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:
 $3 \frac{1}{x} + 3 \frac{1}{x-1}$
 Mike finished the job working by himself in an additional 5 hours:
 $3 \frac{1}{x} + 3 \frac{1}{x-1} + 5 \frac{1}{x-1} = 1$
 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