Solve Tricky Work Problem: Find Combined Effort Time

[vin300]

1. A works alone, takes 4 days more to complete the job than if both A and B do together. If B works alone, he takes 16 more days than the situation of working together. The question asks to find how many days it takes for the combined effort, which being apparently simple has unfortunately made me crazy.2. Eq: m+ 4 n = 0.25

The Attempt at a Solution

: Let m part of the job be completed per day when A works alone, in four days he completes 4m. If B completes n per day, he needs 16n extra work to be done[/B]

 

[HallsofIvy]

When several people work together (or pipes fill or empty a tank, etc.) their rates add. Taking "x" to be the time it takes A to do the job alone and "y" to be the time it takes B to finish the job alone, then when A works alone, his rate is 1/x, when B works alone his rate is 1/y, and when they work together their rate is 1/x+ 1/y= (x+ y)/xy

"When A works alone, takes 4 days more to complete the job than if both A and B do together"
A and B together work at rate (x+ y)/xy so it would take them xy/(x+ y) days to complete the job x= xy/(x+ y)+ 4.

If B works alone, he takes 16 more days than the situation of working together.
y= xy/(x+ y)+ 10

Solve those equations for x and y.
Find how many days it takes for the combined effort.
That is, as above, xy/(x+ y).

 

[haruspex]

Let m part of the job be completed per day when A works alone, in four days he completes 4m. If B completes n per day, he needs 16n extra work to be done

Halls' method certainly works (after correcting a typo), but so does yours, so let's see where it leads.
First, I don't think you mean this: " he needs 16n extra work to be done".
Paralleling what you wrote about A, do you mean "in 16 days he completes 16n"?
Suppose that working together they take x days. How many work units do they complete in x days? What equations can you write for the given information expressed in terms of m, n and x?

 

[Chestermiller]

Yes. The method suggested by haruspex seems more intuitively simple to me. Basically, this is similar to a rate-time-distance problem. In your notation, m = rate at which A does work, in jobs/day (similar to km/hr), and n = rate at which B does work, in jobs/day. The rate at which they work together is (m+n) jobs/day. So, to complete 1 job working together, in terms of m and n, how many days does it take? For a to complete 1 job, in terms of m, how many days does it take him? For B to complete 1 job, in terms of n, how many days does it take him?

Chet