SOLUTION: 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 5 hours until Homer had to go hom

Algebra ->  Human-and-algebraic-language -> SOLUTION: 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 5 hours until Homer had to go hom      Log On


   

Question 1174769: 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 5 hours until Homer had to go home. Mike finished the job working by himself in an additional 2 hours. How long would it have taken Homer to fix the deck​ himself?
Answer by VFBundy(438) About Me  (Show Source):
You can put this solution on YOUR website!
Homer's rate of work = 1%2Fx of the job per hour
Mike's rate of work = 1%2F%28x-1%29 of the job per hour

Rate of work working together = 1%2Fx + 1%2F%28x-1%29 of the job per hour

Simplify:

%28x-1%29%2F%28x%28x-1%29%29 + x%2F%28x%28x-1%29%29

%28x-1%2Bx%29%2F%28x%28x-1%29%29

%282x-1%29%2F%28x%28x-1%29%29

Rate of work working together = %282x-1%29%2F%28x%28x-1%29%29 of the job per hour

After working five hours together, calculate the amount of the job that was completed:

5+%2A+%28%282x-1%29%2F%28x%28x-1%29%29%29

%2810x+-+5%29%2F%28x%28x-1%29%29

Therefore, the amount of the job that is left to be completed is:

1-+%2810x+-+5%29%2F%28x%28x-1%29%29

%28x%28x-1%29%29%2F%28x%28x-1%29%29+-+%2810x+-+5%29%2F%28x%28x-1%29%29

%28x%5E2-x%29%2F%28x%28x-1%29%29+-+%2810x+-+5%29%2F%28x%28x-1%29%29

%28x%5E2+-+x+-+%2810x+-+5%29%29%2F%28x%28x-1%29%29

%28x%5E2+-+x+-+10x+%2B+5%29%2F%28x%28x-1%29%29

%28x%5E2+-+11x+%2B+5%29%2F%28x%28x-1%29%29

Mike finishes the job in two hours. This means Mike's rate of work per hour is:

1%2F2* %28x%5E2+-+11x+%2B+5%29%2F%28x%28x-1%29%29

%28x%5E2+-+11x+%2B+5%29%2F%282x%28x-1%29%29

So, Mike's rate of work is %28x%5E2+-+11x+%2B+5%29%2F%282x%28x-1%29%29 of the job per hour.

From earlier, we know that Mike's rate of work is also 1%2F%28x-1%29 of the job per hour. Therefore:

%28x%5E2+-+11x+%2B+5%29%2F%282x%28x-1%29%29 = 1%2F%28x-1%29

Solve for x.

Cross-multiply:

%28x-1%29%28x%5E2+-+11x+%2B+5%29 = 2x%28x-1%29

%28x%5E2+-+11x+%2B+5%29 = 2x

x%5E2+-+13x+%2B+5 = 0

Use the quadratic formula to solve for x:

x = %2813-sqrt%28149%29%29%2F2 = 0.40
x = %2813%2Bsqrt%28149%29%29%2F2 = 12.60

There are two solutions. Because Mike's rate of work is 1%2F%28x-1%29 of the job per hour, the first solution, x = 0.40, would make Mike's rate of work 1%2F%28-0.60%29. Since this is a negative rate of work, this is impossible.

Therefore, we are left with x = 12.60

We are looking how long it would have taken Homer to fix the deck working alone. Homer's rate of work is 1%2Fx of the job per hour. Plugging in x = 12.60, this means Homer's rate of work is 1%2F12.60 of the job per hour. This means it would have taken Homer 12.60 hours to fix the deck working alone.