How Much Time Until Two Clocks Read the Same Time: A Comprehensive Analysis

Imagine you have two clocks: one clock that runs 15 minutes fast every 60 minutes and another that runs 5 minutes slow every 60 minutes. How much time would you have to wait for these two clocks to read the same time again? This article explores the math behind clock synchronization and demonstrates a step-by-step approach to determine the time until the clocks match.

Understanding the Problem

In this scenario, one clock (Clock A) is running fast, gaining 15 minutes every hour, while the other clock (Clock B) is slow, losing 5 minutes every hour. We need to determine the time it will take for these two clocks to show the same time again, starting from the moment the time difference was 0 minutes. This problem involves understanding the rates at which the clocks change and how these rates affect their time differences over time.

Step-by-Step Analysis

Relative Speed

The first step is to calculate the relative speed between the two clocks. Clock A gains 15 minutes in an hour, while Clock B loses 5 minutes in an hour. Therefore, the effective difference in their rates is:

[ text{Relative speed} 15 text{ minutes/hour} - 5 text{ minutes/hour} 20 text{ minutes/hour} ]

This means that for every hour that passes, the difference between the two clocks increases by 20 minutes. Essentially, Clock A is gaining 20 minutes on Clock B for every real hour that passes.

Initial Difference

Starting from the assumption that both clocks were synchronized to the same time, the initial time difference is 0 minutes. This is a crucial point, as it sets the base for our calculation. If there was an initial difference, we would need that specific value to solve the problem more accurately.

Time Until They Match

To find the time until the clocks read the same time again, we use the concept that Clock A is gaining 20 minutes on Clock B per hour. The goal is to determine the time when the difference accumulates to a full 60 minutes, as 60 minutes is equivalent to an hour on a clock.

[ text{Total Difference} 20t ] [ 20t 60 text{ minutes} ] [ t frac{60}{20} 3 text{ hours} ]

Thus, it will take 3 hours for Clock A to have gained 45 minutes (3 hours × 15 minutes/hour) and Clock B to have lost 15 minutes (3 hours × 5 minutes/hour), resulting in a total difference of 60 minutes. At this point, both clocks will read the same time again.

Miscellaneous Considerations

Mathematical Representation

Mathematically, let ( t ) be the time in hours until the clocks read the same time again.

[ text{Clock A} text{ gains: } 15t text{ minutes} ][ text{Clock B} text{ loses: } 5t text{ minutes} ] [ text{Total Difference} 15t - 5t 20t text{ minutes} ] [ 20t 60 text{ minutes} ] [ t 3 text{ hours} ]

This formula confirms that the clocks will be synchronized again after 3 hours.

Conclusion

In summary, it will take exactly 3 hours for the two clocks to read the same time again. During this time, Clock A will have gained 45 minutes, and Clock B will have lost 15 minutes, effectively resetting the 60-minute difference to zero, aligning both clocks' readings.

Understanding this process helps in solving clock synchronization issues, which is useful in various fields such as time management, scheduling, and even in creating more accurate timekeeping systems.