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Clock Running Slow / Fast

Introduction

Slow या fast चलने वाली clocks वाली problems में faulty clock की reading और true time के बीच conversion करना होता है। यह pattern इसलिए ज़रूरी है क्योंकि कई aptitude questions पूछते हैं कि faulty clock कब सही होगी, किसी समय पर clock क्या दिखाएगी या कितने समय में वह दिया हुआ gain या loss कर लेगी।

Pattern: Clock Running Slow / Fast

Pattern

Key concept: Faulty clock को एक steady rate पर चलता हुआ मानें और elapsed time या error को convert करने के लिए direct proportion (या linear interpolation) का उपयोग करें।

Useful forms (इनको intuitive तरीके से याद रखें):

  • Rate form (per-hour): Per hour gain या loss निकालें, फिर required hours पर scale करें:
    Required time में Gain/Loss = (given gain/loss × desired time) ÷ given time.
  • Elapsed-time ratio (simple): Clock अगर uniformly चल रही है, तो clock-interval और true-interval का ratio लेकर scale करें:
    True elapsed = Clock elapsed × (True time interval ÷ Clock time interval).
  • Interpolation (जब दो errors दिए हों): अगर किसी true time T₁ पर error E₁ है और true time T₂ पर error E₂ है, तो error E कब होगा:
    T = T₁ + (E - E₁) × (T₂ - T₁) ÷ (E₂ - E₁).

Step-by-Step Example

Question

एक clock true time 8:00 AM पर 5 minutes slow है, और उसी दिन true time 8:00 PM पर 5 minutes fast है। Clock बिल्कुल कब सही थी?

Solution

  1. Step 1: Readings और errors को साफ़-साफ़ लिखें

    True time 8:00 AM पर faulty clock 7:55 AM दिखाती है → error = -5 minutes (clock पीछे)। True time 8:00 PM पर faulty clock 8:05 PM दिखाती है → error = +5 minutes (clock आगे)。

  2. Step 2: Total error change और time interval निकालें

    Total error change = +5 - (-5) = 10 minutes. Total true-time interval = 8:00 AM → 8:00 PM = 12 hours.

  3. Step 3: Zero error (correctness) का time निकालें - Linear interpolation

    8:00 AM से correctness तक समय = (initial error magnitude ÷ total error change) × total interval = (5 ÷ 10) × 12 hours = 6 hours.

  4. Final Answer:

    2:00 PM

  5. Quick Check:

    6 hours बाद error -5 से +5 के बीच बिल्कुल बीच में (0) होगा। 8:00 AM + 6 hours = 2:00 PM ✅

Quick Variations

1. Clock की reading पर true time निकालना: Clock का elapsed time किसी known reset से लें, उसे rate से true elapsed में बदलें, फिर known true time में जोड़ें।

2. Clock कब सही होगी: दो दिए गए error points के बीच interpolation करें (जैसे example में)।

3. X minutes gain/lose कब होंगे: दिए गए gain/loss को per-hour rate में बदलकर X को उस rate से divide करें।

Trick to Always Use

  • Step 1 → Errors को हमेशा (Clock - True) के रूप में लिखें। Signs: fast = +, slow = -।
  • Step 2 → Scaling के लिए per-hour rates और “कब सही” type problems के लिए interpolation बेहतर रहता है।

Summary

Summary

  • Reference साफ रखें: time true है या clock की shown reading है, यह हमेशा लिखें।
  • Long intervals के लिए proportional scaling (gain/loss per period) का उपयोग करें।
  • दो known errors के बीच linear interpolation करके intermediate time (जैसे error 0) निकालें।
  • Signs ध्यान से रखें: fast होने पर gain जोड़ें, slow होने पर loss subtract करें।

याद रखने लायक example:
अगर clock 8:00 AM पर 5 minutes slow और 8:00 PM पर 5 minutes fast है, तो वह 2:00 PM पर बिल्कुल सही होगी (8:00 AM से 6 hours बाद)।

Practice

(1/5)
1. A clock is 4 minutes slow at 10:00 AM and 4 minutes fast at 10:00 PM the same day. When will the clock be correct?
easy
A. 4:00 PM
B. 3:00 PM
C. 2:00 PM
D. 5:00 PM

Solution

  1. Step 1: Determine total change

    From -4 (slow) to +4 (fast) → total change = 8 minutes.

  2. Step 2: Interval

    10:00 AM → 10:00 PM = 12 hours.

  3. Step 3: Time to correct

    =(4 ÷ 8)×12 = 6 hours after 10:00 AM → 4:00 PM.

  4. Final Answer:

    4:00 PM → Option A

  5. Quick Check:

    After 6 hours, halfway change reached; clock becomes correct ✅
Hint: Take half the total time when slow and fast magnitudes are equal.
Common Mistakes: Mixing up total change and initial error.
2. A clock gains 5 minutes every 10 hours. How much will it gain in 3 days?
easy
A. 30 minutes
B. 35 minutes
C. 36 minutes
D. 40 minutes

Solution

  1. Step 1: Rate

    Gain = 5 min per 10 hours.

  2. Step 2: Total hours

    3 days = 72 hours.

  3. Step 3: Proportion

    Gain = (5×72) ÷ 10 = 36 minutes.

  4. Final Answer:

    36 minutes → Option C

  5. Quick Check:

    72 ÷ 10 = 7.2 intervals → 7.2×5 = 36 ✅
Hint: Multiply gain per given hours by total hours ÷ given hours.
Common Mistakes: Forgetting to convert days into hours.
3. A clock is 6 minutes slow at 6:00 AM and 6 minutes fast at 6:00 PM. At what time is it correct?
easy
A. 12:00 Noon
B. 1:00 PM
C. 2:00 PM
D. 3:00 PM

Solution

  1. Step 1: Change in error

    Total change = +6 - (-6) = 12 minutes.

  2. Step 2: Interval

    6 AM to 6 PM = 12 hours.

  3. Step 3: Time to correct

    =(6 ÷ 12)×12 = 6 hours → 6 AM + 6 hours = 12 Noon.

  4. Final Answer:

    12 Noon → Option A

  5. Quick Check:

    Halfway between -6 and +6 errors → 12 Noon ✅
Hint: When errors are equal in magnitude, correctness occurs at midpoint of time interval.
Common Mistakes: Using the total 12 instead of initial 6 for interpolation fraction.
4. A clock is 3 minutes fast at 9:00 AM on Monday and 9 minutes slow at 9:00 AM on Tuesday. When was it correct?
medium
A. 1:00 PM Monday
B. 3:00 PM Monday
C. 6:00 PM Monday
D. 9:00 PM Monday

Solution

  1. Step 1: Change in error

    Total change = -9 - (+3) = -12 minutes (magnitude 12).

  2. Step 2: Time interval

    9 AM Mon → 9 AM Tue = 24 hours.

  3. Step 3: Time to correct

    =(3 ÷ 12)×24 = 6 hours after 9 AM = 3:00 PM Monday.

  4. Final Answer:

    3:00 PM Monday → Option B

  5. Quick Check:

    Error decreases 12 min in 24h → after 6h error reaches 0 ✅
Hint: Find fraction = |initial error| ÷ total change × total interval.
Common Mistakes: Forgetting that total change includes both directions of error (fast → slow).
5. A clock gains 8 minutes in 16 hours. How many minutes will it gain in 2 days?
medium
A. 18 minutes
B. 20 minutes
C. 36 minutes
D. 24 minutes

Solution

  1. Step 1: Given rate

    8 min gain in 16 hours.

  2. Step 2: Convert total period

    2 days = 48 hours.

  3. Step 3: Proportion

    Gain = (8 × 48) ÷ 16 = 24 minutes.

  4. Final Answer:

    24 minutes → Option D

  5. Quick Check:

    Rate = 0.5 min/hr → 48×0.5 = 24 ✅
Hint: Use gain × (required hours ÷ given hours).
Common Mistakes: Using 24 instead of 48 for 2 days.

Mock Test

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