Solving the Two Clocks Problem: An SEO-Optimized Guide

This classic problem from the domain of speed-time-distance can seem complex at first, but with the right approach, it becomes straightforward. In this article, we will explore how to solve the problem where one clock gains a minute each day and the other loses a minute each day, and determine when they will next display the same time.

Understanding the Problem

Imagine you have two clocks, both set to 12:00 and started running. One clock gains one minute each day, while the other loses one minute each day. The challenge is to understand when these two clocks will next show the same time. This problem is often referred to as the 'two clocks problem' and is a great way to illustrate concepts from speed, time, and distance.

Solution 1: LCM Method

To solve this, we can use the concept of the Least Common Multiple (LCM).

Consider the following:

Clock 1: Gains 1 minute per day.

24 hours 1440 minutes.

Day to gain 10 minutes 10 minutes / (1 minute/day) 10 days.

Number of days to gain 2460 minutes (1440 minutes for a full clock cycle):

2460 minutes / 10 minutes/day  246 days.

Clock 2: Loses 1 minute per day.

24 hours 1440 minutes.

Day to lose 5 minutes 5 minutes / (1 minute/day) 5 days.

Number of days to lose 2460 minutes (1440 minutes for a full clock cycle):

2460 minutes / 5 minutes/day  2460/5  2460/5  492 days.

We need to find the LCM of 246 and 492. LCM(246, 492) 48, which is the number of days after which both clocks will show the exact time.

Solution 2: Relative Speed Method

We can also treat this problem as a speed-distance problem. Here's the step-by-step reasoning:

Relative Speed: Since the two clocks are moving in opposite directions, their relative speed is the sum of their individual speeds:

Relative speed 1 minute/day (gain) 1 minute/day (loss) 2 minutes/day.

24 hours 1440 minutes. The total distance they need to cover to be 12 hours apart is 720 minutes (12 hours * 60 minutes/hour).

Time Calculation: Time taken Distance / Relative Speed:

Time  720 minutes / 2 minutes/day  360 days / 7.5 days/week  48 hours.

Solution 3: Direct Reasoning

Let's use a more intuitive approach:

Differential Time: The difference between the two clocks accumulates at a rate of 15 minutes per hour. This is because one clock gains 10 minutes and the other loses 5 minutes, resulting in a net gain of 15 minutes per hour.

Total Time to Match: To match the 12-hour clock, they need to have a 12-hour difference, which is 720 minutes (12 * 60).

Time required 720 minutes / 15 minutes/hour 48 hours.

Conclusion

Based on the above solutions, the two clocks will next show the same time 48 hours after they are started. This corresponds to two full days. During this period, the first clock will show 6:00 PM, and the second clock will also show 6:00 PM, although the actual time in the real world would be 10:00 AM.

It's important to note that the 'real world' time is not a factor in this problem, as the question specifically asks for the time when the clocks will match, which can be determined independently of the actual time.

To summarize, the two clocks will show the same time after 48 hours, which is 2 full days, regardless of the time of day or AM/PM distinctions.