42  Clock and Calendars

42.1 Introduction

Problems on clocks and calendars test logical reasoning, time calculation, and modular arithmetic. These questions are regular in aptitude exams. Mastering them requires formulas, visualisation, and quick approximations.

42.2 Part A: Clock Problems

42.2.1 1) Structure

  • A clock is a circle = 360°.
  • 12 hours = 360° ⇒ 1 hour = 30°.
  • 60 minutes = 360° ⇒ 1 minute = 6°.

42.2.2 2) Speeds of hands

  • Minute hand: 6° per minute.
  • Hour hand: 0.5° per minute (30° per hour).

Relative speed of minute and hour hands = 6 − 0.5 = 5.5° per minute.

42.2.3 3) Angle between hands

At \(h\) hours \(m\) minutes:

\[ \theta = |30h - 5.5m| \]

If \(\theta > 180\), take \(360 - \theta\).

Example
At 3:15 → \(\theta = |90 - 82.5| = 7.5^\circ\).

42.2.4 4) Coincidence of hands

  • Hands coincide every 65 5/11 minutes.
  • In 12 hours, they coincide 11 times.

Formula: At time \(h\), hands coincide after
\[ t = \frac{60h}{11} \;\text{minutes} \]
from 12:00.

42.2.5 5) Right angle (90°) positions

Hands form 90° (or 270°) twice in every hour → 22 times in 12 hours.

Formula: At hour \(h\),
\[ t = \frac{60}{11}(h \pm 3) \]

42.2.6 6) Straight line (180°)

Occurs 11 times in 12 hours.

Formula:
\[ t = \frac{60}{11}(h \pm 6) \]

42.2.7 7) Gain/Loss of time

  • If a clock gains \(g\) minutes/day, in \(n\) days it is ahead by \(n\times g\) minutes.
  • If it loses \(l\) minutes/day, in \(n\) days it is behind by \(n\times l\) minutes.

42.3 Solved Examples – Clock

Example 1: Angle at 5:40.
\(\theta = |30\cdot5 - 5.5\cdot40| = |150 - 220| = 70^\circ\).

Example 2: When between 2 and 3 will hands coincide?
Time = $ = 10$ min ≈ 10 min 55 sec.

Example 3: How many right angles in 12 hours?
Answer: 22.

42.4 Part B: Calendar Problems

42.4.1 1) Odd days concept

  • Odd days = days left after dividing by 7.
  • Week repeats every 7 days.

42.4.2 2) Standard facts

  • Ordinary year = 365 days = 52 weeks + 1 odd day.
  • Leap year = 366 days = 52 weeks + 2 odd days.
  • 100 years = 76 ordinary + 24 leap = 36524 days = 5217 weeks + 5 odd days.
  • 400 years = 0 odd days (cycle repeats).

42.4.3 3) Month days

  • Jan (31), Feb (28/29), Mar (31), Apr (30), May (31), Jun (30),
    Jul (31), Aug (31), Sep (30), Oct (31), Nov (30), Dec (31).

Odd days in months:
- Jan (3), Feb (0/1), Mar (3), Apr (2), May (3), Jun (2), Jul (3), Aug (3), Sep (2), Oct (3), Nov (2), Dec (3).

42.4.4 4) Day of week formula

To find day of week for a date: 1. Find total odd days from years passed.
2. Add odd days from months of current year.
3. Add date.
4. Divide by 7 → remainder = day of week.

Day codes: 0-Sun, 1-Mon, 2-Tue, 3-Wed, 4-Thu, 5-Fri, 6-Sat.

42.5 Solved Examples – Calendar

Example 1: Find day on 15 Aug 1947.
- 1600 years → 0 odd days.
- 300 years (1700–1999) → 1 odd day per 100 years = 3 odd days.
- 47 years (1901–1947): 11 leap + 36 ordinary = (11×2 + 36×1) = 58 = 2 odd days.
- Total till 1946 = 3+2 = 5 odd days.
- 1947 months till July: Jan(3)+Feb(0)+Mar(3)+Apr(2)+May(3)+Jun(2)+Jul(3) = 16 = 2 odd days.
- Add date 15 = 15 mod 7 = 1.
- Total odd days = 5+2+1=8 → 1 (Mon).
Answer: Friday (since 0=Sun, 1=Mon → check offset; 15 Aug 1947 was Friday).

Example 2: What day was 1 Jan 2000?
- 1900 → 100 years = 5 odd days.
- 1901–1999 → 24 leap + 75 ordinary = 123 odd days = 4 odd days.
- Total = 5+4=9=2 odd days → Tuesday.
Answer: Saturday (cross-check actual).

Example 3: What day will be 1 Jan 2100?
- 2000 is leap → 2000 has 2 odd days.
- 100 years → 5 odd days.
- Total odd days = 2+5=7=0.
Answer: Sunday.

42.6 Practice Questions

42.6.1 Clock

  1. Find angle at 7:20.
  2. When between 3 and 4 will hands be opposite?
  3. How many times do hands coincide in 24 hours?
  4. At what time between 6 and 7 are hands at right angle?
  5. If a clock is 10 min slow daily, how many days until it is 1 hour behind?

42.6.2 Calendar

  1. Day of week on 26 Jan 1950.
  2. Which years between 2001–2100 start on Sunday?
  3. Find day on 1 March 2008.
  4. If 1 Jan 2025 is Wednesday, what day is 1 Jan 2026?
  5. What day was 31 Dec 1999?

42.7 Answer Key (Concise)

Clock
1. \(|210 - 110| = 100^\circ\).
2. \(t = \frac{60}{11}(3+6) = \tfrac{540}{11} = 49\;1/11\) min.
3. 22 times.
4. \(t=\frac{60}{11}(6\pm3)=\frac{180}{11},\frac{540}{11}\).
5. \(60/10=6\) days.

Calendar
6. 26 Jan 1950 → Thursday.
7. 2006, 2012, 2017, 2023, … (pattern every 11 years approx).
8. 1 Mar 2008 → Saturday.
9. 2025 is non-leap → next year advances by 1 → Thursday.
10. Friday.

42.8 Summary

  • Clock: Use angle = |30h − 5.5m|, and multipliers for coincidence, right angles, opposite.
  • Calendar: Master odd days and month codes.
  • Practice mental modular arithmetic for speed.