Clock

⏰ Clock Problems

🔹 Basic Concepts

- A clock has 12 hours and 360° total. - Each hour = 30° (360 ÷ 12) - Each minute = 6° (360 ÷ 60)

Angle between hour and minute hands:
\[ \theta = \left|30H - \frac{11}{2}M \right| \] Where H = hour, M = minutes

Minute hand moves: 6° per minute \[ \text{Minute hand angle} = 6 \times M \]

Hour hand moves: 0.5° per minute \[ \text{Hour hand angle} = 30 \times H + 0.5 \times M \]

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🧠 Exam Shortcuts & Tricks

1. Angle = \( \left|30H - \frac{11}{2}M \right| \)
2. Reflex angle = 360° − Angle
3. Maximum angle between hands = 180° (take smaller angle usually)
4. Hour hand moves 0.5° per minute
5. Minute hand moves 6° per minute

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✍️ Solved Examples

Example 1:
Find angle at 3:00 \[ \theta = |30 \times 3 - \frac{11}{2} \times 0| = |90 - 0| = 90^\circ \]

Example 2:
Find angle at 4:20 \[ \theta = \left|30 \times 4 - \frac{11}{2} \times 20 \right| = |120 - 110| = 10^\circ \]

Example 3:
Find reflex angle at 2:15 \[ \text{Angle} = \left|30 \times 2 - \frac{11}{2} \times 15\right| = |60 - 82.5| = 22.5^\circ \] \[ \text{Reflex angle} = 360 - 22.5 = 337.5^\circ \]

Example 4:
Minute hand at 6, hour hand at 9:30 \[ \text{Hour hand angle} = 30 \times 9 + 0.5 \times 30 = 270 + 15 = 285^\circ \] \[ \text{Minute hand angle} = 6 \times 30 = 180^\circ \] \[ \theta = |285 - 180| = 105^\circ \]

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