A homeowner estimates that it will take 11 days to roof his house. A professional roofer estimates that he could roof the house in 6 days. How long willl it take if the homeowner helps the roofer?

3 15/17  days

Step-by-step explanation:

let the amount of work done to roof the whole house by x

Time taken by homeowner to roof his house = 11 days

In 11 days homeowner can do x work

in 1 day homeowner can do x/11 work

Time taken by professional to roof the house = 6 days

In 6 days professional can do x work

in 1 day professional can do x/6  work

In 1 day total work done, if both professional and homeowner work together

= x/6 + x/11 = (11x + 6x)/11*6 =  17x/66

17x/66 work is done by both in 1 day

we have to calculate total work done by both , to find this we multiply both side by 66/17 to eliminate it from term containing x(which is the total work done to roof the house)

17x/66 *(66/17)work is done by both in 66/17  days

x work is is done by both in 3 15/17  days.

Hence, it will take  3 15/17  days to roof the house,  if the homeowner helps the roofer.