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Quantitative Aptitude - Clocks

In-depth guide on angle- and error-related clock problems: angles, overlaps, gains/losses.

1. Hour‑Minute Hand Angle

Formula: \( |30H - 5.5M| \). E.g., at 2:20: |60 - 110| = 50°.

2. Hand Overlaps

  • Every \( \frac{720}{11} \approx 65.4545 \) minutes past each hour.
  • nᵗʰ overlap time = \( \frac{720n}{11} \) minutes after 12.

Example: 3rd overlap → 196.36 min → 3h 16m 22s.

3. Opposite Hands (straight 180°)

Equation: |30H - 5.5M| = 180 → Solve for M.

4. Clock Gain/Loss

Daily gain = d seconds/day → rate per minute = d/(24×60×60) sec/sec. Actual time vs indicated calculate via proportional error.

5. Sample Problems

  • Angle at 9:15: |270 - 82.5| = 187.5° → minimal = 360-187.5 = 172.5°.
  • Clock gains 6 min in 24h: After 10h, gain = 6×10/24 = 2.5 min → indicated and real times differ accordingly.

6. Exam Strategy

  • Always convert angles to ≤180°.
  • Use exact fraction (±1 min) for overlap times.
  • For gain/loss, convert to per‑second rate first.

7. Common Errors

  • Applying hour-hand position as 30H without adding 0.5 per minute.
  • Ignoring minute-hand movement during gain/loss problems.

8. Quick Checklist

  • Practice 7 angles, 5 overlaps, 5 gain/loss problems.

Summary: Angle formula + overlaps (720/11) + time‑error rates = fast solutions. Mind units and minimal angles.

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