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Clock & Calendar

Learning Goals

By the end of this chapter, you will be able to:

1. Find the angle between the hour hand and the minute hand at any given time.
2. Find the time when the two hands are together, opposite, or at a right angle.
3. Solve mirror-image and water-image clock problems.
4. Find the day of the week for any date using odd days.
5. Handle leap-year and century-year traps confidently.

The Hook: Two Questions, Zero Formulas Needed (Almost)

"At what time between 4 and 5 o'clock will the hands of a clock be together?"
"What day of the week was 15 August 1947?"

These two questions appear in some form in almost every SSC, Banking, and Railway exam. And here is the good news: both are solved with one formula each. This entire chapter rests on just a handful of ideas. Master them once, and these become your fastest marks in the reasoning section.

Part 1: The Clock — Understand the Machine First

Many students jump straight to formulas and then freeze when the question is slightly twisted. So first, let's understand what the clock is actually doing.

A clock face is a circle = 360°, divided into 12 hours and 60 minutes.

• Minute hand: covers 360° in 60 minutes → 6° per minute.
• Hour hand: covers 360° in 12 hours (720 minutes) → 0.5° per minute.
• So the minute hand gains on the hour hand at 6 − 0.5 = 5.5° per minute.

That single number — 5.5° per minute — is the heart of every clock problem.

🧠 Active Recall: Without looking up — how many degrees does the hour hand move in one minute? And what is the relative speed of the two hands?

Part 2: Angle Between the Hands

Formula:

> Angle = |30H − (11/2)M|

where H = hours, M = minutes. If the answer is more than 180°, subtract it from 360° (the question usually wants the smaller angle).

Example: Angle at 4:20?
= |30×4 − (11/2)×20| = |120 − 110| = 10°

Example: Angle at 2:30?
= |60 − 165| = 105°. (Smaller angle = 105°, since it is already below 180°.)

🧠 Active Recall: Try it yourself — what is the angle at 6:15? (Answer: |180 − 82.5| = 97.5°)

Part 3: Hands Together, Opposite, and at Right Angle

The hands coincide at 12:00. After that, the minute hand must gain a full 360° to catch up again — that takes 360/5.5 = 65(5/11) minutes. So:

• Hands coincide 11 times in 12 hours (22 times in 24 hours) — not 12!
• Hands are opposite (180°) 11 times in 12 hours.
• Hands are at right angle (90°) 22 times in 12 hours (44 times in 24 hours).

Coincidence formula: Between H and H+1 o'clock, hands meet at:

> Time = (60/11) × H minutes past H o'clock

Example: Between 4 and 5, hands meet at (60/11)×4 = 240/11 = 21(9/11) minutes past 4.

Right angle: hands are 15 minute-spaces apart. Between H and H+1, the two right-angle times are at (5H ± 15) × (12/11) minutes past H.

🧠 Active Recall: How many times do the hands coincide in 24 hours — 22, 23, or 24? Why not 24?

Part 4: Mirror and Water Images of Time

Mirror image: Subtract the given time from 11:60 (this simply means 12:00, written in a form that makes subtraction of minutes easy).

Example: Mirror image of 3:40 → 11:60 − 3:40 = 8:20.

Water image: Subtract the given time from 17:90 (this simply means 18:30, written in a form that makes subtraction easy).

Example: Water image of 3:40 → 17:90 − 3:40 = 14:50, i.e., 2:50.

🧠 Mnemonic: *"Mirror is 11-60, Water is 17-90."* Say it three times — it will never leave you.

Part 5: The Calendar — Odd Days Are Everything

The calendar chapter is really just one question: how many "extra" days pile up?

• Ordinary year = 365 days = 52 weeks + 1 odd day.
• Leap year = 366 days = 52 weeks + 2 odd days.

Leap year rule: Divisible by 4. But a century year (1700, 1800, 1900, 2100) is a leap year only if divisible by 400. So 2000 was a leap year; 1900 was not. This is the examiner's favourite trap.

Odd days in centuries:

Day codes: 0 = Sunday, 1 = Monday, 2 = Tuesday, 3 = Wednesday, 4 = Thursday, 5 = Friday, 6 = Saturday.

Reference point: 1 January 1 AD was a Monday — that is why, in the standard code above, Monday corresponds to 1 odd day. When you finish counting total odd days, the number itself (0 to 6) maps directly to the day using the code table.

Month codes (odd days per month): Jan 3, Feb 0 (ordinary) / 1 (leap), Mar 3, Apr 2, May 3, Jun 2, Jul 3, Aug 3, Sep 2, Oct 3, Nov 2, Dec 3.

🧠 Mnemonic for month odd days: *"3-0-3-2-3-2-3-3-2-3-2-3"* — chant it like a phone number.

Part 6: Finding the Day of Any Date

Method: Count total odd days from the reference point, take mod 7, map to the day using the code table (0 = Sunday … 6 = Saturday).

Example: What day was 15 August 1947?

• 1600 years → 0 odd days
• 300 years → 1 odd day
• 46 years = 11 leap + 35 ordinary → 11×2 + 35×1 = 57 → 57 mod 7 = 1 odd day
• Days in 1947 up to 15 Aug: Jan 31 + Feb 28 + Mar 31 + Apr 30 + May 31 + Jun 30 + Jul 31 + 15 = 227 → 227 mod 7 = 3 odd days
• Total = 0 + 1 + 1 + 3 = 5 → code 5 = Friday ✔

🧠 Active Recall: Close the book and list the odd days for 100, 200, 300, and 400 years. Then check.

Part 7: Quick Calendar Facts Examiners Love

• The calendar repeats every 400 years.
• An ordinary year starts and ends on the same day (1 Jan and 31 Dec are the same weekday).
• A leap year's 31 December is one day ahead of its 1 January.
• The first day of a century can only be Monday, Tuesday, Thursday, or Saturday — never Sunday, Wednesday, or Friday.
• The last day of a century cannot be Tuesday, Thursday, or Saturday.

Summary Table

📌 Review this chapter after 2 days. Redo the 15 August 1947 example from memory — if you can do it without peeking, the calendar is yours forever.

⚠️ Caution: Examiner's Trap

Trap 1 — The "12 times" illusion. Questions ask how many times hands coincide or are opposite in 12/24 hours. The tempting answer is 12 (or 24), but the correct answer is 11 in 12 hours (22 in 24) because between 11 and 1 o'clock the hands coincide only once (at 12:00). Examiners always keep 12 and 24 as options.

Trap 2 — The century leap-year trap. "Was 1900 a leap year?" It is divisible by 4, so students say yes — but century years need divisibility by 400, so 1900 was NOT a leap year. This silently shifts every odd-day calculation by one day.

Trap 3 — The bigger-angle trick. The angle formula sometimes gives a value above 180° (e.g., 262.5°). The examiner puts both 262.5° and 97.5° in the options. Unless the question says otherwise, the angle between the hands means the smaller angle — subtract from 360°.

Trap 4 — Confusing mirror and water subtraction bases. Students mix up 11:60 and 17:90, or subtract from 12:00 directly and get an answer that is one minute off. Fix the mnemonic in memory: *Mirror 11-60, Water 17-90.*