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Wrong Reading Clock

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Introduction

"Wrong reading" clock की problems तब आती हैं जब कोई clock गलत time दिखाती है - या तो वह गलत time पर set की गई होती है (एक fixed offset) या फिर उसकी speed गलत होती है (steady तरीके से gain/lose करती है)। यह pattern आपको सिखाता है कि किस case में आप हैं और कैसे clock की reading से actual time (या इसके उल्टा) निकालना है - या तो एक additive (offset) मॉडल से या proportional (rate) मॉडल से।

Pattern: Wrong Reading Clock

Pattern: Wrong Reading Clock

Key concept: पता करें कि error एक fixed offset है (clock गलत set) या rate error है (clock slow/fast चल रही है)। फिर इनका उपयोग करें:

  • Additive model (fixed offset): अगर किसी समय पर clock X minutes slow/fast दिखती है और clock की rate सही है, तो clock की reading = actual time ± X (constant)। सरल addition/subtraction का प्रयोग करें।
  • Proportional model (rate error): अगर clock steady रूप से gain/lose कर रही है, तो rate factor निकालें k = (clock-minutes elapsed)/(real-minutes elapsed). फिर clock = k × real (या real = clock / k) का उपयोग करें, और elapsed minutes नापने के लिए सामान्य origin (आम तौर पर 12:00) का प्रयोग करें।

Step-by-Step Example

Question

एक clock 4:20 दिखाती है जब actual time 5:00 है। जब clock 9:00 दिखाए, तो actual time क्या होगा?

Solution

  1. Step 1: Fixed offset निकालें

    दिए हुए क्षण पर: Actual = 5:00 (12:00 के बाद 300 minutes)। Clock दिखाती है 4:20 (12:00 के बाद 260 minutes)। Offset = Actual - Clock = 300 - 260 = 40 minutes slow।

  2. Step 2: नए reading पर वही offset apply करें

    जब clock 9:00 दिखाती है → Clock reading = 9:00 = 540 minutes। चूँकि clock 40 minutes slow है, Actual = Clock + 40 = 540 + 40 = 580 minutes।

  3. Step 3: वापस H:M में बदलें

    580 ÷ 60 = 9 hours remainder 40 → Actual time = 9:40।

  4. Final Answer:

    9:40

  5. Quick Check:

    अगर actual 9:40 है, तो clock दिखाएगा 9:40 - 40 = 9:00 → प्रश्न के मुताबिक match करता है। ✅

Quick Variations

1. Proportional (rate) example: अगर actual 5:00 पर clock 4:20 दिखाती है और दोनों ने 12:00 से सही शुरू किया था, तो clock slower चल रही है। Rate निकालें k = ClockElapsed/RealElapsed = 260/300 = 13/15। जब clock 9:00 (540) दिखाए तो real = Clock / k = 540 × (15/13) = 8100/13 minutes → H:M में बदलें। (जब problem बताती/इशारा करती है कि clock लगातार gain/lose कर रही है तो proportional method इस्तेमाल करें।)

2. Gain/Loss per period: अगर clock किसी P hours में G minutes gain करती है, तो उसका rate factor per एक real minute के लिए होगा: ClockMinuteElapsed = (1 + (G ÷ (P×60))) × RealMinuteElapsed। Readings map करने के लिए proportion का उपयोग करें।

3. Fixed-set vs running error: अगर problem केवल एक instantaneous mismatch देती है और gain/lose per hour का कोई उल्लेख नहीं है, तो additive (fixed-offset) मॉडल प्राथमिकता दें जब तक context continuous gain/loss का संकेत न दे।

Trick to Always Use

  • Step 1 → मॉडल चुनें: additive (constant minutes) या proportional (rate)। ऐसे शब्द देखें: “gains 2 min/hour” → proportional; केवल “shows 4:20 when actual 5:00” जैसा वाक्य → additive हो सकता है।
  • Step 2 → गणनाएँ करने के लिए times को एक सामान्य origin (आम तौर पर 12:00) से minutes में बदलें ताकि offsets या ratios सही निकलें।
  • Step 3 → मॉडल लागू करें और परिणाम को H:M में बदलें; हमेशा एक quick check करें और अपने computed actual time को original relation में substitute करके देखें।

Summary

  • Key takeaway 1: अगर clock बस गलत set है (fixed offset), तो clock reading में offset add/subtract करके actual time निकालें।
  • Key takeaway 2: अगर clock steady तरीके से gain/lose कर रही है, तो known timestamp pair से rate factor k = (clock-elapsed)/(real-elapsed) निकालें แล้ว फिर real = clock / k (या उल्टा) लगाएँ।
  • Key takeaway 3: साफ़ arithmetic के लिए हमेशा times को एक सामान्य origin (12:00) से minutes में बदलें, फिर H:M में वापस करें।
  • Key takeaway 4: Quick check में अपने computed actual time को original relation में डालें - इससे model-choice (additive vs proportional) की गलती पकड़ में आ जाती है।

याद रखने के लिए example:
यदि clock 4:20 दिखाती है जब actual 5:00 है → fixed offset 40 min slow → जब clock 9:00 दिखाए actual = 9:40।

Practice

(1/5)
1. A clock shows 3:10 when the actual time is 3:40. When the clock shows 6:00, what is the actual time?
easy
A. 6:30
B. 5:50
C. 6:10
D. 5:40

Solution

  1. Step 1: Convert times to minutes from 12:00

    Clock reading = 3:10 → 190 minutes. Actual time = 3:40 → 220 minutes.

  2. Step 2: Compute fixed offset

    Offset = Actual - Clock = 220 - 190 = 30 minutes slow (clock is 30 minutes behind).

  3. Step 3: Apply offset to new clock reading

    When clock shows 6:00 → 360 minutes. Actual = Clock + 30 = 360 + 30 = 390 minutes.

  4. Step 4: Convert back to H:M

    390 ÷ 60 = 6 hours remainder 30 → 6:30.

  5. Final Answer:

    6:30 → Option A

  6. Quick Check:

    If actual = 6:30, clock shows 6:30 - 30 = 6:00 ✅
Hint: Find actual - clock offset once, then add/subtract it for other readings.
Common Mistakes: Treating the offset as clock - actual instead of actual - clock.
2. A clock gains 4 minutes every hour. How much will it gain in 6 hours?
easy
A. 24 minutes
B. 20 minutes
C. 30 minutes
D. 18 minutes

Solution

  1. Step 1: Understand the rate

    The clock gains 4 minutes per 60 real minutes.

  2. Step 2: Scale proportionally

    Gain in 6 hours = 4 minutes/hour × 6 hours = 24 minutes.

  3. Final Answer:

    24 minutes → Option A

  4. Quick Check:

    After 6 real hours the clock will read 6:24 ahead of true time → gain = 24 min ✅
Hint: Multiply per-hour gain by the number of hours.
Common Mistakes: Using clock-hours instead of real hours when a rate is given (they are the same unit here).
3. A clock loses 2 minutes every hour. If the actual time is 5:00, what time will the slow clock show?
easy
A. 4:52
B. 4:58
C. 4:50
D. 4:40

Solution

  1. Step 1: Compute total loss

    The clock loses 2 minutes per hour → over 5 hours it loses 2 × 5 = 10 minutes.

  2. Step 2: Subtract loss from actual time

    Actual 5:00 → 300 minutes. Clock = Actual - Loss = 300 - 10 = 290 minutes.

  3. Step 3: Convert back to H:M

    290 ÷ 60 = 4 hours remainder 50 → 4:50.

  4. Final Answer:

    4:50 → Option C

  5. Quick Check:

    Clock shows 4:50 while real is 5:00 → 10 minutes behind, matches 2 min/hr × 5 hr ✅
Hint: Total loss = rate × time; subtract from real time for the displayed time.
Common Mistakes: Applying loss to clock-time instead of actual-time baseline.
4. A clock (started correct at 12:00) reads 4:30 when the actual time is 5:00. If it continues running at the same rate, when the clock reads 9:00 what will the actual time be?
medium
A. 8:30
B. 9:00
C. 9:30
D. 10:00

Solution

  1. Step 1: Compute elapsed minutes

    Clock elapsed until the observed instant = 4:30 → 270 minutes. Real elapsed = 5:00 → 300 minutes.

  2. Step 2: Find rate factor

    k = ClockElapsed / RealElapsed = 270 / 300 = 9/10.

  3. Step 3: When clock shows 9:00 (540 minutes elapsed), compute real elapsed

    Real = Clock / k = 540 ÷ (9/10) = 540 × 10/9 = 600 minutes.

  4. Step 4: Convert back to H:M

    600 ÷ 60 = 10 hours → 10:00.

  5. Final Answer:

    10:00 → Option D

  6. Quick Check:

    Clock runs slow (9/10); by 9:00 clock, real = 600 min = 10:00, consistent with ratio ✅
Hint: Find k = clock/real using a known pair, then real = clock / k for any future reading.
Common Mistakes: Using addition/subtraction when a proportional rate model applies.
5. A clock is 10 minutes slow at 6:00 a.m. and 14 minutes fast at 6:00 p.m. When (between 6 a.m. and 6 p.m.) was the clock showing correct time?
medium
A. 11:00 a.m.
B. 12:00 p.m.
C. 2:00 p.m.
D. 10:00 a.m.

Solution

  1. Step 1: Compute total change in error

    From 10 minutes slow to 14 minutes fast → total change = 14 - (-10) = 24 minutes over 12 hours (6 a.m. → 6 p.m.).

  2. Step 2: Time to correct from 6:00 a.m.

    Time to cover the initial 10-minute error = (Initial error ÷ Total change) × Total period = (10 ÷ 24) × 12 hours = (5/12)×12 = 5 hours.

  3. Step 3: Add to 6:00 a.m.

    6:00 a.m. + 5 hours = 11:00 a.m. (the clock is correct at this time).

  4. Final Answer:

    11:00 a.m. → Option A

  5. Quick Check:

    After 5 hours the error changes by (5/12)×24 = 10 minutes → initial -10 + 10 = 0 → correct at 11:00 a.m. ✅
Hint: Interpolate proportionally: time to fix = (initial error ÷ total change) × total interval.
Common Mistakes: Using absolute times instead of proportional interpolation across the interval.