Final answer:
Craig takes 32 hours to clean the house alone. We find this by setting up an equation based on the combined work rate of Craig and Tom, and solving for Craig's time using algebra.
Step-by-step explanation:
The question seeks to determine how long it takes for Craig to clean the house alone given that together with Tom they can do it in 8 hours, and Tom takes four times longer than Craig to clean the house by himself.
Let's assign the variable C to represent the time Craig takes to clean the house alone. According to the problem, Tom takes four times as long, so his time would be 4C. When working together, their combined effort leads to the house being cleaned in 8 hours. We can represent their combined work rate as the sum of their individual work rates which results in the completion of one job.
To find their rates, we turn the times into fractions of the job done per hour: Craig's rate would be 1/C and Tom's rate would be 1/(4C). Adding these rates together gives us the rate at which they work together: 1/C + 1/(4C) = 1/8, since together they take 8 hours to clean the house.
To solve for C, we add the fractions: 1/C + 1/(4C) which results in (4+1)/(4C), simplifying to 5/(4C). We then set this equal to 1/8 and solve for C. Multiplying both sides by 4C gets rid of the fraction on the left, leaving us with 5 = 1/2C. Dividing both sides by 5, we find C = 32.
Therefore, Craig alone takes 32 hours to clean the house.