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Clock & Calendar Basics

Mathematics · Algebra, Geometry & Data

Useful for: Railway RRB, OSSC RHT — Regular High School Teacher, OSEPA Primary Teacher / JT Category-I, OSEPA Upper Primary — Arts, OSEPA Upper Primary — Science, OSSSC Govt Physical Education Teacher (PET), OSSSC Govt TGT Arts, OSSSC Govt TGT Science — CBZ

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OSSSC Govt TGT Science — PCM

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Overview

Clock questions use the angle formula and hand speeds; calendar questions count odd days.

Key points

Angle between the hands at H:M = |30H − 5.5M|; if over 180°, subtract from 360°.
The minute hand gains 5.5° per minute over the hour hand.
Hands coincide 11 times in 12 hours (22 in a day) and are at right angles 22 times in 12 hours (44 in a day).
Odd days: an ordinary year has 1 odd day, a leap year 2.
Century odd days: 100 years = 5, 200 = 3, 300 = 1, 400 = 0.

Explanation

A leap year is divisible by 4; century years must be divisible by 400 (2000 is leap, 1900 is not).
Days after today: take the number of days mod 7.
The calendar of a year repeats after 6, 11 or 28 years depending on leap years.
Mirror image of a clock: subtract the time from 11:60.

Step by step

Choose clock or calendar method.
Apply the formula or odd-day count.
Check the reasonable range.

Examples

Angle at 3:40 = |90 − 220| = 130°.
If today is Monday, the day after 61 days is Saturday (61 mod 7 = 5).
If 1 January 2023 was a Sunday, 1 January 2024 was a Monday (2023 has 1 odd day).

Traps and advanced points

Trap: Forgetting to take 360° − θ when θ exceeds 180°.
Trap: Treating 1900 as a leap year.
Trap: Counting the starting day itself.

Quick revision

Θ = |30H − 5.5M|.
Ordinary year 1 odd day, leap 2.
400 years = 0 odd days.

Practice MCQs

If today is Thursday, what day will it be after 87 days?
1. Sunday
2. Wednesday
3. Monday
4. Tuesday
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Answer: (a) Sunday. Only remainder on division by 7 matters: 87%7=3.
If today is Tuesday, what day will it be after 70 days?
1. Tuesday
2. Thursday
3. Monday
4. Wednesday
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Answer: (a) Tuesday. Only remainder on division by 7 matters: 70%7=0.
If today is Wednesday, what day will it be after 47 days?
1. Wednesday
2. Monday
3. Tuesday
4. Thursday
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Answer: (b) Monday. Only remainder on division by 7 matters: 47%7=5.
If today is Friday, what day will it be after 127 days?
1. Tuesday
2. Monday
3. Wednesday
4. Saturday
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Answer: (d) Saturday. Only remainder on division by 7 matters: 127%7=1.
If today is Monday, what day will it be after 100 days?
1. Thursday
2. Monday
3. Tuesday
4. Wednesday
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Answer: (d) Wednesday. Only remainder on division by 7 matters: 100%7=2.
If today is Tuesday, what day will it be after 119 days?
1. Wednesday
2. Thursday
3. Tuesday
4. Monday
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Answer: (c) Tuesday. Only remainder on division by 7 matters: 119%7=0.
If today is Tuesday, what day will it be after 98 days?
1. Monday
2. Tuesday
3. Wednesday
4. Thursday
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Answer: (b) Tuesday. Only remainder on division by 7 matters: 98%7=0.
If today is Wednesday, what day will it be after 75 days?
1. Tuesday
2. Wednesday
3. Thursday
4. Monday
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Answer: (d) Monday. Only remainder on division by 7 matters: 75%7=5.
If today is Friday, what day will it be after 57 days?
1. Tuesday
2. Wednesday
3. Monday
4. Saturday
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Answer: (d) Saturday. Only remainder on division by 7 matters: 57%7=1.
If today is Monday, what day will it be after 128 days?
1. Thursday
2. Monday
3. Wednesday
4. Tuesday
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Answer: (c) Wednesday. Only remainder on division by 7 matters: 128%7=2.
What is the angle between the hands of a clock at 3:00?
1. 60°
2. 90°
3. 120°
4. 180°
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Answer: (b) 90°. Each hour mark is 30°; 3 × 30° = 90°.
How many times do the hour and minute hands overlap in 12 hours?
1. 10
2. 11
3. 12
4. 24
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Answer: (b) 11. They overlap 11 times (not at about 12–1 separately).
What is the angle between the hands of a clock at 3:30?
1. 60°
2. 75°
3. 90°
4. 105°
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Answer: (b) 75°. |30H − 5.5M| = |90 − 165| = 75°.
What is the smaller angle between the clock hands at 1:14?
1. 77°
2. 62°
3. 47°
4. 32°
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Answer: (c) 47°. Angle = |30H − 5.5M|, then take the smaller angle.
What is the smaller angle between the clock hands at 1:28?
1. 109°
2. 124°
3. 154°
4. 139°
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Answer: (b) 124°. Angle = |30H − 5.5M|, then take the smaller angle.