How to solve Work->Break->Work pattern questions?

Question

Machine/Person A: Works for 20 hours and take break for 4 hours (repeated cycle) Machine/Person B: Works for 40 hours and take break for 8 hours (repeated cycle) In X amount of hours ( say x is 24*7= a week ) starting at same time how many days do both machines/person are at break at the same time? Currently I'm solving them by trying out all option but would like to know if there is a faster way to do them. I'm suspecting that you need to find the least common multiple of some sort. Any help is appreciated

Answer 1

For the specific example you gave, you can note that Machine A has a 24-hour cycle, and Machine B has a 48-hour cycle. As such, the cycles will line up again after every 48 hours (this is found by the least common multiple, or LCM, as you noted in your question). While the cycles don't always line up so perfectly, most questions will have them line up at least relatively well for ease of answering. At worst, I would expect something along the lines of a 12-hour cycle and an 8-hour cycle (which would result in them lining up every 24 hours, and thus requiring 2x and 3x individual cycles, respectively. Note that the example you gave requires 2x and 1x individual cycles, and is much easier to work with).

With this in mind, we can find out how often they match within each cycle, apply it to the given duration, and manually solve the remainder if needed. In the given example, we have Machine A with the following (per 48-hour cycle):

1. 00 - 20 ON
2. 20 - 24 OFF
3. 24 - 44 ON
4. 44 - 48 OFF

Additionally, Machine B has the following:

1. 00 - 40 ON
2. 40 - 48 OFF

Looking at it, we can clearly see that for each 48-hour cycle, the machines line up once during breaks - and the synchronized break lasts for 4 hours (44hr to 48hr). If we take a 7-day interval and try to see how many times the breaks synchronized, we would calculate 7 / 2 = 3.5 (7 days total, with 2 day cycles), so it would be 3 full cycles and some extra. This "extra" is an extra half cycle, but regardless of how much extra it is, you would have to calculate this part manually using what you already found out above (it ends up being 0 for our example, because Machine B doesn't take a break from 00 - 24 at all). However, we can take a shortcut on the 3 full cycles (since each full cycle is the same), and simplify work there.

To fully answer your question, the machines are both on break at some point during 3 of the days (1/cycle * 3cycles + 0 extra).