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Gain or Loss of Time

Introduction

Many clock problems ask how much a faulty clock gains or loses over a period, or how long it will take to gain/lose a certain amount. This pattern is important because it uses simple proportional reasoning to convert a known gain/loss over one interval into any other interval.

Pattern: Gain or Loss of Time

Pattern

Key concept: Use direct proportion - if a clock gains (or loses) G minutes in T hours, then in t hours it gains (or loses) (G × t) ÷ T minutes.

Two helpful forms:

  • Gain/Loss in required time = (Given gain or loss) × (desired time) ÷ (given time period).
  • Time to gain/lose X minutes = (Given time period) × X ÷ (given gain or loss).

Step-by-Step Example

Question

A clock gains 2 minutes every 6 hours. How much will it gain in 24 hours?

Solution

  1. Step 1: Identify given rate

    The clock gains 2 minutes in 6 hours.

  2. Step 2: Write proportional relation

    Gain in 24 hours = (Given gain × Desired time) ÷ Given time = (2 × 24) ÷ 6.

  3. Step 3: Compute

    (2 × 24) ÷ 6 = 48 ÷ 6 = 8 minutes.

  4. Final Answer:

    8 minutes

  5. Quick Check:

    6 hours → 2 min, so 24 hours is 4 × 6 hours → 4 × 2 min = 8 min ✅

Quick Variations

1. Clock loses time instead of gains - treat loss as negative gain and apply same proportion.

2. Given rate per hour (e.g., 0.333… min/hr) - multiply by desired hours.

3. Find when a clock will be correct: combine gain/loss with known offsets to find the moment of correctness.

Trick to Always Use

  • Step 1 → Convert the given gain/loss into a per-hour rate if it makes calculations easier (G ÷ T).
  • Step 2 → Multiply the per-hour rate by required hours (or invert the relation to find hours needed for a given gain/loss).

Summary

Summary

  • Convert the given gain/loss into a rate (minutes per hour) when helpful.
  • Use direct proportion: (Given gain × desired time) ÷ given time.
  • To find time for a required gain/loss: (Given time × required gain) ÷ given gain.
  • Always check sign (gain = +, loss = -) and units (hours, minutes) before finalising.

Example to remember:
A clock that gains 2 minutes in 6 hours will gain 8 minutes in 24 hours (2 × 24 ÷ 6 = 8).

Practice

(1/5)
1. A clock loses 3 minutes every 12 hours. How much will it lose in 30 days?
easy
A. 3 hours (180 minutes)
B. 2 hours (120 minutes)
C. 4 hours (240 minutes)
D. 90 minutes

Solution

  1. Step 1: Identify given rate

    The clock loses 3 minutes in 12 hours.

  2. Step 2: Convert to per-hour rate

    Loss per hour = 3 ÷ 12 = 0.25 minutes/hour.

  3. Step 3: Compute for 30 days

    30 days = 30 × 24 = 720 hours → Total loss = 0.25 × 720 = 180 minutes = 3 hours.

  4. Final Answer:

    3 hours (180 minutes) → Option A

  5. Quick Check:

    Every 12 hours → 3 min, so in 720 hours there are 60 such 12-hour blocks: 60×3 = 180 min ✅
Hint: Convert the given period to hours, find per-hour rate, multiply by required hours.
Common Mistakes: Forgetting to convert days to hours or mixing minutes/hr units.
2. A clock gains 5 minutes every 24 hours. How many days will it take to gain 1 hour?
easy
A. 10 days
B. 12 days
C. 15 days
D. 8 days

Solution

  1. Step 1: Identify given rate

    The clock gains 5 minutes in 24 hours (1 day).

  2. Step 2: Use proportional relation

    Time to gain X minutes = (Given time × X) ÷ (Given gain). Here X = 60 minutes.

  3. Step 3: Compute

    Days = (1 day × 60) ÷ 5 = 60 ÷ 5 = 12 days.

  4. Final Answer:

    12 days → Option B

  5. Quick Check:

    5 min/day → in 12 days gain = 12×5 = 60 min = 1 hour ✅
Hint: Use (given time × required gain) ÷ given gain to get needed time.
Common Mistakes: Confusing minutes with hours or forgetting to scale by days.
3. A clock loses 2 minutes every 8 hours. If it is set correct at 6:00 AM, when will it be 10 minutes slow?
easy
A. 8:00 PM the same day
B. 6:00 AM the next day
C. 10:00 PM the next day
D. 4:00 PM the next day

Solution

  1. Step 1: Identify given rate

    The clock loses 2 minutes in 8 hours.

  2. Step 2: Time to lose 10 minutes

    Using proportion: Time = (Given time × Required loss) ÷ Given loss = (8 hours × 10) ÷ 2 = (80) ÷ 2 = 40 hours.

  3. Step 3: Add to starting time

    6:00 AM + 40 hours = 6:00 AM next day + 16 hours = 10:00 PM the next day.

  4. Final Answer:

    10:00 PM the next day → Option C

  5. Quick Check:

    40 hours contains five 8-hour blocks → 5×2 = 10 minutes lost ✅
Hint: Compute number of given-period blocks in required time and multiply the per-block gain/loss.
Common Mistakes: Adding 40 hours incorrectly across days or mixing AM/PM.
4. A slow clock shows 3:00 PM when the correct time is 3:30 PM. Later the clock reads 7:00 PM. What is the true time then?
medium
A. 7:30 PM
B. 8:00 PM
C. 7:50 PM
D. 8:10 PM

Solution

  1. Step 1: Find rate ratio

    When clock shows 3:00, true time = 3:30. So for every 3 hours shown by the clock, real time = 3.5 hours. Rate factor = real/clock = 3.5/3 = 7/6.

  2. Step 2: Find elapsed shown hours

    Clock later reads 7:00 - that is 4 hours after 3:00 by the clock.

  3. Step 3: Convert to real elapsed time

    Real elapsed = 4 × (7/6) = 28/6 = 4 2/3 hours = 4 hours 40 minutes.

  4. Step 4: Add to true time at 3:00 reading

    True time at clock 3:00 was 3:30 PM. Add 4:40 → 3:30 + 4:40 = 8:10 PM.

  5. Final Answer:

    8:10 PM → Option D

  6. Quick Check:

    Clock advanced 4 hours → real advanced 4h40m; 3:30 + 4h40m = 8:10 ✅
Hint: Use ratio real/clock = (true elapsed)÷(clock elapsed) to scale future clock readings.
Common Mistakes: Using absolute readings instead of elapsed-time ratios.
5. A clock gains 10 minutes in 5 hours. It is set correct at 12:00 noon. What will it read when the true time is 8:00 PM the same day?
medium
A. 8:16 PM
B. 8:20 PM
C. 8:10 PM
D. 8:24 PM

Solution

  1. Step 1: Identify gain rate

    Clock gains 10 minutes in 5 hours → gain rate = 10 ÷ 5 = 2 minutes per hour.

  2. Step 2: Time interval

    From 12:00 noon to 8:00 PM = 8 hours.

  3. Step 3: Compute total gain

    Total gain = 2 × 8 = 16 minutes.

  4. Step 4: Clock reading

    True time = 8:00 PM, clock is fast by 16 minutes → clock reads 8:16 PM.

  5. Final Answer:

    8:16 PM → Option A

  6. Quick Check:

    2 min/hr × 8 hr = 16 min gain → 8:00 + 0:16 = 8:16 ✅
Hint: Multiply per-hour gain/loss by hours elapsed, then add/subtract from true time.
Common Mistakes: Forgetting to apply gain as + to the clock reading (fast) or using wrong sign.

Mock Test

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