# A Freeloader

I borrowed the following problem from Alexander Borovik's *Mathematics under the Microscope*

It takes two hours for Tom and Dick to do a job. Tom and Harry take three hours to do the same job. Dick and Harry take six hours for the job. Prove that Harry is a freeloader.

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Copyright © 1996-2018 Alexander Bogomolny

### Solution

It takes two hours for Tom and Dick to do a job. Tom and Harry take three hours to do the same job. Dick and Harry take six hours for the job. Prove that Harry is a freeloader.

We'll see how much can the fellow do in 6 hours, 6 being the Least Common Multiple of the three given durations - 2, 3, 6. In 6 hours, Tom and Dick can do 3 jobs,1 job,2 jobs,3 jobs,6 jobs, Tom and Harry 2 jobs,1 job,2 jobs,3 jobs,6 jobs, and Dick and Harry 1 job,1 job,2 jobs,3 jobs,6 jobs. Looking at the last two pieces of information, and imagining there are two copies of Harry, the team that consisted of Tom and Harry(#1) and Dick and Harry(#2), would, in 6 hours, accomplish 2 + 1 = 3 jobs, exactly as much as Tom and Dick would do without Harry's help. It follows that the result of Harry's effort's is zero, nada, and the caption catchers well on his function - a *freeloader*.

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