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Multi-Clock Comparison

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Introduction

Problems that compare two or more clocks ask how their readings diverge or coincide over time when each clock runs at a different rate (gaining or losing). This pattern is important because many real-world and test questions require reasoning about relative rates and converting gains/losses into time intervals.

Pattern: Multi-Clock Comparison

Pattern: Multi-Clock Comparison

Convert each clock’s error into a common rate (minutes gained or lost per unit time), compute the relative rate between clocks, then use the relation:
Time to achieve required difference = Required difference in minutes ÷ Relative rate (minutes per unit time).

Useful forms:

  • If Clock A gains gA min/hr and Clock B gains gB min/hr, relative rate = |gA - gB| min/hr.
  • To convert gains given over different intervals: scale to same unit (e.g., per hour or per day).

Step-by-Step Example

Question

Two clocks, A and B, are set to the same time at 12:00 noon. Clock A gains 2 minutes every hour. Clock B loses 1 minute every hour. After how long (from noon) will the readings on the two clocks differ by exactly 1 hour (60 minutes)?

Solution

  1. Step 1: Convert each clock’s rate to a common unit

    Clock A gains 2 minutes per hour.
    Clock B loses 1 minute per hour (equivalently, gains -1 min/hr).

  2. Step 2: Compute relative rate

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

  3. Step 3: Use required difference ÷ relative rate

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

  4. Final Answer:

    20 hours

  5. Quick Check:

    In 20 hours Clock A gains 20×2 = 40 minutes; Clock B loses 20×1 = 20 minutes; difference = 40 - (-20) = 60 minutes ✅

Quick Variations

1. One clock gains per day and another per hour → scale both to the same unit (e.g., minutes per hour).

2. Find when a fast clock will be correct again given it is fast by x minutes now → use time = x ÷ (gain per unit time) if compared to true time.

3. Compare more than two clocks by using pairwise relative rates or comparing all to a reference (true) clock.

Trick to Always Use

  • Step 1 → Convert all gains/losses to the same time unit (usually minutes per hour or minutes per day).
  • Step 2 → Compute relative rate (absolute difference of rates) and divide the required minute-difference by this rate.

Summary

  • Convert clock errors to a common rate (e.g., minutes/hour) before any comparison.
  • Relative rate = absolute difference of individual rates; use it to find time for a required difference.
  • Normalize units (hours ↔ minutes) early to avoid arithmetic errors.
  • Quick check by recomputing each clock’s total gain/loss over the found time and verifying the stated difference.

Example to remember:
If A gains 2 min/hr and B loses 1 min/hr, time for 60-min difference = 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.