Final answer:
To determine how long Sabrina takes to clean the villa alone, calculate her hourly cleaning rate (S) using the combined cleaning rate with Gillian and Gillian's individual rate, then find the reciprocal of her rate. It takes Sabrina approximately 3.33 hours to clean the villa by herself.
Step-by-step explanation:
The question at hand involves determining how long it will take for Sabrina to clean a villa on her own, given that together with Gillian, they take 2 hours to clean the villa, and Gillian alone takes 5 hours to clean it. This is a problem that can be solved using the concept of rates in mathematics, specifically involving working together.
Let's define Sabrina's rate of working as the fraction of the villa she can clean per hour, which we will call S. Similarly, Gillian's rate is known; since she can clean the entire villa in 5 hours, her rate G is 1/5 of the villa per hour. When Gillian and Sabrina work together, they combine their rates, meaning that their combined rate is G + S, which is equal to one whole villa in 2 hours, so 1/2 villa per hour.
If G is 1/5 and G + S is 1/2, then we can set up the following equation:
1/5 + S = 1/2.
To solve for S, we subtract Gillian's rate from the combined rate:
S = 1/2 - 1/5 = 5/10 - 2/10 = 3/10.
This means that Sabrina can clean 3/10 of the villa per hour. To find how many hours it takes her to clean the whole villa, we take the reciprocal of her rate:
1 / (3/10) = 10/3 = 3.33 hours.
Therefore, it takes Sabrina approximately 3.33 hours to clean the villa on her own.