Clock

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Clock — Key Facts for SSC CGL • The clock face has 12 hours numbered 1 to 12. The full circle is 360°. • The hour hand moves 360° in 12 hours = 30° per hour = 0.5° per minute. • The minute hand moves 360° in 60 minutes = 6° per minute. • Relative speed of minute hand over hour hand = $6° - 0.5° = 5.5°$ per minute. • Right angle (90°) between hands occurs approximately every 32.7 minutes. • Straight line (180°) between hands occurs approximately every 65.5 minutes. • At 12:00 (both hands together): angle = 0°. • At 6:00 (opposite sides): angle = 180°.

⚡ Exam Tip: The formula $\left|30H - 5.5M\right|$ gives the angle between hands. For angles > 180°, subtract from 360°. This single formula handles 90% of all SSC CGL clock questions.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Clock — SSC CGL Study Guide

Core Concept: Clock problems are essentially relative speed problems on a circular track. The key is understanding that both hands rotate continuously — the hour hand at 0.5°/min and the minute hand at 6°/min. The angle between them changes as time passes.

Key Formulas:

1. Angle between hands at $H$ hours and $M$ minutes: $$\theta = \left|30H - 5.5M\right|$$

2. Minimum angle (always ≤ 180°): $$\theta_{\min} = \min(\theta, 360° - \theta)$$

3. When are hands together? (Minute hand catches hour hand): $$M = \frac{60}{11} \times H \text{ minutes past } H$$

4. When are hands opposite?: $$M = \frac{60}{11} \times (H + 6) \text{ minutes}$$

Worked Example 1: Find the angle at 3:40.

• H = 3, M = 40
• $\theta = |30 \times 3 - 5.5 \times 40| = |90 - 220| = 130°$
• Minimum angle = 130° (already ≤ 180°)
• Answer: 130°

Worked Example 2: At what time between 3 and 4 o’clock are the hands of a clock exactly opposite?

• Opposite means angle = 180°
• $|30 \times 3 - 5.5 \times M| = 180°$
• $|90 - 5.5M| = 180°$ → $5.5M = 90 + 180 = 270°$ (taking positive case)
• $M = 270 / 5.5 = 2700/55 = 540/11 = 49\frac{1}{11}$ minutes
• Time = 3:49 and 1/11 minutes (approximately 3:49:05)

Worked Example 3: How many times do the hands coincide in a day?

• They coincide every ~65.45 minutes (approximately every 1 hour 5 minutes 27 seconds)
• In 12 hours: coincides 11 times (not 12, because at 12 they coincide, then again at ~1:05, 2:10, …, 10:54, 12:00)
• In 24 hours: 22 times
• In 12 hours: opposite 11 times; so in 24 hours: 22 times

Common Student Mistakes: Forgetting to multiply H by 30 (many students use 0.5H which gives wrong angle), not taking the minimum of $\theta$ and 360°−$\theta$, and misremembering that the hour hand moves continuously (not just per hour jump).

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Clock — Comprehensive SSC CGL Notes

Theoretical Foundation: Clock problems are geometric relative motion problems. The face is a 360° circle divided into 12 equal arcs of 30°. The hour hand completes one full rotation (360°) in 12 hours (720 minutes) = 0.5° per minute. The minute hand completes 360° in 60 minutes = 6° per minute. The relative speed difference of 5.5° per minute is the key parameter.

Advanced Clock Concepts:

1. Mirror Image / Reflection Time: If the time shown in a mirror is X, the actual time = $11:60 - X$ (for times between 1:01 and 10:59).

   • Example: Mirror shows 4:20 → Actual = $11:60 - 4:20 = 7:40$

2. Water Image / Mirror in Water (less common in SSC): Water image flips vertically: actual = $23:60 - \text{mirror time}$

3. Incorrect Clock: If a clock gains or loses uniformly:

   • If it gains $G$ minutes in $T$ hours → Rate = $G/T$ minutes per hour
   • To find the correct time when it shows X, calculate total error accumulated

4. Angle chasing (when multiple hands involved): At $H:M$, the positions are:

   • Hour hand: $0.5 \times (60H + M) = 30H + 0.5M$ degrees
   • Minute hand: $6M$ degrees
   • Angle = $|6M - 30H - 0.5M| = |5.5M - 30H|$

Worked Example — Incorrect Clock: A clock loses 4 minutes per hour. It shows 10:00 AM when the correct time is 11:00 AM. How long will it take to show the correct time again?

• Loss per hour = 4 minutes
• Total time to lose 60 minutes = 60/4 = 15 hours
• It will show correct time after 15 hours (i.e., at 1:00 AM the next day)

SSC CGL PYQ Pattern (2019-2023):

• 2023 Tier-I: 1-2 clock questions — angle calculation and mirror time
• 2022 Tier-I: 1 question — hands coincide
• Most common question types: (1) angle at given time, (2) time when angle is X, (3) mirror/reflection time
• Mirror image formula: If mirror shows $T$, real time = $11:60 - T$
• Most difficult: Incorrect clock problems combined with angle

Speed Cheat Sheet:

Relative Speed	Value
Minute vs Hour	5.5°/minute
Right angle occurs	Every 32 $\frac{8}{11}$ minutes
Straight line occurs	Every 65 $\frac{5}{11}$ minutes
Hands together	Every 65 $\frac{5}{11}$ minutes

Practice Strategy: Write out the 5.5°/minute relative speed formula and practice 10 angle questions. Then move to mirror time (straightforward substitution). For incorrect clocks, set up the gain/loss rate equation. Time yourself — target 45 seconds per question.

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