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Time Between Two Positions

Introduction

Clock problems में अक्सर hour और minute hands की दो relative positions के बीच का interval पूछा जाता है - जैसे दो बार hands के साथ आने (coincidence) के बीच का समय, दो बार opposite होने (180° apart) के बीच का समय, या दो लगातार right angles के बीच का समय। यह pattern इसलिए important है क्योंकि इन सभी questions को एक ही concept से हल किया जा सकता है: relative angular speed और proportional reasoning।

Pattern: Time Between Two Positions

Pattern

Key concept: Hour और minute hands एक-दूसरे के मुकाबले 11/2° per minute की relative speed से move करते हैं (minute hand 6°/min - hour hand 0.5°/min)। इसलिए, relative angle में Δ degrees का change होने में:

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

Clarification:
• Relative angle में Δ° change होने का समय = (2/11)×Δ minutes (जैसे 180° apart एक बार)।
• दो identical positions के बीच का समय (जैसे दो बार opposite होना या दो बार coincide होना) → पूरी 360° relative rotation चाहिए = (2/11)×360 = 720/11 minutes.

Common special cases (Δ degrees में):

  • Coincidence (hands साथ हों): Δ = 360° → successive coincidences का समय = (2/11)×360 = 720/11 = 65 5/11 minutes.
  • Opposite (180° apart): साथ से opposite होने तक समय = (2/11)×180 = 360/11 = 32 8/11 minutes.
    Successive identical opposite positions के बीच समय = (2/11)×360 = 720/11 = 65 5/11 minutes.
  • Right angle (90° apart): Δ = 90° → 90° होने तक का समय = (2/11)×90 = 180/11 = 16 4/11 minutes.

Step-by-Step Example

Question

Two successive coincidences (दो बार hands के साथ आने) के बीच कितना समय होता है?

Solution

  1. Step 1: Relative angular change पहचानें

    Hands फिर से साथ तभी आएँगे जब relative angle 360° बढ़ जाएगा।

  2. Step 2: Relative speed निकालें

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

  3. Step 3: Formula लगाएँ

    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:

    12 hours में 11 coincidences होते हैं → (12×60)/11 = 720/11 minutes ✅

Quick Variations

1. Successive coincidences के बीच समय: (2/11)×360 = 720/11 = 65 5/11 minutes.

2. 0° से 180° (opposite) तक समय: (2/11)×180 = 360/11 = 32 8/11 minutes.

3. Successive identical opposite positions का समय: (2/11)×360 = 720/11 = 65 5/11 minutes.

4. Right angle (90°) बनने का समय: (2/11)×90 = 180/11 = 16 4/11 minutes (पहली right angle position)। Successive right angles अक्सर ≈ 32 8/11 minutes apart होते हैं।

Trick to Always Use

  • Step 1 → यह तय करें कि relative angle कितने degrees बदलना चाहिए।
  • Step 2 → Time = (2/11) × (degrees change) formula से minutes निकालें।
  • Step 3 → याद रखें: identical relative positions हर बार 720/11 minutes बाद repeat होती हैं, 360/11 नहीं।

Summary

Summary

  • Key takeaway 1: Relative speed = 11/2°/min; Δ° change का समय = (2/11)×Δ minutes.
  • Key takeaway 2: Coincidences हर 720/11 = 65 5/11 minutes में repeat होती हैं।
  • Key takeaway 3: Opposite positions भी बिल्कुल same interval पर repeat होती हैं - 720/11 minutes (32 8/11 सिर्फ transition time है)।
  • Key takeaway 4: Right angles में पहला 90° = 16 4/11 minutes; successive ones ≈ 32 8/11 minutes apart।

याद रखने लायक example:
Two identical opposite positions के बीच समय = (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).

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