How long does it take Joan to clean the house alone? How about Lyra?

Question

Joan and Lyra can clean the house in 5 hours and 50 minutes if they work together. The time that Joan takes in cleaning the house alone is 4 hours more than the time Lyra takes in cleaning the same house. How long does it take Joan to clean the house? How about Lyra?

Answer 1

Let J=the time it takes Joan to clean

Let L = time for Lyra

1/J+ 1/L = 1/(5 5/6) 5 and 5/6 hours = 5 hours and 50 minutes

1/(L+4) + 1/L = 1/35/6

(2L+4)/(L^2+4L) = 6/35

cross multiply, simplify

3L^2 -23L -70 = 0 Use the quadratic formula

L=10, J=14

That's sort of an average 12 each, divide by 2 to get 6 which is close to 5 5/6. It's in the right ball park