Final answer:
Jake can clean the entire building by himself in 0 hours.
Step-by-step explanation:
To find out how long it takes Jake to clean the building by himself, we need to determine how long it takes Skyler to clean the building. Let's assume it takes Skyler x hours to clean the building. We are given that Jake can clean the building in 3 fewer hours than Skyler, so Jake can clean the building in (x-3) hours.
If Jake and Skyler can clean the building together in 2 hours, their combined work rate is 1/2 of the building per hour. Jake's individual work rate is 1/(x-3) of the building per hour. Since work rates add up when working together, we can set up the equation:
1/2 + 1/(x-3) = 1/x
To solve this equation, we can multiply both sides by 2x(x-3) to get rid of the fractions. Simplifying the equation gives us:
x(x-3) + 2x = 2(x-3)
Simplifying further, we get:
x^2 - 3x + 2x = 2x - 6
x^2 - x - 6 = 0
This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Factoring gives us:
(x-3)(x+2) = 0
So x-3 = 0 or x+2 = 0. Solving for x gives us 2 solutions: x = 3 or x = -2. Since x represents the time it takes Skyler to clean the building, we discard the negative solution. Therefore, it takes Skyler 3 hours to clean the building.
Since Jake can clean the building in 3 fewer hours than Skyler, it takes Jake 3 - 3 = 0 hours to clean the building by himself.