Final answer:
By calculating the combined work rates of the homeowner and the professional roofer, we determine it will take about 2.5455 days for them to roof the house together.
Step-by-step explanation:
The question asks us to calculate how long it will take if a homeowner and a professional roofer work together to roof a house. Since the homeowner estimates that it will take 7 days to complete the job by himself and the roofer can do it in 4 days, we calculate the combined work rate by adding their reciprocal work rates together. If the homeowner's work rate is 1/7 of the job per day, and the roofer's work rate is 1/4 of the job per day, their combined work rate when working together is (1/7 + 1/4) of the job per day.
To find the least common denominator, we calculate (1/7) * (4/4) + (1/4) * (7/7) = (4/28) + (7/28) = 11/28 of the roofing job per day. To determine the number of days it will take both of them working together, we take the reciprocal of the combined work rate: 28/11 days. Therefore, it will take approximately 2.5455 days for the homeowner and the professional roofer to complete the roofing job together.