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Multi-Clock Comparison

Introduction

दो या उससे अधिक clocks की तुलना वाले प्रश्न पूछते हैं कि जब हर clock अलग rate से चले (gain/lose कर रही हो) तो उनकी readings समय के साथ कैसे अलग होंगी या कब मेल खाएँगी। यह pattern महत्वपूर्ण है क्योंकि कई वास्तविक और टेस्ट वाले सवालों में relative rates के बारे में सोचना पड़ता है और gains/losses को time intervals में बदलना पड़ता है।

Pattern: Multi-Clock Comparison

Pattern

प्रत्येक clock के error को एक सामान्य rate में बदलें (minutes gained या lost प्रति unit time), clocks के बीच relative rate निकालें, और फिर यह relation उपयोग करें:
Time to achieve required difference = Required difference in minutes ÷ Relative rate (minutes per unit time).

उपयोगी रूप (Useful forms):

  • यदि Clock A gA min/hr gain करता है और Clock B gB min/hr gain करता है, तो relative rate = |gA - gB| min/hr.
  • अगर gains अलग-अलग intervals में दिए हों: दोनों को एक ही unit (जैसे प्रति hour या प्रति day) में scale करें।

Step-by-Step Example

Question

दो clocks, A और B, noon 12:00 पर एक ही time पर set किए गए। Clock A हर hour में 2 minutes gain करता है। Clock B हर hour में 1 minute lose करता है। noon से कितने समय बाद (from noon) दोनों clocks की readings ठीक 1 hour (60 minutes) से अलग होंगी?

Solution

  1. Step 1: प्रत्येक clock के rate को एकเดียว unit में बदलें

    Clock A gains 2 minutes per hour.
    Clock B loses 1 minute per hour (इसे equivalently gain = -1 min/hr माना जा सकता है)।

  2. Step 2: Relative rate निकालें

    Relative rate = |gain of A - gain of B| = |2 - (-1)| = 3 minutes per hour।

  3. Step 3: Required difference ÷ relative rate लागू करें

    Required difference = 60 minutes।
    Time = 60 ÷ 3 = 20 hours।

  4. Final Answer:

    20 hours

  5. Quick Check:

    20 hours में Clock A कुल 20×2 = 40 minutes gain करेगा; Clock B कुल 20×1 = 20 minutes lose करेगा; difference = 40 - (-20) = 60 minutes ✅

Quick Variations

1. एक clock प्रति day gain देता है और दूसरा प्रति hour - दोनों को पहले same unit (जैसे minutes/hour) में बदलें।

2. अगर किसी fast clock को फिर से सही होने में कितना समय लगेगा पता करना हो और वह अब x minutes fast है → time = x ÷ (gain per unit time) (true clock के सापेक्ष)।

3. दो से ज़्यादा clocks की तुलना करने पर pairwise relative rates या एक reference (true) clock के सापेक्ष सभी clocks की तुलना करें।

Trick to Always Use

  • Step 1 → सभी gains/losses को एक ही time unit में बदलें (आमतौर पर minutes per hour या minutes per day)।
  • Step 2 → Relative rate निकालें (rates का absolute difference) और required minute-difference को उस rate से divide करें।

Summary

Summary

  • Clock errors को तुलना से पहले एक सामान्य rate (उदा., minutes/hour) में बदलें।
  • Relative rate = व्यक्तिगत rates का absolute difference; इसे required difference का समय निकालने के लिए उपयोग करें।
  • Units को पहले normalize (hours ↔ minutes) कर लें ताकि arithmetic त्रुटियाँ न हों।
  • Quick check के लिए निकले हुए time पर हर clock का total gain/loss फिर से निकालें और बताया गया difference verify करें।

याद रखने के लिए example:
यदि A हर hour में 2 min gain करता है और B हर hour में 1 min lose करता है, तो 60-min difference के लिए समय = 60 ÷ (2 - (-1)) = 60 ÷ 3 = 20 hours।

Practice

(1/5)
1. Two clocks A and B are set to the same time at 12:00 noon. Clock A gains 3 minutes every hour, and Clock B loses 2 minutes every hour. After how many hours will the difference between the two clocks be 1 hour?
easy
A. 12 hours
B. 15 hours
C. 20 hours
D. 10 hours

Solution

  1. Step 1: Define the variables

    Clock A gains +3 min/hr. Clock B loses -2 min/hr (equivalently gains -2 min/hr).

  2. Step 2: Compute relative rate

    Relative rate = |3 - (-2)| = 5 minutes per hour.

  3. Step 3: Compute required time

    Required difference = 60 minutes → Time = 60 ÷ 5 = 12 hours.

  4. Final Answer:

    12 hours → Option A

  5. Quick Check:

    After 12 hr, A gains 36 min and B loses 24 min → difference = 36 - (-24) = 60 min ✅
Hint: Add gains (treat losses as negative gains) to get relative rate; divide required minutes by that rate.
Common Mistakes: Forgetting to treat a loss as negative gain or using wrong sign when computing relative rate.
2. Two clocks show the same time at 9 a.m. Clock A gains 1 minute in 2 hours, while Clock B loses 1 minute in 3 hours. When will they differ by 10 minutes?
easy
A. 12 hours
B. 20 hours
C. 30 hours
D. 36 hours

Solution

  1. Step 1: Define the variables

    Clock A rate = +1 minute per 2 hours = +1/2 = 0.5 min/hr. Clock B rate = -1 minute per 3 hours = -1/3 ≈ -0.3333 min/hr.

  2. Step 2: Compute relative rate

    Relative rate = 0.5 - (-1/3) = 0.5 + 0.333333… = 5/6 min/hr.

  3. Step 3: Compute time for 10-min difference

    Time = 10 ÷ (5/6) = 10 × 6/5 = 12 hours.

  4. Final Answer:

    12 hours → Option A

  5. Quick Check:

    Per hour difference grows by 5/6 min → in 12 hours difference = 12 × 5/6 = 10 min ✅
Hint: Convert both rates to min/hr first, then divide required minutes by their difference.
Common Mistakes: Using rates in different units without converting to a common unit.
3. Clock A is 5 minutes fast and gains 2 minutes per hour. Clock B is 10 minutes slow and loses 1 minute per hour. How long after they are set will the difference be 1 hour (60 minutes)?
easy
A. 15 hours
B. 18 hours
C. 20 hours
D. 25 hours

Solution

  1. Step 1: Define the variables

    Initial difference = A is 5 min fast and B is 10 min slow → initial difference = 5 + 10 = 15 minutes (A ahead of B).

  2. Step 2: Compute relative rate

    A gains +2 min/hr; B loses -1 min/hr → relative rate = |2 - (-1)| = 3 min/hr.

  3. Step 3: Compute additional time needed

    We need total difference 60 min, so additional required = 60 - 15 = 45 minutes.
    Time = 45 ÷ 3 = 15 hours.

  4. Final Answer:

    15 hours → Option A

  5. Quick Check:

    After 15 hr A gains 30 min, B loses 15 → net change = 45; initial 15 + 45 = 60 min ✅
Hint: Subtract initial offset from target, then divide by relative rate.
Common Mistakes: Failing to include the initial offset (existing difference) before computing time.
4. Two clocks are set together at midnight. Clock A gains 5 minutes in 12 hours, while Clock B loses 3 minutes in 8 hours. After how many hours will they show a 30-minute difference?
medium
A. 36 hours 59 minutes
B. 40 hours 28 minutes
C. 37 hours 53 minutes
D. 48 hours 11 minutes

Solution

  1. Step 1: Define the variables

    Clock A: 5 minutes in 12 hr → rate = 5/12 min/hr. Clock B: -3 minutes in 8 hr → rate = -3/8 min/hr.

  2. Step 2: Compute relative rate

    Relative rate = (5/12) - (-3/8) = 5/12 + 3/8 = (10 + 9)/24 = 19/24 min/hr.

  3. Step 3: Compute time for 30-min difference

    Time = 30 ÷ (19/24) = 30 × 24 / 19 = 720/19 hours = 37 17/19 hours (exact). Converting exactly: 17/19 hour = 53 minutes 41.05 seconds, so the time ≈ 37 hours 53 minutes 41 seconds (≈ 37 hours 53 minutes when rounded to minutes).

  4. Final Answer:

    37 hours 53 minutes → Option C

  5. Quick Check:

    (720/19) × (19/24) = 30 minutes - exact difference achieved ✅
Hint: Convert the interval gains/losses to per-hour, find the relative rate, then divide required minutes by that rate.
Common Mistakes: Mixing fraction and decimal formats or rounding too early in the calculation.
5. Clock A is 8 minutes fast and gains 4 minutes per hour. Clock B is 2 minutes slow and loses 2 minutes per hour. When will their readings differ by exactly 1 hour?
medium
A. 7 hours 20 minutes
B. 9 hours
C. 10 hours
D. 8 hours 20 minutes

Solution

  1. Step 1: Define the variables

    Initial difference = 8 + 2 = 10 minutes (A ahead of B). Rates: A = +4 min/hr, B = -2 min/hr.

  2. Step 2: Compute relative rate

    Relative rate = 4 - (-2) = 6 min/hr.

  3. Step 3: Compute additional time needed

    Needed additional = 60 - 10 = 50 minutes.
    Time = 50 ÷ 6 = 25/3 hours = 8 hours 20 minutes.

  4. Final Answer:

    8 hours 20 minutes → Option D

  5. Quick Check:

    After 8h20m, A gains 33.333… min, B loses 16.666… min → net gain ≈ 50 min; initial 10 + 50 = 60 min ✅
Hint: Subtract the initial offset from 60, divide by combined rate; convert fractional hours to minutes.
Common Mistakes: Rounding too early or using integer-hour answers when fraction is exact.

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