[ASK] Paint Problem

Algebra

[ASK] Paint Problem

Postby Monox D. I-Fly » Fri Dec 04, 2020 3:05 am

3 friends, Alan, Brian and Chester, paint a house. If Alan had to paint it on his own, it would take him one hour more than the time it would take for all three to paint it together. If Brian had to paint it on his own, it would take him five hours more than the time it would take for all three to paint it together, and Chester 8 hours more.

How much time would it take for Alan and Brian to paint it together?


If Alan had to paint it on his own, it would take him one hour more than the time it would take for all three to paint it together.

Would it be like this?

[tex]\frac1{A+1}=\frac1A+\frac1B+\frac1C[/tex]?

Will it have something to do with quadratic equation?
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Re: [ASK] Paint Problem

Postby HallsofIvy » Sun Jan 03, 2021 5:02 pm

You should realize that it is impossible to say whether your equation is correct or not because you did not say what "A", "B", and "C" represent! I could assume that "A" is the number of hours it would take to do the job, "B" the number of hours for Brian, and "C" the number of hours for Chester but you should say that!
You do seem to be correctly using that fact that when different people or machines work together their rates of work add. But I would NOT use the number of hours required for the individual people as variables! We are told that if Alan had to paint the house alone it would take one hour more than if all three worked together, that Brian would take 5 hours more, and that Chester would take 8 hours more. I would take, say, x to be the time that would be required for all three, working together to do the job. Then Alan would take x+1 hours so his rate is 1/(x+ 1). Brian takes x+ 5 hours so his rate is 1/(x+ 5). Chester takes x+ 8 so his rate is 1/(x+ 8 ).

Working together their rate would be 1/(x+ 1)+ 1/(x+ 5)+ 1/(x+ 8 ) and that is, by definition of x, 1/x.
Solve 1/(x+ 1)+ 1/(x+ 5)+ 1/(x+ 8 )= 1/x for x. Multiplying both sides of the equation by x(x+ 1)(x+ 5)(x+ 8 ) give the cubic equation x(x+ 5)(x+ 8 )+ x(x+ 1)(x+ 8 )+ x(x+ 1)(x+ 5)= (x+ 1)(x+ 5)(x+ 8 ).

You can then get the rates for Alan and Brian separately and combine them to find the time for Alan and Brian working together.
Last edited by HallsofIvy on Sun Jan 03, 2021 5:06 pm, edited 1 time in total.

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