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Hands Opposite Each Other

Introduction

ऐसे प्रश्न जो पूछते हैं कि hour और minute की सूइयाँ ठीक भी opposite (180° apart) हों, reasoning tests में आम होते हैं। यह पैटर्न महत्वपूर्ण है क्योंकि opposite स्थिति सीधे straight-line अवधारणाओं से जुड़ी होती है और occurrence गिनने व timing पहेलियाँ हल करने में मदद करती है।

Pattern: Hands Opposite Each Other

Pattern

हाथ तब opposite होते हैं जब उनका angle difference 180° हो। hour = 30H + 0.5m और minute = 6m लगाकर, सेट करें: |(30 × H) - (11/2 × m)| = 180.

हल करने पर दो linear रूप और मिनट के सूत्र मिलते हैं:

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

केवल वे मिनट मान स्वीकार करें जहाँ 0 ≤ m < 60 होता है - नकारात्मक या ≥60 परिणाम पास के घंटे में आते हैं और उसी हिसाब से interpret करना चाहिए।

Step-by-Step Example

Question

1 और 2 बजे के बीच सूइयाँ एक-दूसरे के opposite कब होंगी?

Solution

  1. Step 1: opposite की condition लिखें

    उपयोग करें |30H - (11/2)m| = 180.

  2. Step 2: H = 1 दोनों linear रूपों में रखें

    (a) (11/2)m = 30×1 + 180 = 210 → m = 420/11 = 38 2/11 minutes.
    (b) (11/2)m = 30×1 - 180 = -150 → m = -300/11 = -27 3/11 minutes (1-2 के लिए discard करें).

  3. Step 3: घंटे के भीतर मान्य root interpret करें

    1:00 और 2:00 के बीच केवल वैध समय है 1:38 2/11. नकारात्मक root पिछले घंटे का समय दर्शाती है।

  4. Final Answer:

    1:38 2/11

  5. Quick Check:

    Hour position = 30×1 + 0.5×38.181... ≈ 49.091°. Minute = 6×38.181... ≈ 229.091°. अंतर ≈ 180° ✅

Quick Variations

1. किसी घंटे H के लिए मापें: m = (60/11)(H ± 6) और 0-59.999… के बीच मान स्वीकार करें।

2. यदि (H + 6) पर m = 60 बना तो opposite ठीक अगली घड़ी के शुरू में होगा (उदा., कुछ मामलों में 6:00 या 12:00)।

3. (H - 6) से नकारात्मक m मिले तो opposite पिछले घंटे में हुआ था।

Trick to Always Use

  • Step 1: उपयोग करें m = (60/11)(H + 6) और m = (60/11)(H - 6).
  • Step 2: केवल वही m रखें जहाँ 0 ≤ m < 60; अन्यथा adjacent-hour मान लें।
  • Step 3: सत्यापित करने के लिए hour = 30H + 0.5m और minute = 6m निकालकर 180° difference जाँचें।

Summary

Summary

  • Opposite condition: |30H - (11/2)m| = 180.
  • मिनट के सूत्र: m = (60/11)(H ± 6); केवल 0 ≤ m < 60 स्वीकार करें उस घंटे के लिए।
  • Opposites 12 घंटे में 11 बार होते हैं (coincidences जितनी बार)।
  • हमेशा substitution से verify करें - नकारात्मक या ≥60 roots adjacent hours के times हैं।

याद रखने के लिए उदाहरण:
1 और 2 के बीच → 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.

Mock Test

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