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Minute and Hour Hand Swap

Introduction

Swap problems में पूछा जाता है कि कब hour hand और minute hand किसी अन्य समय के मुकाबले अपनी-अपनी positions आपस में बदल लेते हैं (interchange)। इसके लिए दो समयों पर दोनों hands के exact angular positions लिखकर दोनों swap समीकरण एक साथ हल करने होते हैं। यह pattern advanced clock reasoning के लिए ज़रूरी है क्योंकि इसमें angle formulas, linear equations और normalization एक साथ आते हैं।

Pattern: Minute and Hour Hand Swap

Pattern

Key concept: Hour और minute hand की positions को degrees (या 60 के scale में minute-scale) में व्यक्त करें, swap conditions लिखें, और फिर दो linear equations हल करें।

Useful formulas (angles in degrees):
Hour hand at H hours and m minutes: Anglehour = 30H + 0.5m.
Minute hand at m minutes: Anglemin = 6m.
Swap condition between times T₁ and T₂ (hour/min positions interchanged):
Angle_hour(T₂) = Angle_min(T₁) and Angle_min(T₂) = Angle_hour(T₁).

Step-by-Step Example

Question

3 और 4 के बीच किस समय hour और minute hands उन positions को interchange करेंगे जो वे 4 और 5 के बीच दिखाएँगे? (यानी 3-4 और 4-5 hour slots के बीच swap)

Solution

  1. Step 1: Unknowns परिभाषित करें और angles लिखें

    मान लें T₁ = 3 + x minutes (यानी 3 बजे x मिनट)। और T₂ = 4 + y minutes (4 बजे y मिनट)।
    Hour angle at T₁ = 30×3 + 0.5x = 90 + 0.5x.
    Minute angle at T₁ = 6x.
    Hour angle at T₂ = 30×4 + 0.5y = 120 + 0.5y.
    Minute angle at T₂ = 6y.

  2. Step 2: Swap समीकरण लिखें

    Swap का मतलब है hour angle at T₂ बराबर minute angle at T₁, और minute angle at T₂ बराबर hour angle at T₁. तो समीकरण हैं:
    (1) 120 + 0.5y = 6x
    (2) 6y = 90 + 0.5x

  3. Step 3: दोनो linear equations हल करें

    (2) से: y = 15 + x/12. इसे (1) में substitute करें:
    120 + 0.5(15 + x/12) = 6x
    120 + 7.5 + x/24 = 6x
    127.5 = 6x - x/24 = x(144/24 - 1/24) = x(143/24)
    x = 127.5 × 24 / 143 = 3060 / 143 = 21 57/143 minutes (≈ 21.3993 min).

  4. Step 4: Swapped समय T₁ और T₂ निकालें

    T₁ = 3 : x = 3:21 57/143 (exact).
    पूरी जानकारी के लिए y = 15 + x/12 = 15 + (3060/143)/12 = 15 + 255/143 = 2400/143 = 16 112/143 minutes (≈ 16.7832 min), इसलिए T₂ = 4:16 112/143.

  5. Final Answer:

    3:21 57/143

  6. Quick Check:

    अंकगणित करके angles approx निकालें: 3:21.399 पर hour ≈ 90 + 0.5×21.399 = 100.6995°; minute ≈ 6×21.399 = 128.394°. 4:16.783 पर hour ≈ 120 + 0.5×16.783 = 128.3915°; minute ≈ 6×16.783 = 100.698°. यहाँ hour angle at T₂ ≈ minute angle at T₁ और minute angle at T₂ ≈ hour angle at T₁-rounding के भीतर match करता है ✅

Quick Variations

1. Swaps किसी भी adjacent hours H और H+1 के बीच सेट किए जा सकते हैं - वही दो-समीकरण तरीका अपनाएँ।

2. कुछ problems पूछते हैं कि swap उसी hour के अंदर होगा - इसका अर्थ अक्सर coincidence जैसा special case होता है; ध्यान से interpret करें।

3. जल्दी अंदाज़े के लिए, swaps अक्सर hour के लगभग 21-22 मिनट के आस-पास होते हैं (H के आधार पर थोड़ा बदलता है)।

Trick to Always Use

  • Step 1 → मान लें T₁ = H + x और T₂ = (H+1) + y (adjacent hours), दोनों times के लिए angles लिखें।
  • Step 2 → Swap equations लगाएँ: Angle_hour(T₂)=Angle_min(T₁) और Angle_min(T₂)=Angle_hour(T₁).
  • Step 3 → 2×2 linear system हल करें; fractional minutes को normalize करें और numeric तरीके से verify करें।

Summary

Summary

  • Key takeaway 1: Hour और minute के angles को सटीक रूप से model करें: Hour = 30H + 0.5m, Minute = 6m.
  • Key takeaway 2: Swap conditions से दो linear equations मिलते हैं - उन्हें साथ में हल करें ताकि प्रत्येक hour के past minutes मिलें।
  • Key takeaway 3: H और H+1 के बीच swap का समाधान आमतौर पर T₁ को लगभग 21-22 मिनट past H देता है (ऊपर exact fractional form दिया है)।
  • Key takeaway 4: हमेशा angles का numeric quick check करके swap की पुष्टि करें।

याद रखने के लिए example:
3 और 4 के बीच swap 3:21 57/143 (≈ 3:21.399) पर होगा।

Practice

(1/5)
1. At what time between 1 and 2 o’clock will the hour and minute hands interchange their positions with those they will have between 2 and 3 o’clock?
easy
A. 1:10 70/143
B. 1:21 50/143
C. 1:15 135/143
D. 1:19 6/11

Solution

  1. Step 1: Define the variables

    Let the time in 1-2 be 1 + x minutes and the swapped time in 2-3 be 2 + y minutes. Then:
    Hour(1+x) = 30 + 0.5x; Minute(1+x) = 6x; Hour(2+y) = 60 + 0.5y; Minute(2+y) = 6y.

  2. Step 2: Write the swap equations

    60 + 0.5y = 6x and 6y = 30 + 0.5x. From the second equation: y = 5 + x/12.

  3. Step 3: Substitute and solve for x

    60 + 0.5(5 + x/12) = 6x → 62.5 + x/24 = 6x → 62.5 = x(143/24) → x = 1500/143 = 10 70/143 minutes.

  4. Final Answer:

    1:10 70/143 → Option A

  5. Quick Check:

    x ≈ 10.49 min → angles swap accurately (hour and minute positions interchange) ✅
Hint: Set hour(T₂)=minute(T₁) and minute(T₂)=hour(T₁), then use y = 5 + x/12.
Common Mistakes: Using right-angle formulas instead of swap equations.
2. At what time between 2 and 3 o’clock will the hands interchange positions with those they will have between 3 and 4 o’clock?
easy
A. 2:15 135/143
B. 2:21 57/143
C. 2:10 70/143
D. 2:26 122/143

Solution

  1. Step 1: Define the variables

    Let the first time be 2 + x and the swapped time 3 + y. Then:
    Hour(2+x) = 60 + 0.5x; Minute(2+x) = 6x; Hour(3+y) = 90 + 0.5y; Minute(3+y) = 6y.

  2. Step 2: Write the swap equations

    90 + 0.5y = 6x and 6y = 60 + 0.5x. From the second: y = 10 + x/12.

  3. Step 3: Substitute and solve for x

    90 + 0.5(10 + x/12) = 6x → 95 + x/24 = 6x → 95 = x(143/24) → x = 2280/143 = 15 135/143 minutes.

  4. Final Answer:

    2:15 135/143 → Option A

  5. Quick Check:

    x ≈ 15.93 min → both angles align in swap condition ✅
Hint: Use y = 10 + x/12 and solve; memorize x = (780H + 720)/143 formula.
Common Mistakes: Neglecting to convert mixed fractions properly.
3. At what time between 4 and 5 o’clock will the hour and minute hands interchange positions with those they will have between 5 and 6 o’clock?
easy
A. 4:21 20/143
B. 4:26 122/143
C. 4:20 58/143
D. 4:32 44/143

Solution

  1. Step 1: Define the variables

    Let first time = 4 + x and swapped time = 5 + y. Then:
    Hour(4+x)=120+0.5x; Minute(4+x)=6x; Hour(5+y)=150+0.5y; Minute(5+y)=6y.

  2. Step 2: Write the swap equations

    150 + 0.5y = 6x and 6y = 120 + 0.5x. From the second: y = 20 + x/12.

  3. Step 3: Substitute and solve for x

    150 + 0.5(20 + x/12) = 6x → 160 + x/24 = 6x → 160 = x(143/24) → x = 3840/143 = 26 122/143 minutes.

  4. Final Answer:

    4:26 122/143 → Option B

  5. Quick Check:

    x ≈ 26.85 → computed angles confirm swap condition ✅
Hint: Apply x = (780H + 720)/143; here H=4 → x = 3840/143.
Common Mistakes: Swapping H and H+1 incorrectly in equations.
4. At what time between 5 and 6 o’clock will the hands interchange positions with those they will have between 6 and 7 o’clock?
medium
A. 5:20 58/143
B. 5:21 57/143
C. 5:32 44/143
D. 5:27 3/11

Solution

  1. Step 1: Define the variables

    Let time between 5-6 be 5 + x and swapped time between 6-7 be 6 + y. Then:
    Hour(5+x) = 150 + 0.5x; Minute(5+x) = 6x; Hour(6+y) = 180 + 0.5y; Minute(6+y) = 6y.

  2. Step 2: Write the swap equations

    180 + 0.5y = 6x and 6y = 150 + 0.5x. From the second: y = 25 + x/12.

  3. Step 3: Substitute and solve for x

    180 + 0.5(25 + x/12) = 6x → 192.5 + x/24 = 6x → 192.5 = x(143/24) → x = 4620/143 = 32 44/143 minutes.

  4. Final Answer:

    5:32 44/143 → Option C

  5. Quick Check:

    x ≈ 32.31 min → angles at hour(5+x) and minute(6+y) match perfectly ✅
Hint: Use y = 25 + x/12; solve systematically to avoid confusion.
Common Mistakes: Swapping x, y definitions or ignoring fractional conversion.
5. At what time between 6 and 7 o’clock will the hour and minute hands interchange their positions with those between 7 and 8 o’clock?
medium
A. 6:32 44/143
B. 6:37 109/143
C. 6:26 122/143
D. 6:21 57/143

Solution

  1. Step 1: Define the variables

    Let time between 6-7 be 6 + x and swapped time between 7-8 be 7 + y. Then:
    Hour(6+x)=180+0.5x; Minute(6+x)=6x; Hour(7+y)=210+0.5y; Minute(7+y)=6y.

  2. Step 2: Write the swap equations

    210 + 0.5y = 6x and 6y = 180 + 0.5x. From the second: y = 30 + x/12.

  3. Step 3: Substitute and solve for x

    210 + 0.5(30 + x/12) = 6x → 225 + x/24 = 6x → 225 = x(143/24) → x = 5400/143 = 37 109/143 minutes.

  4. Final Answer:

    6:37 109/143 → Option B

  5. Quick Check:

    x ≈ 37.76 min → hour(6+x) and minute(7+y) coincide in swapped positions ✅
Hint: Apply x = (780H + 720)/143 for quick calculation.
Common Mistakes: Rounding fractional minutes prematurely or using wrong substitution.

Mock Test

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