🕐 Clocks & Calendars

Day 3 | ⏱️ Part of 3-Hour Advanced Quant Block | TCS Interview Prep

📌 Subtopics

Clocks

• Angle between hour and minute hands
• Speed of hands (minute hand: 6°/min, hour hand: 0.5°/min)
• Overlapping/right angle/opposite direction of hands
• Faulty/gaining/losing clock problems

Calendars

• Odd days concept
• Day of the week for a given date
• Leap year rules
• Number of odd days in a century

📖 Comprehensive Notes

1. Clocks

A clock is a circle of $360^\circ$ divided into $12$ major divisions (hours) and $60$ minor divisions (minutes).

• Speed of Hands:
  • Minute Hand: covers $360^\circ$ in $60\text{ minutes} \implies 6^\circ/\text{minute}$.
  • Hour Hand: covers $360^\circ$ in $12\text{ hours} \implies 30^\circ/\text{hour} \implies 0.5^\circ/\text{minute}$.
  • Relative Speed of Minute Hand with respect to Hour Hand: $6^\circ - 0.5^\circ = 5.5^\circ/\text{minute}$ (or $\frac{11}{2}^\circ/\text{minute}$).
• Angle between Hands: $\text{Angle } (\theta) = \left|30H - \frac{11}{2}M\right|$ where $H$ is the hour value and $M$ is the minutes value.
• Special Positions in 12 Hours:
  • Coincide / Overlap ($0^\circ$ angle): Occurs $11$ times.
  • Opposite Directions ($180^\circ$ angle): Occurs $11$ times.
  • Right Angle ($90^\circ$ angle): Occurs $22$ times.
  • Note: To find the count in $24$ hours, double these values ($22$, $22$, $44$ respectively).

Example: Find the angle between the hands of a clock at $8:20$. $H = 8, M = 20$. $\text{Angle} = \left|30(8) - 5.5(20)\right| = \left|240 - 110\right| = 130^\circ$.

2. Calendars

The calendar system relies on counting cycles of weeks, where any remainder is called an odd day.

• Odd Days: The number of days remaining after dividing the total number of days by $7$. $\text{Odd Days} = \text{Total Days} \pmod 7$
• Year Types:
  • Ordinary Year (365 days): $52 \text{ weeks} + 1 \text{ day} \implies 1\text{ odd day}$.
  • Leap Year (366 days): $52 \text{ weeks} + 2 \text{ days} \implies 2\text{ odd days}$.
• Leap Year Rule:
  • A non-century year is a leap year if it is divisible by $4$.
  • A century year (ending in $00$) is a leap year only if it is divisible by $400$.
  • Example: $2000$ and $2400$ are leap years, but $1900$ and $2100$ are not.

3. Odd Days in Centuries & Days Reference

• Odd Days in Centuries:
  • $100$ years $= 5$ odd days
  • $200$ years $= 3$ odd days
  • $300$ years $= 1$ odd day
  • $400$ years $= 0$ odd days (and multiples: $800, 1200, 1600, 2000$ have $0$ odd days).
• Day Codes (Standard Reference):
  • Sunday $= 0$, Monday $= 1$, Tuesday $= 2$, Wednesday $= 3$, Thursday $= 4$, Friday $= 5$, Saturday $= 6$.

Example: What day of the week was June 15, 2026? Let’s split $2025$ completed years $+$ days of $2026$ up to June 15:

• $2000$ completed years $= 0$ odd days.
• Remaining $25$ completed years: contains $25 / 4 = 6$ leap years and $19$ ordinary years.
  • Odd days $= 6 \times 2 + 19 \times 1 = 12 + 19 = 31$ days $\equiv 31 \pmod 7 = 3$ odd days.
• Days in 2026 up to June 15:
  • Jan ($31$) + Feb ($28$, non-leap) + Mar ($31$) + Apr ($30$) + May ($31$) + June ($15$).
  • Jan ($3$) + Feb ($0$) + Mar ($3$) + Apr ($2$) + May ($3$) + June ($15 \equiv 1$) $= 12$ days $\equiv 5$ odd days.
• Total odd days $= 0 + 3 + 5 = 8 \text{ days} \equiv 1$ odd day.
• Day Code $1$ corresponds to Monday.

🌐 Learning Resources

✅ Practice Links

📝 Key Formulas

CLOCKS:
  Angle = |30H - 5.5M| (where H=hour, M=minutes)
  Hands overlap 22 times in 24 hours (11 times in 12 hours)
  Hands are at right angle 44 times in 24 hours
  Relative speed = 5.5° per minute

CALENDARS:
  Odd Days: Number of days beyond complete weeks
  Leap Year: Divisible by 4, but NOT by 100, unless also by 400
  Ordinary year = 1 odd day
  Leap year = 2 odd days
  100 years = 5 odd days
  200 years = 3 odd days
  300 years = 1 odd day
  400 years = 0 odd days

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

🔗 Related Topics

• Time Speed and Distance — Relative speed concept is identical to relative speed of clock hands
• Directions and Distance — Angle deviations (30 degrees, 90 degrees) on a clock face are direction sweeps
• Equations — Forming equations to solve clock gain/loss or leap year distributions
• Permutations Combinations and Probability — Probability questions involving selection of dates/days
• Number System HCF LCM — Modular arithmetic used in calendar cycle remainder calculations