Question

Please help step by step!
Three painters, beth, billl, and edie, working together , can paint the exterior of a home in 10 hours. Bill and Edie together have painted a similar house in 15 hours. One day, all thee worked on this same kind of house for 8 hours, after which Edie left. Beth and Bill required 4 more hours to finish. Assuming no gain or loss in efficiency, how long should it take ah person to complete such a job alone? Answer: It would take beth 30 hours, bill 60 hours, and edie 20 individually complete the job. How to they get that answer?

Answer 1

What level class is this?

Answer 2

college precalc

Answer 3

I haven't solved it all, but I think you can get going by setting up a system of equations to describe each of the "knowns" they give you. Then, you can solve for each of the unknowns, which at the end is the number of hours each person needs if working alone.

Answer 4

Ok, sorry, guess I'm still thinking on this one. The real issue is that each works at different efficiencies. The number of hours worked divided by the person's efficiency equals the "real" number of hours the project would take for a 100% efficient worker. So a 50% worker takes twices as long, a 90% worker takes 100/90 hours, etc.

Answer 5

If B is Beth's efficiency, L is Bill's, and E is Edith's, then working together, they finish the job in 10 hours, so 10(B + L + E) = the total "real" worker-hours needed to finish the job.

Answer 6

When just Bill & Edie work, it takes 15 hours for the same size project, so 15(B+E) = "real" worker-hours (let's call this "P" for project... )

Answer 7

So P = 10(B+E+L) and P = 15(B+E)

Answer 8

The last part is the part you have to be careful setting up. Same project, so it also equals P in total work.

They all work together for 8 hours, so that part of the expression is 8(B+E+L)

then Beth and Bill take 4 more hours to finish, or 4(B + L)

The total project P takes 8(B+E+L) + 4(B+L)

Answer 9

Using these 3 equations of P = ...

You have 3 equations, 3 unknows, and you can solve for each person's efficiency

Answer 10

Then, because you know the project size, P, you can find each person's required hours by dividing P by that person's efficiency.

Answer 11

Does that approach make sense? There may be another way to do it, but the real thing to make sure you include is the fact that an hour of Bill's work is a different amount of actual work than for Edie and for Beth... the real output of work varies based on their efficiency. Other than that, it's just a basic system of 3 equations, like in Algebra 2 (not that those are always fun...)

Answer 12

Ugh yes...long process but very helpful. Thanks!

Answer 13

Believe me, I understand the "ugh" factor. Just get your mind tuned that a problem that has 3 different people doing things in different ways with different results is a classic set-up for a 3-variable system of equations. The whole trick is to "translate" the words into a standard system of equations that you already know.

Good luck on these... hope some are more straightforward :)