands of a clock at 5:48.

  1. A246°
  2. B138°
  3. C114°
  4. D120°
Show answer & solution

Answer: C. 114°

Solution

Angle = |30h - 5.5m| = |150 - 264| = 114° Answer: 114°

Q20.

At what time between 2 and 3 o'clock will the hands of a clock coincide?

  1. A10 minutes past 2
  2. B11 5/11 minutes past 2
  3. C11 minutes past 2
  4. D10 10/11 minutes past 2
Show answer & solution

Answer: D. 10 10/11 minutes past 2

Solution

At 2:00 the hour hand is 5 x 2 = 10 minute-spaces ahead of the minute hand. Minute hand gains 55/60 = 11/12 minute-space per minute. Time = 10 / (11/12) = 120/11 = 10 10/11 minutes Answer: 10 10/11 minutes past 2

Shortcut: Coincide at 60h/11 minutes past h.

Q21.

At what time between 4 and 5 o'clock will the hands of a clock coincide?

  1. A21 9/11 minutes past 4
  2. B20 minutes past 4
  3. C22 4/11 minutes past 4
  4. D21 minutes past 4
Show answer & solution

Answer: A. 21 9/11 minutes past 4

Solution

At 4:00 the hour hand is 5 x 4 = 20 minute-spaces ahead of the minute hand. Minute hand gains 55/60 = 11/12 minute-space per minute. Time = 20 / (11/12) = 240/11 = 21 9/11 minutes Answer: 21 9/11 minutes past 4

Shortcut: Coincide at 60h/11 minutes past h.

Q22.

At what time between 7 and 8 o'clock will the hands of a clock coincide?

  1. A35 minutes past 7
  2. B38 2/11 minutes past 7
  3. C38 8/11 minutes past 7
  4. D36 minutes past 7
Show answer & solution

Answer: B. 38 2/11 minutes past 7

Solution

At 7:00 the hour hand is 5 x 7 = 35 minute-spaces ahead of the minute hand. Minute hand gains 55/60 = 11/12 minute-space per minute. Time = 35 / (11/12) = 420/11 = 38 2/11 minutes Answer: 38 2/11 minutes past 7

Shortcut: Coincide at 60h/11 minutes past h.

Q23.

At what time between 2 and 3 o'clock are the hands of a clock exactly opposite each other (180 degrees apart)?

  1. A48 7/11 minutes past 2
  2. B10 10/11 minutes past 2
  3. C43 7/11 minutes past 2
  4. D43 10/11 minutes past 2
Show answer & solution

Answer: C. 43 7/11 minutes past 2

Solution

Hour hand is 60 degrees from 12; the gap must become 180 degrees. |60 - 5.5m| = 180, so 5.5m = 60 + 180 = 240 (the other option is negative) m = 240/5.5 = 43 7/11 Answer: 43 7/11 minutes past 2

Q24.

At what time between 3 and 4 o'clock are the hands of a clock exactly opposite each other (180 degrees apart)?

  1. A54 1/11 minutes past 3
  2. B16 4/11 minutes past 3
  3. C49 4/11 minutes past 3
  4. D49 1/11 minutes past 3
Show answer & solution

Answer: D. 49 1/11 minutes past 3

Solution

Hour hand is 90 degrees from 12; the gap must become 180 degrees. |90 - 5.5m| = 180, so 5.5m = 90 + 180 = 270 (the other option is negative) m = 270/5.5 = 49 1/11 Answer: 49 1/11 minutes past 3

Q25.

At what time after 4 o'clock will the hands of a clock first be at right angles?

  1. A5 5/11 minutes past 4
  2. B10 5/11 minutes past 4
  3. C5 minutes past 4
  4. D21 9/11 minutes past 4
Show answer & solution

Answer: A. 5 5/11 minutes past 4

Solution

At 4:00 the hour hand is 120 degrees ahead. The gap must reduce to 90 degrees. 120 - 5.5m = 90, so m = 30/5.5 = 60/11 = 5 5/11 Answer: 5 5/11 minutes past 4

Q26.

After how many minutes do the hands of a correct clock coincide again?

  1. A60 minutes
  2. B65 5/11 minutes
  3. C66 minutes
  4. D64 6/11 minutes
Show answer & solution

Answer: B. 65 5/11 minutes

Solution

Minute hand gains 5.5 degrees per minute on the hour hand. For a coincidence again it must gain 360 degrees. Time = 360 / 5.5 = 720/11 = 65 5/11 minutes Answer: 65 5/11 minutes

Q27.

A clock loses 3 minutes every hour. It is set right at 8:00 AM. What time will it show when the correct time is 2:00 PM the same day?

  1. A2:18 PM
  2. B1:57 PM
  3. C1:42 PM
  4. D1:54 PM
Show answer & solution

Answer: C. 1:42 PM

Solution

Time elapsed = 6 hours Loss = 3 x 6 = 18 minutes Clock shows 2:00 PM - 18 min = 1:42 PM Answer: 1:42 PM

Q28.

A clock gains 15 minutes per day. It is set right at 12:00 noon on Monday. What time will it show at 12:00 noon (correct time) on Thursday of the same week?

  1. A12:15 PM
  2. B1:00 PM
  3. C12:30 PM
  4. D12:45 PM
Show answer & solution

Answer: D. 12:45 PM

Solution

Noon Monday to noon Thursday = 3 days Gain = 3 x 15 = 45 minutes Clock shows 12:45 PM Answer: 12:45 PM

Q29.

A clock shows 7:45. What time will its mirror image show?

  1. A4:15
  2. B5:15
  3. C4:45
  4. D3:15
Show answer & solution

Answer: A. 4:15

Solution

Mirror image = 11:60 - 7:45 = 4:15 Answer: 4:15

Q30.

A clock takes 6 seconds to strike 4 o'clock (4 strokes). How long will it take to strike 8 o'clock, assuming equal gaps between strokes?

  1. A12 seconds
  2. B14 seconds
  3. C16 seconds
  4. D13 seconds
Show answer & solution

Answer: B. 14 seconds

Solution

4 strokes have 3 gaps, so 1 gap = 6 / 3 = 2 seconds. 8 strokes have 7 gaps. Time = 7 x 2 = 14 seconds Answer: 14 seconds

Q31.

A wall clock strikes the number of hours at each hour (once at 1 o'clock, twice at 2 o'clock, and so on). How many times does it strike from 1 o'clock to 12 o'clock, both included?

  1. A72
  2. B66
  3. C78
  4. D84
Show answer & solution

Answer: C. 78

Solution

Total strokes = 1 + 2 + 3 + ... + 12 = 12 x 13 / 2 = 78 Answer: 78

Q32.

India became independent on 15 August 1947. On which day of the week was that?

  1. AThursday
  2. BSaturday
  3. CSunday
  4. DFriday
Show answer & solution

Answer: D. Friday

Solution

Odd days: 1600 years = 0, 300 years = 1. The 46 years 1901-1946 have 11 leap years: odd days = 11 x 2 + 35 x 1 = 57, and 57 mod 7 = 1. Days in 1947 up to 15 Aug = 31+28+31+30+31+30+31+15 = 227; 227 mod 7 = 3. Total odd days = 0 + 1 + 1 + 3 = 5, and 5 means Friday (0 = Sunday). Answer: Friday

Q33.

Today is Wednesday. What day of the week will it be after 100 days?

  1. AFriday
  2. BThursday
  3. CSaturday
  4. DMonday
Show answer & solution

Answer: A. Friday

Solution

100 = 14 weeks + 2 days, so 100 mod 7 = 2. Wednesday + 2 days = Friday Answer: Friday

Q34.

The calendar of the year 2018 will be exactly the same as the calendar of which later year?

  1. A2024
  2. B2029
  3. C2027
  4. D2028
Show answer & solution

Answer: B. 2029

Solution

2018 is an ordinary year. We need the odd days from 2018 up to the year before the target to total a multiple of 7. 2018:1, 2019:1, 2020:2, 2021:1, 2022:1, 2023:1, 2024:2, 2025:1, 2026:1, 2027:1, 2028:2; total = 14, a multiple of 7. So 2029 starts on the same day as 2018, and both are ordinary years. Answer: 2029

Q35.

A month has 31 days and its 1st day is a Friday. How many Sundays does it have?

  1. A4
  2. B3
  3. C5
  4. D6
Show answer & solution

Answer: C. 5

Solution

31 days = 4 weeks + 3 extra days. The extra days are the first three days of the month: Friday, Saturday, Sunday. So Friday, Saturday and Sunday occur 5 times. Sundays = 5 Answer: 5

Q36.

In a month whose 1st day is a Sunday, on which date does the third Tuesday fall?

  1. A15th
  2. B16th
  3. C18th
  4. D17th
Show answer & solution

Answer: D. 17th

Solution

1st is Sunday, so the first Tuesday is the 3rd. Tuesdays: 3rd, 10th, 17th Third Tuesday = 17th Answer: 17th

Q37.

How many days are there between 12 March 2023 and 5 July 2023, excluding both dates?

  1. A114
  2. B115
  3. C116
  4. D113
Show answer & solution

Answer: A. 114

Solution

Days left in March after the 12th = 19 April = 30, May = 31, June = 30 Days of July before the 5th = 4 Total = 19 + 30 + 31 + 30 + 4 = 114 Answer: 114

Hard questions

Q38.

The hands of a clock coincide every 66 minutes of the time shown by this faulty clock. How much time does the clock gain or lose in 24 hours?

  1. ALoses 12 minutes
  2. BGains 12 minutes
  3. CGains 6 minutes
  4. DLoses 10 5/11 minutes
Show answer & solution

Answer: B. Gains 12 minutes

Solution

In a correct clock the hands coincide every 720/11 minutes of true time. The faulty clock shows 66 minutes in that true time, so it runs fast. Gain per coincidence = 66 - 720/11 = 6/11 minute. Number of coincidence intervals in 24 h = 1440 / (720/11) = 22 Gain in 24 hours = 22 x 6/11 = 12 minutes Answer: Gains 12 minutes

Q39.

A clock loses 1 minute every hour. It is set right at 6:00 AM. At some time later the same day it shows 2:51 PM. What is the correct time then?

  1. A2:59 PM
  2. B2:51 PM
  3. C3:00 PM
  4. D3:09 PM
Show answer & solution

Answer: C. 3:00 PM

Solution

Clock shows 59 minutes for every 60 true minutes. Time elapsed on the clock = 6:00 AM to 2:51 PM = 8 h 51 min = 531 minutes True time = 531 x 60/59 = 540 minutes = 9 hours Correct time = 6:00 AM + 9 h = 3:00 PM Answer: 3:00 PM

Q40.

How many times in a day (24 hours) are the hands of a clock at right angles to each other?

  1. A22
  2. B48
  3. C24
  4. D44
Show answer & solution

Answer: D. 44

Solution

In every 12 hours the hands are at right angles 22 times. So in 24 hours: 22 x 2 = 44 Answer: 44

Q41.

1 January 2026 is a Thursday. In how many months of the year 2026 does the 13th fall on a Friday?

  1. A3
  2. B1
  3. C2
  4. D4
Show answer & solution

Answer: A. 3

Solution

13 Jan = 12 days after Thursday; 12 mod 7 = 5, so 13 Jan is Tuesday. Each next 13th moves by the odd days of the previous month (31 days: +3, 30: +2, 28: +0): Jan Tue, Feb Fri (+3), Mar Fri (+0), Apr Mon (+3), May Wed (+2), Jun Sat (+3), Jul Mon (+2), Aug Thu (+3), Sep Sun (+3), Oct Tue (+2), Nov Fri (+3), Dec Sun (+2). Fridays on the 13th: Feb, Mar, Nov = 3 Answer: 3

Q42.

1 January 2000 was a Saturday. What day of the week will 1 January 2100 be?

  1. AThursday
  2. BFriday
  3. CSaturday
  4. DSunday
Show answer & solution

Answer: B. Friday

Solution

From 1 Jan 2000 to 1 Jan 2100 = 100 years. Leap years in between: 2000, 2004, ..., 2096 = 25 Odd days = 100 + 25 = 125; 125 mod 7 = 6 Saturday + 6 days = Friday Answer: Friday

Q43.

At what time after 7 o'clock will the angle between the hands of a clock first be 24 degrees?

  1. A42 6/11 minutes past 7
  2. B38 2/11 minutes past 7
  3. C33 9/11 minutes past 7
  4. D32 9/11 minutes past 7
Show answer & solution

Answer: C. 33 9/11 minutes past 7

Solution

At 7:00 the hour hand is 210 degrees ahead of the minute hand. Minute hand gains 5.5 degrees per minute, so 210 - 5.5m = 24 5.5m = 186, m = 372/11 = 33 9/11 minutes (The other case, 5.5m - 210 = 24, gives 42 6/11, which is later.) Answer: 33 9/11 minutes past 7

Q44.

A watch gains uniformly. It is 3 minutes slow at noon on Monday and 5 minutes fast at noon on Wednesday. When was it correct?

  1. ANoon on Tuesday
  2. B6:00 PM on Monday
  3. C6:00 PM on Tuesday
  4. D6:00 AM on Tuesday
Show answer & solution

Answer: D. 6:00 AM on Tuesday

Solution

From noon Monday to noon Wednesday = 48 hours. Total swing in 48 hours = 3 + 5 = 8 minutes. To make up the 3 minutes it was behind: time = 3/8 x 48 = 18 hours after noon Monday. 12:00 noon + 18 h = 6:00 AM Tuesday Answer: 6:00 AM on Tuesday

Q45.

A clock gains 5 minutes every hour. It is set right at 6:00 AM. At some time later it shows 7:00 PM. What is the correct time?

  1. A6:00 PM
  2. B6:05 PM
  3. C5:55 PM
  4. D6:30 PM
Show answer & solution

Answer: A. 6:00 PM

Solution

Clock shows 65 minutes for every 60 true minutes (ratio 13 : 12). Time shown elapsed = 6:00 AM to 7:00 PM = 13 hours True time elapsed = 13 x 12/13 = 12 hours Correct time = 6:00 AM + 12 h = 6:00 PM Answer: 6:00 PM

Q46.

15 August 2024 was a Thursday. What day of the week was 26 January 2025?

  1. ASaturday
  2. BSunday
  3. CMonday
  4. DFriday
Show answer & solution

Answer: B. Sunday

Solution

Days from 15 Aug 2024 to 26 Jan 2025 = 16 (rest of Aug) + 30 (Sep) + 31 (Oct) + 30 (Nov) + 31 (Dec) + 26 (Jan) = 164 164 = 23 x 7 + 3, so 3 odd days Thursday + 3 = Sunday Answer: Sunday

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