how would you solve this?

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 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 1

The workrates for Mike and Homer are

1/x and 1/(x+1)

The entire job was finished in 5 hours working together

and 2 hours of Mike...

5[ 1/x + 1/(x+1)] + 2/x = 1

5/x + 5/(x+1) + 2/x = 1

5/(x+1) + 7/x = 1

5x + 7(x+1) = x(x+1)

5x + 7x +7 = x^2 + x

12x + 7 = x^2 + x

0 = x^2 -11x - 7

x = [ 11 + sqrt( 121 + 28)]/2

= [ 11 + sqrt(149)]/2

Mike 11.6 hours working alone

Home 12.6 hours working alone

0.1655719759168035030103995621237 is

the combined work rate, which means

working together, they will finish the job

is 6.05 hours