Clocks

Module 13 — Lesson 0013 · In Depth · 45 min

A clock is a circular race — minute hand at 6°/min, hour hand at 0.5°/min, relative speed 5.5°/min. Every angle problem is |30H − 5.5M| with a reflex check; every faulty-clock problem is a ratio of real vs indicated time.

1. The Geometry of a Clock

• Total: 360°
• Hour space (12 divisions): 360/12 = 30° per hour number
• Minute space (60 divisions): 360/60 = 6° per minute mark

2. Speed of the Hands

Relative speed is the lap speed — minute hand laps the hour hand every 720/11 ≈65.45 min (not 60 — because hour hand moves).

3. Finding the Angle Between Hands

Standard formula: at H hours and M minutes:

θ = |30H − 5.5M| = |30H − (11/2)M|

H is hour digit (12 → 0), M is minutes. θ is internal (smaller) angle. If θ >180°, smaller angle = 360 − θ (reflex is the larger).

4. Finding the Time for a Specific Angle

Given angle (0° overlap, 180° straight, 90° right), find M between H and H+1:

1. Start at M=0: initial angle = 30H
2. Desired angle = θ_des (e.g., 0, 90, 180)
3. Distance to cover = |θ_des − 30H| (handling wrap) or more generally 5.5M = |30H − θ_des|
4. M = Distance / 5.5

Example: Between 3 and 4, when are hands at 90°? At 3:00 angle 90° already (so M=0 is one solution). Second solution: need angle 90° on the other side → distance = 180−90=90° beyond? Actually between 3–4, after 3:00 the minute gains: solve |90−5.5M|=90 → M=0 or M=180/5.5≈32.73 min → times 3:00 and ~3:32:43.

5. Standard Frequency Observations

Missing instance: No overlap between 11:00 and 1:00 — they coincide at exactly 12:00, counted once, so 11 not 12.

6. Gain and Loss of Time (Faulty Clocks)

Method 1: Ratio Method

Correct time : faulty time → find real duration for a given indicated duration.

Method 2: True Time from Gain

If a clock gains x minutes per 24h of real time:

Faulty covers 24h + x in 24h real → True Time = Indicated Duration × 24 / (24 + Gain)

Loses x → denominator 24 − x (since faulty covers less).

7. Advanced / Custom Clocks

If hour hand speed is not 0.5°/min (e.g., 0.7°/min as in puzzle): redefine relative speed = |MinuteSpeed − HourSpeed|, then reuse overlap/angle logic: time to next overlap = 360 / RelativeSpeed.

8. Summary of Key Variables

Notes

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