Episode 8 — Aptitude and Reasoning / 8.19 — Clocks

8.19 Quick Revision - Clocks

Speed of Hands

Hand	Speed
Minute hand	6 degrees/min
Hour hand	0.5 degrees/min
Relative speed	5.5 degrees/min

The Master Formula

Angle at H:M = |30H - 5.5M| degrees

If result > 180: Angle = 360 - result

Special Positions

Position	Formula for M (between H and H+1)
Overlap (0 degrees)	`M = 60H/11`
Right angle (90 degrees)	`M = (60H +/- 180)/11`
Straight line (180 degrees)	`M = (60H - 360)/11`
Any angle X	`M = (30H +/- X)/5.5`

Frequency Table

Event	In 12 hrs	In 24 hrs	Gap between events
Overlap (0 deg)	11	22	65 + 5/11 min
Right angle (90 deg)	22	44	32 + 8/11 min
Opposite (180 deg)	11	22	65 + 5/11 min
Straight line (0 or 180)	22	44	32 + 8/11 min

Angles at Exact Hours

12:00 = 0       1:00 = 30      2:00 = 60
3:00  = 90      4:00 = 120     5:00 = 150
6:00  = 180     7:00 = 150     8:00 = 120
9:00  = 90      10:00 = 60     11:00 = 30

Mirror Image Formula

Actual time = 11:60 - Mirror time

If mirror time > 11:60:
Actual time = 23:60 - Mirror time

Faulty Clocks

Clock GAINS g min/hr:
  Shows (60+g) min for every real 60 min
  Real time = Clock time * 60/(60+g)
  Shows correct again after: 720/(g*24) days

Clock LOSES l min/hr:
  Shows (60-l) min for every real 60 min
  Real time = Clock time * 60/(60-l)
  Shows correct again after: 720/(l*24) days

Two clocks (gain a, lose b):
  Same time after: 720/(a+b) hours

Angle Traced

Minute hand in t minutes: 6t degrees
Hour hand in t minutes: 0.5t degrees (= t/2 degrees)

Quick Mental Math

5/11 = 0.4545... (about 27 seconds)
8/11 = 0.7272... (about 44 seconds)
60/11 = 5.4545... (5 min 27 sec)
720/11 = 65.4545... (65 min 27 sec)

Common Mistakes to Avoid

1. Always check if angle > 180 (take 360 - angle)
2. Overlaps in 12 hrs = 11, NOT 12
3. Between H and H+1: check 0 <= M < 60
4. Mirror formula: 11:60, not 12:00
5. Reflex angle = 360 - normal angle
6. Clock striking: n strikes = (n-1) intervals

Back to Section 8.19