Solving the Clock Problem: Tricks and Techniques for Calculating Time Differences
Have you ever encountered the challenge of figuring out what time two differently adjusted clocks will both show after a specified period? This intriguing problem involves understanding the rates at which two clocks gain or lose time over a specific duration. Let's explore this question: Two clocks are set at 10 o'clock in the morning. One clock gains 5 minutes every hour, while the other loses 3 minutes every hour. What will be the time shown by the two clocks at 3:30 in the afternoon?
Understanding Time Elapsed
To solve this problem, we must first calculate the total time that has passed from 10:00 AM to 3:30 PM. This period is 5 hours and 30 minutes, which can be expressed as 5.5 hours.
Calculating the Gain and Loss for Each Clock
Clock That Gains Time
The first clock gains 5 minutes every hour. To find out how much time it gains in 5.5 hours, we multiply the rate of gain by the number of hours:
[ text{Total Gain} 5 text{ minutes/hour} times 5.5 text{ hours} 27.5 text{ minutes} ]
Starting from 10:00 AM, by 3:30 PM, the first clock will have gained 27.5 minutes. Therefore, the time displayed on this clock will be:
[ 10:00 text{ AM} 5 text{ hours} 27.5 text{ minutes} 3:27.5 text{ PM} ] or 3:28 PM (rounded to the nearest minute).
Clock That Loses Time
The second clock loses 3 minutes every hour. To find out how much time it loses in 5.5 hours, we multiply the rate of loss by the number of hours:
[ text{Total Loss} 3 text{ minutes/hour} times 5.5 text{ hours} 16.5 text{ minutes} ]
Starting from 10:00 AM, by 3:30 PM, the second clock will have lost 16.5 minutes. Therefore, the time displayed on this clock will be:
[ 10:00 text{ AM} 5 text{ hours} - 16.5 text{ minutes} 3:30 text{ PM} - 16.5 text{ minutes} 2:43.5 text{ PM} ] or 2:44 PM (rounded to the nearest minute).
Summary of the Problem and Solution
In summary, at 3:30 PM:
The first clock that gains time will show approximately 3:28 PM. The second clock that loses time will show approximately 2:44 PM.Additional Examples and Applications
Understanding how to calculate time differences between clocks with different adjustment rates can be applied in various real-world scenarios. For example:
Quick Solutions: Short Computation Methods
Using a more streamlined method, we can break down the time differences into smaller segments. For the first clock:
Over the 5.5 hours:
5 hours and 30 minutes equals 5 * 5 2.5 27.5 minutes (since 30 minutes is half an hour and thus 2.5 minutes of gain).Thus, the first clock will show 3:27.5 PM (or 3:28 PM when rounded).
For the second clock:
5 hours and 30 minutes equals 5 * 3 1.5 16.5 minutes (since 30 minutes is half an hour and thus 1.5 minutes of loss).Therefore, the second clock will show 2:43.5 PM (or 2:44 PM when rounded).
These examples demonstrate the importance of understanding how to adjust time calculations accurately, ensuring that your solutions are precise and efficient.
Conclusion
By mastering these techniques, you can solve complex clock problems quickly and accurately. Whether in academic settings or real-world applications, understanding how to calculate time differences with different rates of adjustment is a valuable skill.