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Hands Opposite Each Other

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Introduction

கடிகாரத்தில் மணி ஊசி (hour hand) மற்றும் நிமிட ஊசி (minute hand) சரியாக எதிரெதிராக (180° apart) இருக்கும் நேரத்தை கண்டுபிடிக்கும் கேள்விகள் reasoning exams-ல் அடிக்கடி கேட்கப்படும்.
இந்த pattern-ஐ நன்றாக புரிந்துகொண்டால், straight-line, opposite, frequency போன்ற clock problems அனைத்தையும் எளிதாக தீர்க்க முடியும்.

Pattern: Hands Opposite Each Other

Pattern: Hands Opposite Each Other

Key idea: மணி ஊசி மற்றும் நிமிட ஊசி இடையிலான கோணம் 180° ஆக இருந்தால் அவை opposite நிலையில் இருக்கும்.

Hour hand position = 30H + 0.5m
Minute hand position = 6m

ஆகவே condition: |(30 × H) - (11/2 × m)| = 180

இதை தீர்க்கும்போது இரண்டு linear forms கிடைக்கும்:

  • (11/2)m = 30H + 180 → m = (60/11)(H + 6)
  • (11/2)m = 30H - 180 → m = (60/11)(H - 6)

கொடுக்கப்பட்ட hour-க்கு, 0 ≤ m < 60 என்ற நிமிட மதிப்புகள் மட்டுமே valid. Negative அல்லது ≥ 60 வந்தால், அந்த opposite நிகழ்வு முந்தைய அல்லது அடுத்த hour-ல் நடைபெறும்.

Step-by-Step Example

Question

1 மணி முதல் 2 மணி வரை, கடிகார ஊசிகள் எப்போது சரியாக opposite ஆக இருக்கும்?

Solution

  1. Step 1: Opposite condition எழுதவும்

    |30H - (11/2)m| = 180

  2. Step 2: H = 1 substitute செய்து இரண்டு roots கணக்கிடவும்

    (a) (11/2)m = 30×1 + 180 = 210
    m = 420/11 = 38 2/11 நிமிடங்கள் ✔️

    (b) (11/2)m = 30×1 - 180 = -150
    m = -300/11 = -27 3/11 ❌ (previous hour)

  3. Step 3: Valid root தேர்ந்தெடுக்கவும்

    0 ≤ m < 60 உள்ளதால், 1-2 மணி இடைவெளிக்குள் valid நேரம்: 1:38 2/11

  4. Final Answer:

    1:38 2/11

  5. Quick Check:

    Hour position = 30×1 + 0.5×38.18… ≈ 49.09°
    Minute position = 6×38.18… ≈ 229.09°
    Difference = 229.09 - 49.09 = 180° ✅

Quick Variations

1. எந்த hour H-க்கும், m = (60/11)(H ± 6) கணக்கிட்டு, 0 ≤ m < 60 உள்ளதை தேர்வு செய்யவும்.

2. (H + 6) வைத்து m = 60 வந்தால், opposite சரியாக அடுத்த hour-ல் இருக்கும் (எ.கா: 6:00, 12:00).

3. (H - 6) negative வந்தால், அந்த opposite முந்தைய hour-ல் நடந்தது.

Trick to Always Use

  • Step 1 → m = (60/11)(H + 6) மற்றும் m = (60/11)(H - 6) இரண்டையும் கணக்கிடவும்.
  • Step 2 → 0 ≤ m < 60 உள்ளதை மட்டும் அந்த hour-க்கு எடுத்துக்கொள்ளவும்.
  • Step 3 → hour = 30H + 0.5m, minute = 6m என verify செய்யவும் (difference = 180°).

Summary

  • Opposite condition: |30H - (11/2)m| = 180
  • Minute formulas: m = (60/11)(H ± 6)
  • 12 மணி நேரத்தில், ஊசிகள் 11 முறை opposite ஆக இருக்கும்.
  • Negative அல்லது ≥60 roots வந்தால், அது adjacent hour-ஐ குறிக்கும் - எப்போதும் substitute செய்து check செய்யவும்.

Example to remember:
1-2 மணி → 1:38 2/11 (opposite)

Practice

(1/5)
1. When are the hour and minute hands opposite each other between 2 and 3 o’clock?
easy
A. 2:43 7/11
B. 2:21 9/11
C. 2:27 3/11
D. 2:49 1/11

Solution

  1. Step 1: Use opposite condition

    The hands are opposite when |30H - (11/2)m| = 180. This gives m = (60/11)(H ± 6).

  2. Step 2: Substitute H = 2

    Compute m = (60/11)(2 + 6) = (60/11)×8 = 480/11 = 43 7/11 minutes. The other root (H - 6) is negative and discarded for 2-3.

  3. Step 3: Write the time

    The hands are opposite at 2:43 7/11.

  4. Final Answer:

    2:43 7/11 → Option A

  5. Quick Check:

    Hour ≈ 60 + 0.5×43.636 = 81.818°; minute ≈ 6×43.636 = 261.818°; difference = 180° ✅
Hint: Use m = (60/11)(H + 6) for the later opposite in the hour; discard negative roots.
Common Mistakes: Accepting negative minute values inside the same hour.
2. When are the hands opposite each other between 4 and 5 o’clock?
easy
A. 4:54 6/11
B. 4:49 1/11
C. 4:38 2/11
D. 4:21 9/11

Solution

  1. Step 1: Use the formula

    m = (60/11)(H ± 6).

  2. Step 2: Substitute H = 4

    m = (60/11)(4 + 6) = (60/11)×10 = 600/11 = 54 6/11 minutes. The (H - 6) root is negative and discarded for 4-5.

  3. Step 3: Write the time

    The hands are opposite at 4:54 6/11.

  4. Final Answer:

    4:54 6/11 → Option A

  5. Quick Check:

    Hour ≈ 120 + 0.5×54.545 = 147.273°; minute ≈ 327.273°; difference mod 360 = 180° ✅
Hint: Compute (H + 6) first for opposites; if result ≤ 60 it is the later opposite in that hour.
Common Mistakes: Confusing which algebraic root maps to the current hour.
3. At what time between 9 and 10 o’clock will the hands be opposite each other?
easy
A. 9:27 3/11
B. 9:16 4/11
C. 9:43 7/11
D. 9:32 8/11

Solution

  1. Step 1: Use opposite formula

    m = (60/11)(H ± 6).

  2. Step 2: Substitute H = 9

    m = (60/11)(9 - 6) = (60/11)×3 = 180/11 = 16 4/11 minutes. The (H + 6) root exceeds 60 and is invalid for 9-10.

  3. Step 3: Write the time

    The hands are opposite at 9:16 4/11.

  4. Final Answer:

    9:16 4/11 → Option B

  5. Quick Check:

    Hour ≈ 270 + 0.5×16.364 = 278.182°; minute ≈ 98.182°; difference = 180° ✅
Hint: If (H - 6) gives a small positive value, that gives the earlier opposite inside the hour.
Common Mistakes: Not checking which root stays within 0-60 minutes.
4. When are the hands opposite each other between 11 and 12 o’clock?
medium
A. 11:21 9/11
B. 11:38 2/11
C. 11:27 3/11
D. 11:05 5/11

Solution

  1. Step 1: Use m = (60/11)(H ± 6)

  2. Step 2: Substitute H = 11

    m = (60/11)(11 - 6) = (60/11)×5 = 300/11 = 27 3/11 minutes. The (H + 6) root > 60 and is invalid for 11-12.

  3. Step 3: Write the time

    The hands are opposite at 11:27 3/11.

  4. Final Answer:

    11:27 3/11 → Option C

  5. Quick Check:

    Hour ≈ 330 + 0.5×27.273 = 343.636°; minute ≈ 163.636°; difference = 180° ✅
Hint: Try (H - 6) first for the earlier opposite time; verify 0 ≤ m < 60.
Common Mistakes: Selecting the root that lies outside the hour range.
5. How many times do the hands of a clock stand opposite each other in 24 hours?
medium
A. 11
B. 22
C. 24
D. 23

Solution

  1. Step 1: Opposites in 12 hours

    The hour and minute hands are opposite each other 11 times in 12 hours.

  2. Step 2: Extend to 24 hours

    Since the same cycle repeats twice in a full day (24 hours), total opposites = 11 × 2 = 22 times.

  3. Step 3: Verify logic

    Each 12-hour cycle (AM or PM) produces 11 opposite positions, not 12.

  4. Final Answer:

    22 → Option B

  5. Quick Check:

    11 opposites in 12 hours × 2 = 22 in 24 hours ✅
Hint: Opposites occur 11 times in 12 hours → multiply by 2 for 24 hours = 22 times.
Common Mistakes: Mistaking the count as 24 or 44 by assuming one per hour.