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The problem is:

It takes Danny 18 hours to complete a job alone, Ian 24 hours alone, and Roy 30 hours alone. If the three people work in the following seqence: Danny => Ian => Roy => Ian => Roy => Danny => Roy => Danny => Ian. Each person works for 1 hour in a session. This schedule repeats every 9 turns until the job is done. At the end, how many hours did Danny, Ian and Roy work when the job is done respectively?

I have only thought of a brute-force approach to this problem, which is rather slow.

Any other way to solve this problem?

Edit: I used some hints given in the comments, below is my work:

Each nine-turn round would give us

$\frac{20+15+12}{360}\times3 = \frac{141}{360}$ of the total work if the whole job is "one"

Then after two rounds of work, we would have $\frac{78}{360}$ of the total work left to be done. Now all three workers have 6 hours of work time.

Now note Danny gets $\frac{20}{360}$ or total work done each turn(per hour), Ian gets $\frac{15}{360}$ and Roy gets $\frac{12}{360}$

Cycle from the start. After three turns(hours) we'd have $\frac{31}{360}$ more. All three workers have 7 hours of work time

Now after the next person Ian we'd have $\frac{16}{360}$ more...

Next is Roy, after him is $\frac{4}{360}$ more.

Lastly, it only takes Danny 4/20 = 0.2 hours to finish off all the work.

So we have it! 8 hours for Ian and Roy while Danny works for $7+0.2 = 7.2$ hours

Cyh1368
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    Which contest is this for? Also, is this from an on-going contest? – Soham Konar Aug 03 '20 at 14:01
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    Hint: think about what proportion of the job each of them gets done in an hour. For example, if it takes me $x$ hours to do a job, I can get $\frac{1}{x}$ of the job done in 1 hour. – Soham Konar Aug 03 '20 at 14:04
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    I'd first figure out how much work the triple gets done in a $3$ hour cycle. Then figure out how many who cycles are needed, and adjust the stub. – lulu Aug 03 '20 at 14:10
  • @SohamKonar No worries, it's not on-going. This is a 7th grader WMI competition problem a few years ago. – Cyh1368 Aug 04 '20 at 07:54
  • @Cyh1368 Have you gotten it yet? I can post my solution if you want. – Soham Konar Aug 04 '20 at 21:46
  • @SohamKonar I have used some hints given by you guys and got a solution: 8, $7\frac{11}{24}$, 7 hours for Danny, Ian and Roy respectively. But that wasn't the answer given by the contest officials (7.2, 8, 8 hours respectively) – Cyh1368 Aug 05 '20 at 11:28
  • Maybe you're looking at the wrong cycle because there are three different cycles (D-I-R, I-R-D, R-D-I) and that would affect the end of the answer and judging from your answer it looks like you did choose the wrong cycle but I can also tell that you are definitely on the right path. – Soham Konar Aug 05 '20 at 14:25
  • @SohamKonar I checked my original solution and I found that I mis-calculated the work hours and the cycle. I have posted my solution in the post, which got the right answer. – Cyh1368 Aug 06 '20 at 10:33
  • @Cyh1368 Well done! That's exactly how I did it :) – Soham Konar Aug 06 '20 at 13:19

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