Question

Homer and Mike were replacing the boards on​ Mike's old deck. Mike can do the job alone in 1 hour less time than Homer. They worked together for 3 hours until Homer had to go home. Mike finished the job working by himself in an additional 5 hours. How long would it have taken Homer to fix the deck​ himself?

Answer 1

For this case, the first thing we are going to do is define variables.
We have then:
x: number of hours it takes Homer to finish work alone.
They worked together for 3 hours until Homer had to go home:

`3 (1)/(x) + 3 (1)/(x-1)`
Mike finished the job working by himself in an additional 5 hours:

`3 (1)/(x) + 3 (1)/(x-1) + 5 (1)/(x-1) = 1`
From here, we must clear the value of x.
We then have to multiply both sides by x (x-1):

`3 (x-1) + 3x + 5x = x (x-1)`
Rewriting we have:

`3x - 3 + 3x + 5x = x ^ 2 - x x ^ 2 - 12x + 3 = 0`
The solutions to the equation are:

`x1 = 0.25 (discarded) x2 = 11.75`
Answer:
it has taken Homer to fix the deck himself about:
x2 = 11.75 hours