घड़ी और कैलेंडर

हिंदी में पढ़ें

Clock & Calendar

Why This Chapter Can Win You the Exam

Every year, lakhs of aspirants lose 2–3 marks in SSC CGL, IBPS PO, and RRB NTPC not because Clock & Calendar is hard — but because they *think* it is hard and skip it. Here is the truth: this chapter runs on barely five formulas and two ideas. Once those click, every question becomes a 30-second calculation. Toppers don't solve these questions faster because they are smarter — they solve them faster because they stopped memorising and started understanding *why* the formulas work. That is exactly what we will do here.

Learning Goals

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

1. Find the angle between the hour and minute hands at any given time.
2. Find the time when the two hands are together, opposite, or at a given angle.
3. Solve "faulty clock" (gain/loss) problems.
4. Find the day of the week for any date using odd days.
5. Handle leap-year traps and century-year traps confidently.

The Common Struggle — Let's Name It

Most students struggle here for three reasons:

• They memorise the angle formula without knowing where it comes from, so a twisted question breaks them.
• They count odd days wrongly around century years (Is 1900 a leap year? No! Is 2000? Yes!).
• They panic when the question asks "when will the hands be at right angles between 4 and 5?" — because there are *two* answers, not one.

We will fix all three, one chunk at a time.

Part 1: The Clock — Speeds of the Hands

Everything in clock problems comes from one fact: the two hands move at different speeds.

• The minute hand covers 360° in 60 minutes → 6° per minute.
• The hour hand covers 360° in 12 hours (720 minutes) → 0.5° per minute.

So the minute hand gains on the hour hand at:

Relative speed = 6 − 0.5 = 5.5° per minute

This single number — 5.5°/min — is the engine of every clock question.

🧠 Active Recall: Without looking up — how many degrees does the hour hand move in 20 minutes? And what is the relative speed of the two hands?

Part 2: The Angle Formula

At time H : M (H hours, M minutes):

• Minute hand position (from 12) = 6M degrees
• Hour hand position (from 12) = 30H + 0.5M degrees

Angle between hands = |30H − 5.5M|

If this comes out more than 180°, subtract it from 360° (the angle between two hands is always taken as the smaller one).

Example: Angle at 4:20?

Example: Angle at 2:50?

🧠 Active Recall: Find the angle at 7:15 before reading further. (Answer: |210 − 82.5| = 127.5°)

Part 3: When Are the Hands Together or Opposite?

Together (0° apart): The hands coincide 11 times in 12 hours (not 12 — between 11 and 1 they meet only once, at 12:00). Gap between two coincidences = 12/11 hours = 65⁵⁄₁₁ minutes.

To find when hands coincide between H and H+1 o'clock:
Time = H : (60H/11) minutes — e.g., between 4 and 5: 4 : 240/11 = 4 : 21⁹⁄₁₁.

Opposite (180° apart): Also 11 times in 12 hours, gap of 65⁵⁄₁₁ minutes. The logic: the minute hand must be 30 minutes' worth (180°) ahead of the hour hand.

• For H < 6: Time = H : (60H + 360)/11 minutes — e.g., between 4 and 5: 4 : 600/11 = 4 : 54⁶⁄₁₁.
• For H > 6: Time = H : (60H − 360)/11 minutes — e.g., between 8 and 9: 8 : 120/11 = 8 : 10¹⁰⁄₁₁.

At right angles (90°): This happens 22 times in 12 hours — twice every hour (except the 2–3 and 8–9 windows, where it happens only once per gap, which is why 22 and not 24). This is the examiner's favourite trap.

🧠 Active Recall: How many times in 24 hours do the hands (a) coincide, (b) stand opposite, (c) form a right angle? (Answers: 22, 22, 44.)

Part 4: Faulty Clocks (Gain & Loss)

Idea: if a clock gains 5 minutes every hour, then when true time advances 60 minutes, this clock shows 65 minutes. Set up a proportion.

Example: A clock gains 10 minutes every 24 hours. It is set right at 8 AM Monday. What will it show at 8 PM Tuesday (true time)?

• True elapsed = 36 hours.
• Clock shows 24h10m for every true 24h → in 36 true hours it shows 36 × (24⅙/24) = 36h 15m.
• So it shows 8:15 PM.

Mnemonic 🧠: *"Gain goes ahead, Loss lags behind — proportion is the spine."* Whatever the fault, convert everything into "faulty-clock minutes per true minutes" and scale.

Part 5: The Calendar — Odd Days

The whole calendar chapter is one question: how many "extra" days beyond complete weeks?

• 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.

Odd days in centuries (memorise this table):

Why? 100 years = 76 ordinary + 24 leap = 76 + 48 = 124 odd days → 124 mod 7 = 5. The rest follow by addition mod 7.

Day codes: Sunday = 0, Monday = 1, … Saturday = 6.

Month codes (odd days per month, cumulative): Jan 3, Feb 0 (non-leap) / 1 (leap), Mar 3, Apr 2, May 3, Jun 2, Jul 3, Aug 3, Sep 2, Oct 3, Nov 2, Dec 3 — these are just each month's days mod 7, added up.

🧠 Active Recall: How many odd days in 300 years? Is 2100 a leap year? (Answers: 1; No.)

Part 6: Finding the Day for Any Date

Method: Odd days = (odd days in completed centuries) + (odd days in remaining years) + (odd days in months of current year) + (date). Take mod 7, map to day.

Example: 15 August 1947.

• 1600 years → 0 odd days
• 300 years → 1
• 46 years = 35 ordinary + 11 leap → 35 + 22 = 57 → 57 mod 7 = 1
• Months Jan–Jul 1947 (1947 not leap): 3+0+3+2+3+2+3 = 16 → 2
• Date 15 → 15 mod 7 = 1
• Total = 0+1+1+2+1 = 5 → Friday ✓

Handy anchor: 1 January 2000 was a Saturday. You can count forward/backward from it for nearby years.

Part 7: Calendar Short Facts

• A calendar repeats when total odd days over the gap = 0 (mod 7) and leap-year pattern matches. Ordinary-year calendar repeats after 6 or 11 years; leap-year calendar after 28 years (within the same century pattern).
• The first day of a century (1 Jan 2001, etc.) can only be Monday, Tuesday, Thursday, or Saturday — never Sunday, Wednesday, or Friday. Examiners love this.
• The last day of a century cannot be Tuesday, Thursday, or Saturday.

🧠 Active Recall: Can 1 January 2101 be a Sunday? Why not?

⚠️ Caution: Examiner's Trap

Trap 1 — "How many times at right angle?" with a twisted range.
The answer 22 times in 12 hours is famous, so examiners ask: *"How many times between 3 PM and 9 PM?"* Now you must actually count (right angles occur at ~3:00, ~3:33, ~4:05, ~4:38, ~5:11, ~5:43, ~6:16, ~6:49, ~7:22, ~7:54, ~8:25 → 11 times). Blindly halving 22 gives 11 here by luck — but between 2 PM and 4 PM the answer is 3, not 4. Never apply the "22/44" shortcut to partial ranges.

Trap 2 — The century-year leap trap dressed in difficult language.
Questions phrase it as: *"If 1 January 1900 was a Monday, what day was 1 January 2000?"* Students who treat 1900 as a leap year (it is NOT — not divisible by 400) get every odd-day count in between wrong. The examiner deliberately picks century boundaries. Rule: check divisibility by 400 for any year ending in 00.

Trap 3 — Double-relation / mirror-time questions.
*"At what time between 4 and 5 will the hands be at right angles?"* has two answers (4 : 60/11 ≈ 4:05⁵⁄₁₁ and 4 : 420/11 ≈ 4:38²⁄₁₁). Options often list only one correct value plus a plausible wrong one, or ask "which of the following times" with both listed separately. Similarly, mirror-image questions ("the mirror shows 3:40 — real time?") use 11:60 − mirror time (or 12:00 − time if seconds are zero and you prefer), and students forget that 12:00 − 3:40 = 8:20 works only via 11:60. Always compute both right-angle times and check which option is present.

Summary Table

📌 Review this chapter after 2 days. Redo the 15 August 1947 example and the 4:20 angle example from memory — if both come out right without notes, this chapter is permanently yours.