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Time Between Two Positions

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Introduction

Many clock problems ask for the interval between two occurrences of a relative position of the hour and minute hands - for example, the time between successive coincidences (hands together), between successive opposite positions (180° apart), or between consecutive right angles. This pattern is important because it reduces all such questions to one key concept: the relative angular speed of the two hands and proportional reasoning.

Pattern: Time Between Two Positions

Pattern: Time Between Two Positions

Key concept: The hour and minute hands move relative to each other at 11/2° per minute (minute hand 6°/min - hour hand 0.5°/min). So, the time required for the relative angle to change by Δ degrees is:

Time (minutes) = Δ ÷ (11/2) = (2/11) × Δ.

Clarification:
• Time to change the relative angle by Δ° = (2/11)×Δ minutes (e.g., 180° apart once).
• Time between successive identical positions (e.g., next time they are again opposite or coincide) requires a full 360° relative rotation = (2/11)×360 = 720/11 minutes.

Common special cases (use Δ in degrees):

  • Coincidence (hands together): Δ = 360° → Time between successive coincidences = (2/11)×360 = 720/11 = 65 5/11 minutes.
  • Opposite (180° apart): Time to go from together (0°) to opposite = (2/11)×180 = 360/11 = 32 8/11 minutes.
    Time between successive identical opposite positions = (2/11)×360 = 720/11 = 65 5/11 minutes.
  • Right angle (90° apart): Δ = 90° → Time to reach 90° = (2/11)×90 = 180/11 = 16 4/11 minutes.

Step-by-Step Example

Question

What is the time between two successive coincidences (two successive moments when the hands are together)?

Solution

  1. Step 1: Identify relative angular change

    The hands will be together again only after the relative angle increases by 360°.

  2. Step 2: Find relative speed

    Relative speed = minute hand - hour hand = 6 - 0.5 = 11/2° per minute.

  3. Step 3: Apply formula

    Time = Δ ÷ (11/2) = 360 ÷ (11/2) = 360 × (2/11) = 720/11 minutes.

  4. Final Answer:

    720/11 minutes = 65 5/11 minutes ≈ 1 hour 5 minutes 27 seconds

  5. Quick Check:

    In 12 hours, there are 11 coincidences → (12×60)/11 = 720/11 minutes ✅

Quick Variations

1. Time between successive coincidences: (2/11)×360 = 720/11 = 65 5/11 minutes.

2. Time to change relative angle by 180° (e.g., from 0° to opposite): (2/11)×180 = 360/11 = 32 8/11 minutes.

3. Time between successive identical opposite positions: (2/11)×360 = 720/11 = 65 5/11 minutes.

4. Time to form right angle (90°): (2/11)×90 = 180/11 = 16 4/11 minutes (for the first right angle). Successive right-angle intervals (e.g., 90° to next 90°) often use 32 8/11 minutes.

Trick to Always Use

  • Step 1 → Determine how many degrees the relative angle must change.
  • Step 2 → Use Time = (2/11) × (degrees change) to find minutes.
  • Step 3 → Remember: identical relative positions repeat every 720/11 minutes, not 360/11.

Summary

  • Key takeaway 1: Relative speed = 11/2°/min; time to change Δ° = (2/11)×Δ minutes.
  • Key takeaway 2: Coincidences repeat every 720/11 = 65 5/11 minutes.
  • Key takeaway 3: Opposite positions (180° apart) recur identically every 720/11 = 65 5/11 minutes (not 32 8/11, which is only the transition time).
  • Key takeaway 4: For right angles, first 90° = 16 4/11 minutes; successive ones ≈ 32 8/11 minutes apart.

Example to remember:
Time between two identical opposite positions = (2/11)×360 = 720/11 = 65 5/11 minutes.

Practice

(1/5)
1. How many minutes are required for the hour and minute hands to change their relative angle by 90°?
easy
A. 16 4/11 minutes
B. 32 8/11 minutes
C. 65 5/11 minutes
D. 5 5/11 minutes

Solution

  1. Step 1: Identify relative speed

    The relative angular speed = 6°/min - 0.5°/min = 11/2 °/min.

  2. Step 2: Use formula Time = (2/11) × Δ

    For Δ = 90°, Time = (2/11) × 90 = 180/11 minutes.

  3. Step 3: Convert to mixed number

    180 ÷ 11 = 16 remainder 4 → 16 4/11 minutes.

  4. Final Answer:

    16 4/11 minutes → Option A

  5. Quick Check:

    16 4/11 ≈ 16.3636 min; (11/2)×16.3636 ≈ 90° ✅
Hint: Use Time = (2/11)×Δ for any Δ° required in relative angle.
Common Mistakes: Confusing the hour-hand rate (0.5°/min) with 0.05° or forgetting to use relative speed.
2. How many minutes are needed for the relative angle between the hands to change by 30°?
easy
A. 10 10/11 minutes
B. 5 5/11 minutes
C. 20 2/11 minutes
D. 8 8/11 minutes

Solution

  1. Step 1: Apply Time = (2/11)×Δ

    For Δ = 30°, Time = (2/11) × 30 = 60/11 minutes.

  2. Step 2: Convert to mixed number

    60 ÷ 11 = 5 remainder 5 → 5 5/11 minutes.

  3. Final Answer:

    5 5/11 minutes → Option B

  4. Quick Check:

    5.4545 min × (11/2) ≈ 30° ✅
Hint: Multiply Δ by 2/11 to get minutes directly.
Common Mistakes: Using minute hand speed alone (6°/min) instead of relative speed.
3. If the relative angle must change by 180° (for example, from together to opposite), how many minutes does that change take?
easy
A. 16 4/11 minutes
B. 65 5/11 minutes
C. 32 8/11 minutes
D. 5 5/11 minutes

Solution

  1. Step 1: Use formula Time = (2/11)×Δ

    For Δ = 180°, Time = (2/11) × 180 = 360/11 minutes.

  2. Step 2: Convert to mixed number

    360 ÷ 11 = 32 remainder 8 → 32 8/11 minutes.

  3. Final Answer:

    32 8/11 minutes → Option C

  4. Quick Check:

    32.7273 min × (11/2) ≈ 180° ✅
Hint: Double Δ then divide by 11 (i.e., 2Δ/11).
Common Mistakes: Mixing up ‘time to change by 180°’ with ‘time between identical opposite occurrences’ (the latter is 720/11).
4. How many minutes will pass between two identical occurrences when the hands are at 60° (i.e., the next time they attain the same 60° orientation)?
medium
A. 65 5/11 minutes
B. 32 8/11 minutes
C. 16 4/11 minutes
D. 5 5/11 minutes

Solution

  1. Step 1: Clarify 'identical occurrence'

    The next identical occurrence of a given orientation requires the relative angle to complete a full 360° cycle.

  2. Step 2: Use full-cycle formula

    Time for full 360° relative rotation = (2/11) × 360 = 720/11 minutes.

  3. Step 3: Convert to mixed number

    720 ÷ 11 = 65 remainder 5 → 65 5/11 minutes.

  4. Final Answer:

    65 5/11 minutes → Option A

  5. Quick Check:

    All identical orientations (any θ) repeat after 720/11 minutes (≈65.45 min) ✅
Hint: Identical orientation repeats after a full 360° relative gain → use 720/11 minutes.
Common Mistakes: Using 360/11 (time to change by 180°) instead of the full 720/11 cycle for identical repeats.
5. If at a given moment the hands are at a right angle, after how many minutes will they next be at a right angle again (the next successive right-angle event)?
medium
A. 16 4/11 minutes
B. 65 5/11 minutes
C. 5 5/11 minutes
D. 32 8/11 minutes

Solution

  1. Step 1: Understand successive right-angle events

    Right angles (±90°) occur twice per 360° relative sweep: after 90° and after 270° relative change. The gap between successive right-angle events is often the larger step of 180° in relative angle.

  2. Step 2: Compute time for 180° relative change

    Time = (2/11) × 180 = 360/11 = 32 8/11 minutes.

  3. Final Answer:

    32 8/11 minutes → Option D

  4. Quick Check:

    First right angle after a coincidence occurs at 16 4/11 min; the next right-angle event (the successive one) is 32 8/11 min later ✅
Hint: Use 16 4/11 for the first 90°; successive right-angle gap typically = 32 8/11 minutes.
Common Mistakes: Confusing the first 90° (16 4/11) with the successive right-angle gap (32 8/11).