Correcting a Faulty Clock: A Severe Time-Loss Analysis

In this article, we will delve into a detailed analysis of a faulty clock that loses 20 minutes every hour. By understanding the mechanics of this time loss and calculating the exact point when the clock will show the correct time again, we'll provide a step-by-step guide that you can apply to similar scenarios. We'll also explore an alternative method to determine the time it takes for the clock to show the correct time again, along with a comprehensive day-by-day breakdown of the faulty clock's behavior.

Understanding the Clock Loss Rate

A clock that loses 20 minutes every hour will not only be off by a significant amount but also require a deep understanding of its behavior to find the correct time. For every hour of actual time that passes, the clock only shows 40 minutes. Let's break down the math to see why this happens and how to calculate the correct time again.

Calculating the Clock's Rate of Loss

In one hour of actual time, the clock only shows 40 minutes. This is because 60 minutes (actual time) minus 20 minutes (loss) equals 40 minutes shown on the clock. To express this mathematically, we can say the clock runs at a rate of (frac{40}{60} frac{2}{3}) of the actual time.

Setting Up the Equation for Correct Time

To find the actual time t when the clock will show the correct time, we can set up an equation based on the clock's loss. Let t be the actual time in hours that passes. The time shown on the clock after t hours will be frac{2}{3}t. We want to find t such that the time shown on the clock equals the actual time. When the clock has completed a full cycle of losing time, the equation is:

t - frac{2}{3}t 20k
This simplifies to frac{1}{3}t 20k, which means:

t 60k

This means that for every hour k that passes, the actual time t will be 60 hours.

Finding the Next Correct Time

To find the next time the clock shows the correct time, we set k 1:

t 60 times 1 60 text{ hours}

Calculating the Actual Time

Starting from Thursday at 7:00 AM:

24 hours later: Friday at 7:00 AM Another 24 hours later: Saturday at 7:00 AM 12 more hours: Saturday at 7:00 PM

Thus, the clock will again show the correct time at Saturday at 7:00 PM.

Alternative Method: Day-by-Day Breakdown

An alternative way to look at this problem is to consider how the faulty clock behaves in terms of actual days. Given that the clock loses 20 minutes every hour, which equates to 120 minutes (2 hours) every 24 hours, we can determine how long it takes for the clock to show the correct time again.

Without counting the day the clock was set to the correct time, the faulty watch loses 2 hours every day. Thus, it will take frac{24 text{ hours}}{2 text{ hours per day}} 12 text{ days}end{math} to show the correct time again. However, if you count the day the clock was set to the correct time, it will take 13 days to show the correct time again.

Day-by-Day Analysis

DAY CORRECT TIME FAULTY TIME Mon 6:00am 6:00am 6:00am Tue 6:00am 6:00am 4:00am Wed 6:00am 6:00am 2:00am Thu 6:00am 6:00am 12:00am Fri 6:00am 6:00am 10:00pm Sat 6:00am 6:00am 8:00pm Sun 6:00am 6:00am 6:00pm Mon 6:00am 6:00am 4:00pm Tue 6:00am 6:00am 2:00pm Wed 6:00am 6:00am 12:00pm Thu 6:00am 6:00am 10:00am Fri 6:00am 6:00am 8:00am Sat 6:00am 6:00am 6:00am

In conclusion, whether you use the equation method or the day-by-day breakdown, the faulty clock will show the correct time again at Saturday at 7:00 PM if set correctly at Thursday 7:00 AM. Understanding these calculations can help in troubleshooting and setting up clocks to maintain accuracy.