Francois, I think you should consider posting the answer choices when you post your problem solving Qs for the reasons below.
1) Some people want to use the questions posted and the clock tool to test themselves as if they were on the GMAT. With no answer choices, one cannot simulate a GMAT question.
2) By hiding the answer choices, you force every solution provided to be an algebraic solution. On the GMAT however, the algebraic solution is not always the most efficient solution. Sometimes plug-in, reverse engineering, ballparking... are better option than the traditional solution. Since you do not post answer choices, we cannot always tell you the best way to approach a question.
That said, below is the solution
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Concept 1: Rate and time it takes to complete 1 job are inverses.
Concept 2: Combined rate is the sum of individual rates.
Concept 3: work = rate * time --> t=w/r
Since Jose and Jane take 20 days to complete 1 job, their combined rate is the inverse, so Jose_rate + Jane_rate = 1/20. Thus we can say that Jose's rate is 'r', and Jane's rate is (1/20 - r)
Jose does half the job. time = work/rate, so the time it took him is (1/2)/r = (1/2r)
Jane does the other half of the job. The time it took her (work/rate) is (1/2)/(1/20 - r) = 10/(1-20r)
We know that the total time is 45 days, so the sum of these times is 45: (1/2r) + 10/(1-20r) = 45. We can solve for r, Jose's rate. You will find that r = 1/30 or 1/60
Since we know that Jane is more efficient, Jose's rate will be the lower rate, or 1/60.
According to concept 1, if Jose's rate is 1/60, the time it would take him to do the job is 60 days. This is the answer
Hope that helps,
-Patrick
